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Causal research: definition, examples and how to use it.

16 min read Causal research enables market researchers to predict hypothetical occurrences & outcomes while improving existing strategies. Discover how this research can decrease employee retention & increase customer success for your business.

What is causal research?

Causal research, also known as explanatory research or causal-comparative research, identifies the extent and nature of cause-and-effect relationships between two or more variables.

It’s often used by companies to determine the impact of changes in products, features, or services process on critical company metrics. Some examples:

• How does rebranding of a product influence intent to purchase?
• How would expansion to a new market segment affect projected sales?
• What would be the impact of a price increase or decrease on customer loyalty?

To maintain the accuracy of causal research, ‘confounding variables’ or influences — e.g. those that could distort the results — are controlled. This is done either by keeping them constant in the creation of data, or by using statistical methods. These variables are identified before the start of the research experiment.

As well as the above, research teams will outline several other variables and principles in causal research:

• Independent variables

The variables that may cause direct changes in another variable. For example, the effect of truancy on a student’s grade point average. The independent variable is therefore class attendance.

• Control variables

These are the components that remain unchanged during the experiment so researchers can better understand what conditions create a cause-and-effect relationship.  

This describes the cause-and-effect relationship. When researchers find causation (or the cause), they’ve conducted all the processes necessary to prove it exists.

• Correlation

Any relationship between two variables in the experiment. It’s important to note that correlation doesn’t automatically mean causation. Researchers will typically establish correlation before proving cause-and-effect.

• Experimental design

Researchers use experimental design to define the parameters of the experiment — e.g. categorizing participants into different groups.

• Dependent variables

These are measurable variables that may change or are influenced by the independent variable. For example, in an experiment about whether or not terrain influences running speed, your dependent variable is the terrain.  

Why is causal research useful?

It’s useful because it enables market researchers to predict hypothetical occurrences and outcomes while improving existing strategies. This allows businesses to create plans that benefit the company. It’s also a great research method because researchers can immediately see how variables affect each other and under what circumstances.

Also, once the first experiment has been completed, researchers can use the learnings from the analysis to repeat the experiment or apply the findings to other scenarios. Because of this, it’s widely used to help understand the impact of changes in internal or commercial strategy to the business bottom line.

Some examples include:

• Understanding how overall training levels are improved by introducing new courses
• Examining which variations in wording make potential customers more interested in buying a product
• Testing a market’s response to a brand-new line of products and/or services

So, how does causal research compare and differ from other research types?

Well, there are a few research types that are used to find answers to some of the examples above:

1. Exploratory research

As its name suggests, exploratory research involves assessing a situation (or situations) where the problem isn’t clear. Through this approach, researchers can test different avenues and ideas to establish facts and gain a better understanding.

Researchers can also use it to first navigate a topic and identify which variables are important. Because no area is off-limits, the research is flexible and adapts to the investigations as it progresses.

Finally, this approach is unstructured and often involves gathering qualitative data, giving the researcher freedom to progress the research according to their thoughts and assessment. However, this may make results susceptible to researcher bias and may limit the extent to which a topic is explored.

2. Descriptive research

Descriptive research is all about describing the characteristics of the population, phenomenon or scenario studied. It focuses more on the “what” of the research subject than the “why”.

For example, a clothing brand wants to understand the fashion purchasing trends amongst buyers in California — so they conduct a demographic survey of the region, gather population data and then run descriptive research. The study will help them to uncover purchasing patterns amongst fashion buyers in California, but not necessarily why those patterns exist.

As the research happens in a natural setting, variables can cross-contaminate other variables, making it harder to isolate cause and effect relationships. Therefore, further research will be required if more causal information is needed.

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How is causal research different from the other two methods above?

Well, causal research looks at what variables are involved in a problem and ‘why’ they act a certain way. As the experiment takes place in a controlled setting (thanks to controlled variables) it’s easier to identify cause-and-effect amongst variables.

Furthermore, researchers can carry out causal research at any stage in the process, though it’s usually carried out in the later stages once more is known about a particular topic or situation.

Finally, compared to the other two methods, causal research is more structured, and researchers can combine it with exploratory and descriptive research to assist with research goals.

Summary of three research types

What are the advantages of causal research?

• Improve experiences

By understanding which variables have positive impacts on target variables (like sales revenue or customer loyalty), businesses can improve their processes, return on investment, and the experiences they offer customers and employees.

• Help companies improve internally

By conducting causal research, management can make informed decisions about improving their employee experience and internal operations. For example, understanding which variables led to an increase in staff turnover.

• Repeat experiments to enhance reliability and accuracy of results

When variables are identified, researchers can replicate cause-and-effect with ease, providing them with reliable data and results to draw insights from.

• Test out new theories or ideas

If causal research is able to pinpoint the exact outcome of mixing together different variables, research teams have the ability to test out ideas in the same way to create viable proof of concepts.

• Fix issues quickly

Once an undesirable effect’s cause is identified, researchers and management can take action to reduce the impact of it or remove it entirely, resulting in better outcomes.

What are the disadvantages of causal research?

• Provides information to competitors

If you plan to publish your research, it provides information about your plans to your competitors. For example, they might use your research outcomes to identify what you are up to and enter the market before you.

• Difficult to administer

Causal research is often difficult to administer because it’s not possible to control the effects of extraneous variables.

• Time and money constraints

Budgetary and time constraints can make this type of research expensive to conduct and repeat. Also, if an initial attempt doesn’t provide a cause and effect relationship, the ROI is wasted and could impact the appetite for future repeat experiments.

• Requires additional research to ensure validity

You can’t rely on just the outcomes of causal research as it’s inaccurate. It’s best to conduct other types of research alongside it to confirm its output.

• Trouble establishing cause and effect

Researchers might identify that two variables are connected, but struggle to determine which is the cause and which variable is the effect.

• Risk of contamination

There’s always the risk that people outside your market or area of study could affect the results of your research. For example, if you’re conducting a retail store study, shoppers outside your ‘test parameters’ shop at your store and skew the results.

How can you use causal research effectively?

To better highlight how you can use causal research across functions or markets, here are a few examples:

Market and advertising research

A company might want to know if their new advertising campaign or marketing campaign is having a positive impact. So, their research team can carry out a causal research project to see which variables cause a positive or negative effect on the campaign.

For example, a cold-weather apparel company in a winter ski-resort town may see an increase in sales generated after a targeted campaign to skiers. To see if one caused the other, the research team could set up a duplicate experiment to see if the same campaign would generate sales from non-skiers. If the results reduce or change, then it’s likely that the campaign had a direct effect on skiers to encourage them to purchase products.

Improving customer experiences and loyalty levels

Customers enjoy shopping with brands that align with their own values, and they’re more likely to buy and present the brand positively to other potential shoppers as a result. So, it’s in your best interest to deliver great experiences and retain your customers.

For example, the Harvard Business Review found that an increase in customer retention rates by 5% increased profits by 25% to 95%. But let’s say you want to increase your own, how can you identify which variables contribute to it?Using causal research, you can test hypotheses about which processes, strategies or changes influence customer retention. For example, is it the streamlined checkout? What about the personalized product suggestions? Or maybe it was a new solution that solved their problem? Causal research will help you find out.

Improving problematic employee turnover rates

If your company has a high attrition rate, causal research can help you narrow down the variables or reasons which have the greatest impact on people leaving. This allows you to prioritize your efforts on tackling the issues in the right order, for the best positive outcomes.

For example, through causal research, you might find that employee dissatisfaction due to a lack of communication and transparency from upper management leads to poor morale, which in turn influences employee retention.

To rectify the problem, you could implement a routine feedback loop or session that enables your people to talk to your company’s C-level executives so that they feel heard and understood.

How to conduct causal research first steps to getting started are:

1. Define the purpose of your research

What questions do you have? What do you expect to come out of your research? Think about which variables you need to test out the theory.

2. Pick a random sampling if participants are needed

Using a technology solution to support your sampling, like a database, can help you define who you want your target audience to be, and how random or representative they should be.

3. Set up the controlled experiment

Once you’ve defined which variables you’d like to measure to see if they interact, think about how best to set up the experiment. This could be in-person or in-house via interviews, or it could be done remotely using online surveys.

4. Carry out the experiment

Make sure to keep all irrelevant variables the same, and only change the causal variable (the one that causes the effect) to gather the correct data. Depending on your method, you could be collecting qualitative or quantitative data, so make sure you note your findings across each regularly.

5. Analyze your findings

Either manually or using technology, analyze your data to see if any trends, patterns or correlations emerge. By looking at the data, you’ll be able to see what changes you might need to do next time, or if there are questions that require further research.

6. Verify your findings

Your first attempt gives you the baseline figures to compare the new results to. You can then run another experiment to verify your findings.

7. Do follow-up or supplemental research

You can supplement your original findings by carrying out research that goes deeper into causes or explores the topic in more detail. One of the best ways to do this is to use a survey. See ‘Use surveys to help your experiment’.

Identifying causal relationships between variables

To verify if a causal relationship exists, you have to satisfy the following criteria:

• Nonspurious association

A clear correlation exists between one cause and the effect. In other words, no ‘third’ that relates to both (cause and effect) should exist.

• Temporal sequence

The cause occurs before the effect. For example, increased ad spend on product marketing would contribute to higher product sales.

• Concomitant variation

The variation between the two variables is systematic. For example, if a company doesn’t change its IT policies and technology stack, then changes in employee productivity were not caused by IT policies or technology.

How surveys help your causal research experiments?

There are some surveys that are perfect for assisting researchers with understanding cause and effect. These include:

• Employee Satisfaction Survey – An introductory employee satisfaction survey that provides you with an overview of your current employee experience.
• Manager Feedback Survey – An introductory manager feedback survey geared toward improving your skills as a leader with valuable feedback from your team.
• Net Promoter Score (NPS) Survey – Measure customer loyalty and understand how your customers feel about your product or service using one of the world’s best-recognized metrics.
• Employee Engagement Survey – An entry-level employee engagement survey that provides you with an overview of your current employee experience.
• Customer Satisfaction Survey – Evaluate how satisfied your customers are with your company, including the products and services you provide and how they are treated when they buy from you.
• Employee Exit Interview Survey – Understand why your employees are leaving and how they’ll speak about your company once they’re gone.
• Product Research Survey – Evaluate your consumers’ reaction to a new product or product feature across every stage of the product development journey.
• Brand Awareness Survey – Track the level of brand awareness in your target market, including current and potential future customers.
• Online Purchase Feedback Survey – Find out how well your online shopping experience performs against customer needs and expectations.

That covers the fundamentals of causal research and should give you a foundation for ongoing studies to assess opportunities, problems, and risks across your market, product, customer, and employee segments.

If you want to transform your research, empower your teams and get insights on tap to get ahead of the competition, maybe it’s time to leverage Qualtrics CoreXM.

Qualtrics CoreXM provides a single platform for data collection and analysis across every part of your business — from customer feedback to product concept testing. What’s more, you can integrate it with your existing tools and services thanks to a flexible API.

Qualtrics CoreXM offers you as much or as little power and complexity as you need, so whether you’re running simple surveys or more advanced forms of research, it can deliver every time.

Get started on your market research journey with CoreXM

Related resources

Market intelligence 10 min read, marketing insights 11 min read, ethnographic research 11 min read, qualitative vs quantitative research 13 min read, qualitative research questions 11 min read, qualitative research design 12 min read, primary vs secondary research 14 min read, request demo.

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Methodology

• Correlation vs. Causation | Difference, Designs & Examples

Correlation vs. Causation | Difference, Designs & Examples

Published on July 12, 2021 by Pritha Bhandari . Revised on June 22, 2023.

Correlation means there is a statistical association between variables. Causation means that a change in one variable causes a change in another variable.

In research, you might have come across the phrase “correlation doesn’t imply causation.” Correlation and causation are two related ideas, but understanding their differences will help you critically evaluate sources and interpret scientific research.

Table of contents

What’s the difference, why doesn’t correlation mean causation, correlational research, third variable problem, regression to the mean, spurious correlations, directionality problem, causal research, other interesting articles, frequently asked questions about correlation and causation.

Correlation describes an association between types of variables : when one variable changes, so does the other. A correlation is a statistical indicator of the relationship between variables. These variables change together: they covary. But this covariation isn’t necessarily due to a direct or indirect causal link.

Causation means that changes in one variable brings about changes in the other; there is a cause-and-effect relationship between variables. The two variables are correlated with each other and there is also a causal link between them.

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There are two main reasons why correlation isn’t causation. These problems are important to identify for drawing sound scientific conclusions from research.

The third variable problem means that a confounding variable affects both variables to make them seem causally related when they are not. For example, ice cream sales and violent crime rates are closely correlated, but they are not causally linked with each other. Instead, hot temperatures, a third variable, affects both variables separately. Failing to account for third variables can lead research biases to creep into your work.

The directionality problem occurs when two variables correlate and might actually have a causal relationship, but it’s impossible to conclude which variable causes changes in the other. For example, vitamin D levels are correlated with depression, but it’s not clear whether low vitamin D causes depression, or whether depression causes reduced vitamin D intake.

You’ll need to use an appropriate research design to distinguish between correlational and causal relationships:

• Correlational research designs can only demonstrate correlational links between variables.
• Experimental designs can test causation.

In a correlational research design, you collect data on your variables without manipulating them.

Correlational research is usually high in external validity , so you can generalize your findings to real life settings. But these studies are low in internal validity , which makes it difficult to causally connect changes in one variable to changes in the other.

These research designs are commonly used when it’s unethical, too costly, or too difficult to perform controlled experiments. They are also used to study relationships that aren’t expected to be causal.

Without controlled experiments, it’s hard to say whether it was the variable you’re interested in that caused changes in another variable. Extraneous variables are any third variable or omitted variable other than your variables of interest that could affect your results.

Limited control in correlational research means that extraneous or confounding variables serve as alternative explanations for the results. Confounding variables can make it seem as though a correlational relationship is causal when it isn’t.

When two variables are correlated, all you can say is that changes in one variable occur alongside changes in the other.

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Regression to the mean is observed when variables that are extremely higher or extremely lower than average on the first measurement move closer to the average on the second measurement. Particularly in research that intentionally focuses on the most extreme cases or events, RTM should always be considered as a possible cause of an observed change.

Players or teams featured on the cover of SI have earned their place by performing exceptionally well. But athletic success is a mix of skill and luck, and even the best players don’t always win.

Chances are that good luck will not continue indefinitely, and neither can exceptional success.

A spurious correlation is when two variables appear to be related through hidden third variables or simply by coincidence.

The Theory of the Stork draws a simple causal link between the variables to argue that storks physically deliver babies. This satirical study shows why you can’t conclude causation from correlational research alone.

When you analyze correlations in a large dataset with many variables, the chances of finding at least one statistically significant result are high. In this case, you’re more likely to make a type I error . This means erroneously concluding there is a true correlation between variables in the population based on skewed sample data.

To demonstrate causation, you need to show a directional relationship with no alternative explanations. This relationship can be unidirectional, with one variable impacting the other, or bidirectional, where both variables impact each other.

A correlational design won’t be able to distinguish between any of these possibilities, but an experimental design can test each possible direction, one at a time.

• Physical activity may affect self esteem
• Self esteem may affect physical activity
• Physical activity and self esteem may both affect each other

In correlational research, the directionality of a relationship is unclear because there is limited researcher control. You might risk concluding reverse causality, the wrong direction of the relationship.

Causal links between variables can only be truly demonstrated with controlled experiments . Experiments test formal predictions, called hypotheses , to establish causality in one direction at a time.

Experiments are high in internal validity , so cause-and-effect relationships can be demonstrated with reasonable confidence.

You can establish directionality in one direction because you manipulate an independent variable before measuring the change in a dependent variable.

In a controlled experiment, you can also eliminate the influence of third variables by using random assignment and control groups.

Random assignment helps distribute participant characteristics evenly between groups so that they’re similar and comparable. A control group lets you compare the experimental manipulation to a similar treatment or no treatment (or a placebo, to control for the placebo effect ).

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis
• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

A correlation reflects the strength and/or direction of the association between two or more variables.

• A positive correlation means that both variables change in the same direction.
• A negative correlation means that the variables change in opposite directions.
• A zero correlation means there’s no relationship between the variables.

Correlation describes an association between variables : when one variable changes, so does the other. A correlation is a statistical indicator of the relationship between variables.

Causation means that changes in one variable brings about changes in the other (i.e., there is a cause-and-effect relationship between variables). The two variables are correlated with each other, and there’s also a causal link between them.

While causation and correlation can exist simultaneously, correlation does not imply causation. In other words, correlation is simply a relationship where A relates to B—but A doesn’t necessarily cause B to happen (or vice versa). Mistaking correlation for causation is a common error and can lead to false cause fallacy .

The third variable and directionality problems are two main reasons why correlation isn’t causation .

The third variable problem means that a confounding variable affects both variables to make them seem causally related when they are not.

The directionality problem is when two variables correlate and might actually have a causal relationship, but it’s impossible to conclude which variable causes changes in the other.

Controlled experiments establish causality, whereas correlational studies only show associations between variables.

• In an experimental design , you manipulate an independent variable and measure its effect on a dependent variable. Other variables are controlled so they can’t impact the results.
• In a correlational design , you measure variables without manipulating any of them. You can test whether your variables change together, but you can’t be sure that one variable caused a change in another.

In general, correlational research is high in external validity while experimental research is high in internal validity .

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What is causal research design?

Last updated

14 May 2023

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Examining these relationships gives researchers valuable insights into the mechanisms that drive the phenomena they are investigating.

Organizations primarily use causal research design to identify, determine, and explore the impact of changes within an organization and the market. You can use a causal research design to evaluate the effects of certain changes on existing procedures, norms, and more.

This article explores causal research design, including its elements, advantages, and disadvantages.

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• Components of causal research

You can demonstrate the existence of cause-and-effect relationships between two factors or variables using specific causal information, allowing you to produce more meaningful results and research implications.

These are the key inputs for causal research:

The timeline of events

Ideally, the cause must occur before the effect. You should review the timeline of two or more separate events to determine the independent variables (cause) from the dependent variables (effect) before developing a hypothesis. 

If the cause occurs before the effect, you can link cause and effect and develop a hypothesis .

For instance, an organization may notice a sales increase. Determining the cause would help them reproduce these results. 

Upon review, the business realizes that the sales boost occurred right after an advertising campaign. The business can leverage this time-based data to determine whether the advertising campaign is the independent variable that caused a change in sales. 

Evaluation of confounding variables

In most cases, you need to pinpoint the variables that comprise a cause-and-effect relationship when using a causal research design. This uncovers a more accurate conclusion. 

Co-variations between a cause and effect must be accurate, and a third factor shouldn’t relate to cause and effect. 

Observing changes

Variation links between two variables must be clear. A quantitative change in effect must happen solely due to a quantitative change in the cause. 

You can test whether the independent variable changes the dependent variable to evaluate the validity of a cause-and-effect relationship. A steady change between the two variables must occur to back up your hypothesis of a genuine causal effect. 

• Why is causal research useful?

Causal research allows market researchers to predict hypothetical occurrences and outcomes while enhancing existing strategies. Organizations can use this concept to develop beneficial plans. 

Causal research is also useful as market researchers can immediately deduce the effect of the variables on each other under real-world conditions. 

Once researchers complete their first experiment, they can use their findings. Applying them to alternative scenarios or repeating the experiment to confirm its validity can produce further insights. 

Businesses widely use causal research to identify and comprehend the effect of strategic changes on their profits. 

• How does causal research compare and differ from other research types?

Other research types that identify relationships between variables include exploratory and descriptive research . 

Here’s how they compare and differ from causal research designs:

Exploratory research

An exploratory research design evaluates situations where a problem or opportunity's boundaries are unclear. You can use this research type to test various hypotheses and assumptions to establish facts and understand a situation more clearly.

You can also use exploratory research design to navigate a topic and discover the relevant variables. This research type allows flexibility and adaptability as the experiment progresses, particularly since no area is off-limits.

It’s worth noting that exploratory research is unstructured and typically involves collecting qualitative data . This provides the freedom to tweak and amend the research approach according to your ongoing thoughts and assessments. 

Unfortunately, this exposes the findings to the risk of bias and may limit the extent to which a researcher can explore a topic. 

This table compares the key characteristics of causal and exploratory research:

Descriptive research

This research design involves capturing and describing the traits of a population, situation, or phenomenon. Descriptive research focuses more on the " what " of the research subject and less on the " why ."

Since descriptive research typically happens in a real-world setting, variables can cross-contaminate others. This increases the challenge of isolating cause-and-effect relationships. 

You may require further research if you need more causal links. 

This table compares the key characteristics of causal and descriptive research.  

Causal research examines a research question’s variables and how they interact. It’s easier to pinpoint cause and effect since the experiment often happens in a controlled setting. 

Researchers can conduct causal research at any stage, but they typically use it once they know more about the topic.

In contrast, causal research tends to be more structured and can be combined with exploratory and descriptive research to help you attain your research goals. 

• How can you use causal research effectively?

Here are common ways that market researchers leverage causal research effectively:

Market and advertising research

Do you want to know if your new marketing campaign is affecting your organization positively? You can use causal research to determine the variables causing negative or positive impacts on your campaign. 

Improving customer experiences and loyalty levels

Consumers generally enjoy purchasing from brands aligned with their values. They’re more likely to purchase from such brands and positively represent them to others. 

You can use causal research to identify the variables contributing to increased or reduced customer acquisition and retention rates. 

Could the cause of increased customer retention rates be streamlined checkout? 

Perhaps you introduced a new solution geared towards directly solving their immediate problem. 

Whatever the reason, causal research can help you identify the cause-and-effect relationship. You can use this to enhance your customer experiences and loyalty levels.

Improving problematic employee turnover rates

Is your organization experiencing skyrocketing attrition rates? 

You can leverage the features and benefits of causal research to narrow down the possible explanations or variables with significant effects on employees quitting. 

This way, you can prioritize interventions, focusing on the highest priority causal influences, and begin to tackle high employee turnover rates. 

• Advantages of causal research

The main benefits of causal research include the following:

Effectively test new ideas

If causal research can pinpoint the precise outcome through combinations of different variables, researchers can test ideas in the same manner to form viable proof of concepts.

Achieve more objective results

Market researchers typically use random sampling techniques to choose experiment participants or subjects in causal research. This reduces the possibility of exterior, sample, or demography-based influences, generating more objective results. 

Improved business processes

Causal research helps businesses understand which variables positively impact target variables, such as customer loyalty or sales revenues. This helps them improve their processes, ROI, and customer and employee experiences.

Guarantee reliable and accurate results

Upon identifying the correct variables, researchers can replicate cause and effect effortlessly. This creates reliable data and results to draw insights from. 

Internal organization improvements

Businesses that conduct causal research can make informed decisions about improving their internal operations and enhancing employee experiences. 

• Disadvantages of causal research

Like any other research method, casual research has its set of drawbacks that include:

Extra research to ensure validity

Researchers can't simply rely on the outcomes of causal research since it isn't always accurate. There may be a need to conduct other research types alongside it to ensure accurate output.

Coincidence

Coincidence tends to be the most significant error in causal research. Researchers often misinterpret a coincidental link between a cause and effect as a direct causal link. 

Administration challenges

Causal research can be challenging to administer since it's impossible to control the impact of extraneous variables . 

Giving away your competitive advantage

If you intend to publish your research, it exposes your information to the competition. 

Competitors may use your research outcomes to identify your plans and strategies to enter the market before you. 

• Causal research examples

Multiple fields can use causal research, so it serves different purposes, such as. 

Customer loyalty research

Organizations and employees can use causal research to determine the best customer attraction and retention approaches. 

They monitor interactions between customers and employees to identify cause-and-effect patterns. That could be a product demonstration technique resulting in higher or lower sales from the same customers. 

Example: Business X introduces a new individual marketing strategy for a small customer group and notices a measurable increase in monthly subscriptions. 

Upon getting identical results from different groups, the business concludes that the individual marketing strategy resulted in the intended causal relationship.

Advertising research

Businesses can also use causal research to implement and assess advertising campaigns. 

Example: Business X notices a 7% increase in sales revenue a few months after a business introduces a new advertisement in a certain region. The business can run the same ad in random regions to compare sales data over the same period. 

This will help the company determine whether the ad caused the sales increase. If sales increase in these randomly selected regions, the business could conclude that advertising campaigns and sales share a cause-and-effect relationship. 

Educational research

Academics, teachers, and learners can use causal research to explore the impact of politics on learners and pinpoint learner behavior trends. 

Example: College X notices that more IT students drop out of their program in their second year, which is 8% higher than any other year. 

The college administration can interview a random group of IT students to identify factors leading to this situation, including personal factors and influences. 

With the help of in-depth statistical analysis, the institution's researchers can uncover the main factors causing dropout. They can create immediate solutions to address the problem.

Is a causal variable dependent or independent?

When two variables have a cause-and-effect relationship, the cause is often called the independent variable. As such, the effect variable is dependent, i.e., it depends on the independent causal variable. An independent variable is only causal under experimental conditions. 

What are the three criteria for causality?

The three conditions for causality are:

Temporality/temporal precedence: The cause must precede the effect.

Rationality: One event predicts the other with an explanation, and the effect must vary in proportion to changes in the cause.

Control for extraneous variables: The covariables must not result from other variables.  

Is causal research experimental?

Causal research is mostly explanatory. Causal studies focus on analyzing a situation to explore and explain the patterns of relationships between variables. 

Further, experiments are the primary data collection methods in studies with causal research design. However, as a research design, causal research isn't entirely experimental.

What is the difference between experimental and causal research design?

One of the main differences between causal and experimental research is that in causal research, the research subjects are already in groups since the event has already happened. 

On the other hand, researchers randomly choose subjects in experimental research before manipulating the variables.

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Causal Research: Definition, Design, Tips, Examples

Appinio Research · 21.02.2024 · 34min read

Ever wondered why certain events lead to specific outcomes? Understanding causality—the relationship between cause and effect—is crucial for unraveling the mysteries of the world around us. In this guide on causal research, we delve into the methods, techniques, and principles behind identifying and establishing cause-and-effect relationships between variables. Whether you're a seasoned researcher or new to the field, this guide will equip you with the knowledge and tools to conduct rigorous causal research and draw meaningful conclusions that can inform decision-making and drive positive change.

What is Causal Research?

Causal research is a methodological approach used in scientific inquiry to investigate cause-and-effect relationships between variables. Unlike correlational or descriptive research, which merely examine associations or describe phenomena, causal research aims to determine whether changes in one variable cause changes in another variable.

Importance of Causal Research

Understanding the importance of causal research is crucial for appreciating its role in advancing knowledge and informing decision-making across various fields. Here are key reasons why causal research is significant:

• Establishing Causality:  Causal research enables researchers to determine whether changes in one variable directly cause changes in another variable. This helps identify effective interventions, predict outcomes, and inform evidence-based practices.
• Guiding Policy and Practice:  By identifying causal relationships, causal research provides empirical evidence to support policy decisions, program interventions, and business strategies. Decision-makers can use causal findings to allocate resources effectively and address societal challenges.
• Informing Predictive Modeling :  Causal research contributes to the development of predictive models by elucidating causal mechanisms underlying observed phenomena. Predictive models based on causal relationships can accurately forecast future outcomes and trends.
• Advancing Scientific Knowledge:  Causal research contributes to the cumulative body of scientific knowledge by testing hypotheses, refining theories, and uncovering underlying mechanisms of phenomena. It fosters a deeper understanding of complex systems and phenomena.
• Mitigating Confounding Factors:  Understanding causal relationships allows researchers to control for confounding variables and reduce bias in their studies. By isolating the effects of specific variables, researchers can draw more valid and reliable conclusions.

Causal Research Distinction from Other Research

Understanding the distinctions between causal research and other types of research methodologies is essential for researchers to choose the most appropriate approach for their study objectives. Let's explore the differences and similarities between causal research and descriptive, exploratory, and correlational research methodologies .

Descriptive vs. Causal Research

Descriptive research  focuses on describing characteristics, behaviors, or phenomena without manipulating variables or establishing causal relationships. It provides a snapshot of the current state of affairs but does not attempt to explain why certain phenomena occur.

Causal research , on the other hand, seeks to identify cause-and-effect relationships between variables by systematically manipulating independent variables and observing their effects on dependent variables. Unlike descriptive research, causal research aims to determine whether changes in one variable directly cause changes in another variable.

Similarities:

• Both descriptive and causal research involve empirical observation and data collection.
• Both types of research contribute to the scientific understanding of phenomena, albeit through different approaches.

Differences:

• Descriptive research focuses on describing phenomena, while causal research aims to explain why phenomena occur by identifying causal relationships.
• Descriptive research typically uses observational methods, while causal research often involves experimental designs or causal inference techniques to establish causality.

Exploratory vs. Causal Research

Exploratory research  aims to explore new topics, generate hypotheses, or gain initial insights into phenomena. It is often conducted when little is known about a subject and seeks to generate ideas for further investigation.

Causal research , on the other hand, is concerned with testing hypotheses and establishing cause-and-effect relationships between variables. It builds on existing knowledge and seeks to confirm or refute causal hypotheses through systematic investigation.

• Both exploratory and causal research contribute to the generation of knowledge and theory development.
• Both types of research involve systematic inquiry and data analysis to answer research questions.
• Exploratory research focuses on generating hypotheses and exploring new areas of inquiry, while causal research aims to test hypotheses and establish causal relationships.
• Exploratory research is more flexible and open-ended, while causal research follows a more structured and hypothesis-driven approach.

Correlational vs. Causal Research

Correlational research  examines the relationship between variables without implying causation. It identifies patterns of association or co-occurrence between variables but does not establish the direction or causality of the relationship.

Causal research , on the other hand, seeks to establish cause-and-effect relationships between variables by systematically manipulating independent variables and observing their effects on dependent variables. It goes beyond mere association to determine whether changes in one variable directly cause changes in another variable.

• Both correlational and causal research involve analyzing relationships between variables.
• Both types of research contribute to understanding the nature of associations between variables.
• Correlational research focuses on identifying patterns of association, while causal research aims to establish causal relationships.
• Correlational research does not manipulate variables, while causal research involves systematically manipulating independent variables to observe their effects on dependent variables.

How to Formulate Causal Research Hypotheses?

Crafting research questions and hypotheses is the foundational step in any research endeavor. Defining your variables clearly and articulating the causal relationship you aim to investigate is essential. Let's explore this process further.

1. Identify Variables

Identifying variables involves recognizing the key factors you will manipulate or measure in your study. These variables can be classified into independent, dependent, and confounding variables.

• Independent Variable (IV):  This is the variable you manipulate or control in your study. It is the presumed cause that you want to test.
• Dependent Variable (DV):  The dependent variable is the outcome or response you measure. It is affected by changes in the independent variable.
• Confounding Variables:  These are extraneous factors that may influence the relationship between the independent and dependent variables, leading to spurious correlations or erroneous causal inferences. Identifying and controlling for confounding variables is crucial for establishing valid causal relationships.

2. Establish Causality

Establishing causality requires meeting specific criteria outlined by scientific methodology. While correlation between variables may suggest a relationship, it does not imply causation. To establish causality, researchers must demonstrate the following:

• Temporal Precedence:  The cause must precede the effect in time. In other words, changes in the independent variable must occur before changes in the dependent variable.
• Covariation of Cause and Effect:  Changes in the independent variable should be accompanied by corresponding changes in the dependent variable. This demonstrates a consistent pattern of association between the two variables.
• Elimination of Alternative Explanations:  Researchers must rule out other possible explanations for the observed relationship between variables. This involves controlling for confounding variables and conducting rigorous experimental designs to isolate the effects of the independent variable.

3. Write Clear and Testable Hypotheses

Hypotheses serve as tentative explanations for the relationship between variables and provide a framework for empirical testing. A well-formulated hypothesis should be:

• Specific:  Clearly state the expected relationship between the independent and dependent variables.
• Testable:  The hypothesis should be capable of being empirically tested through observation or experimentation.
• Falsifiable:  There should be a possibility of proving the hypothesis false through empirical evidence.

For example, a hypothesis in a study examining the effect of exercise on weight loss could be: "Increasing levels of physical activity (IV) will lead to greater weight loss (DV) among participants (compared to those with lower levels of physical activity)."

By formulating clear hypotheses and operationalizing variables, researchers can systematically investigate causal relationships and contribute to the advancement of scientific knowledge.

Causal Research Design

Designing your research study involves making critical decisions about how you will collect and analyze data to investigate causal relationships.

Experimental vs. Observational Designs

One of the first decisions you'll make when designing a study is whether to employ an experimental or observational design. Each approach has its strengths and limitations, and the choice depends on factors such as the research question, feasibility , and ethical considerations.

• Experimental Design: In experimental designs, researchers manipulate the independent variable and observe its effects on the dependent variable while controlling for confounding variables. Random assignment to experimental conditions allows for causal inferences to be drawn. Example: A study testing the effectiveness of a new teaching method on student performance by randomly assigning students to either the experimental group (receiving the new teaching method) or the control group (receiving the traditional method).
• Observational Design: Observational designs involve observing and measuring variables without intervention. Researchers may still examine relationships between variables but cannot establish causality as definitively as in experimental designs. Example: A study observing the association between socioeconomic status and health outcomes by collecting data on income, education level, and health indicators from a sample of participants.

Control and Randomization

Control and randomization are crucial aspects of experimental design that help ensure the validity of causal inferences.

• Control: Controlling for extraneous variables involves holding constant factors that could influence the dependent variable, except for the independent variable under investigation. This helps isolate the effects of the independent variable. Example: In a medication trial, controlling for factors such as age, gender, and pre-existing health conditions ensures that any observed differences in outcomes can be attributed to the medication rather than other variables.
• Randomization: Random assignment of participants to experimental conditions helps distribute potential confounders evenly across groups, reducing the likelihood of systematic biases and allowing for causal conclusions. Example: Randomly assigning patients to treatment and control groups in a clinical trial ensures that both groups are comparable in terms of baseline characteristics, minimizing the influence of extraneous variables on treatment outcomes.

Internal and External Validity

Two key concepts in research design are internal validity and external validity, which relate to the credibility and generalizability of study findings, respectively.

• Internal Validity: Internal validity refers to the extent to which the observed effects can be attributed to the manipulation of the independent variable rather than confounding factors. Experimental designs typically have higher internal validity due to their control over extraneous variables. Example: A study examining the impact of a training program on employee productivity would have high internal validity if it could confidently attribute changes in productivity to the training intervention.
• External Validity: External validity concerns the extent to which study findings can be generalized to other populations, settings, or contexts. While experimental designs prioritize internal validity, they may sacrifice external validity by using highly controlled conditions that do not reflect real-world scenarios. Example: Findings from a laboratory study on memory retention may have limited external validity if the experimental tasks and conditions differ significantly from real-life learning environments.

Types of Experimental Designs

Several types of experimental designs are commonly used in causal research, each with its own strengths and applications.

• Randomized Control Trials (RCTs): RCTs are considered the gold standard for assessing causality in research. Participants are randomly assigned to experimental and control groups, allowing researchers to make causal inferences. Example: A pharmaceutical company testing a new drug's efficacy would use an RCT to compare outcomes between participants receiving the drug and those receiving a placebo.
• Quasi-Experimental Designs: Quasi-experimental designs lack random assignment but still attempt to establish causality by controlling for confounding variables through design or statistical analysis . Example: A study evaluating the effectiveness of a smoking cessation program might compare outcomes between participants who voluntarily enroll in the program and a matched control group of non-enrollees.

By carefully selecting an appropriate research design and addressing considerations such as control, randomization, and validity, researchers can conduct studies that yield credible evidence of causal relationships and contribute valuable insights to their field of inquiry.

Causal Research Data Collection

Collecting data is a critical step in any research study, and the quality of the data directly impacts the validity and reliability of your findings.

Choosing Measurement Instruments

Selecting appropriate measurement instruments is essential for accurately capturing the variables of interest in your study. The choice of measurement instrument depends on factors such as the nature of the variables, the target population , and the research objectives.

• Surveys :  Surveys are commonly used to collect self-reported data on attitudes, opinions, behaviors, and demographics . They can be administered through various methods, including paper-and-pencil surveys, online surveys, and telephone interviews.
• Observations:  Observational methods involve systematically recording behaviors, events, or phenomena as they occur in natural settings. Observations can be structured (following a predetermined checklist) or unstructured (allowing for flexible data collection).
• Psychological Tests:  Psychological tests are standardized instruments designed to measure specific psychological constructs, such as intelligence, personality traits, or emotional functioning. These tests often have established reliability and validity.
• Physiological Measures:  Physiological measures, such as heart rate, blood pressure, or brain activity, provide objective data on bodily processes. They are commonly used in health-related research but require specialized equipment and expertise.
• Existing Databases:  Researchers may also utilize existing datasets, such as government surveys, public health records, or organizational databases, to answer research questions. Secondary data analysis can be cost-effective and time-saving but may be limited by the availability and quality of data.

Ensuring accurate data collection is the cornerstone of any successful research endeavor. With the right tools in place, you can unlock invaluable insights to drive your causal research forward. From surveys to tests, each instrument offers a unique lens through which to explore your variables of interest.

At Appinio , we understand the importance of robust data collection methods in informing impactful decisions. Let us empower your research journey with our intuitive platform, where you can effortlessly gather real-time consumer insights to fuel your next breakthrough.   Ready to take your research to the next level? Book a demo today and see how Appinio can revolutionize your approach to data collection!

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Sampling Techniques

Sampling involves selecting a subset of individuals or units from a larger population to participate in the study. The goal of sampling is to obtain a representative sample that accurately reflects the characteristics of the population of interest.

• Probability Sampling:  Probability sampling methods involve randomly selecting participants from the population, ensuring that each member of the population has an equal chance of being included in the sample. Common probability sampling techniques include simple random sampling , stratified sampling, and cluster sampling .
• Non-Probability Sampling:  Non-probability sampling methods do not involve random selection and may introduce biases into the sample. Examples of non-probability sampling techniques include convenience sampling, purposive sampling, and snowball sampling.

The choice of sampling technique depends on factors such as the research objectives, population characteristics, resources available, and practical constraints. Researchers should strive to minimize sampling bias and maximize the representativeness of the sample to enhance the generalizability of their findings.

Ethical Considerations

Ethical considerations are paramount in research and involve ensuring the rights, dignity, and well-being of research participants. Researchers must adhere to ethical principles and guidelines established by professional associations and institutional review boards (IRBs).

• Informed Consent:  Participants should be fully informed about the nature and purpose of the study, potential risks and benefits, their rights as participants, and any confidentiality measures in place. Informed consent should be obtained voluntarily and without coercion.
• Privacy and Confidentiality:  Researchers should take steps to protect the privacy and confidentiality of participants' personal information. This may involve anonymizing data, securing data storage, and limiting access to identifiable information.
• Minimizing Harm:  Researchers should mitigate any potential physical, psychological, or social harm to participants. This may involve conducting risk assessments, providing appropriate support services, and debriefing participants after the study.
• Respect for Participants:  Researchers should respect participants' autonomy, diversity, and cultural values. They should seek to foster a trusting and respectful relationship with participants throughout the research process.
• Publication and Dissemination:  Researchers have a responsibility to accurately report their findings and acknowledge contributions from participants and collaborators. They should adhere to principles of academic integrity and transparency in disseminating research results.

By addressing ethical considerations in research design and conduct, researchers can uphold the integrity of their work, maintain trust with participants and the broader community, and contribute to the responsible advancement of knowledge in their field.

Causal Research Data Analysis

Once data is collected, it must be analyzed to draw meaningful conclusions and assess causal relationships.

Causal Inference Methods

Causal inference methods are statistical techniques used to identify and quantify causal relationships between variables in observational data. While experimental designs provide the most robust evidence for causality, observational studies often require more sophisticated methods to account for confounding factors.

• Difference-in-Differences (DiD):  DiD compares changes in outcomes before and after an intervention between a treatment group and a control group, controlling for pre-existing trends. It estimates the average treatment effect by differencing the changes in outcomes between the two groups over time.
• Instrumental Variables (IV):  IV analysis relies on instrumental variables—variables that affect the treatment variable but not the outcome—to estimate causal effects in the presence of endogeneity. IVs should be correlated with the treatment but uncorrelated with the error term in the outcome equation.
• Regression Discontinuity (RD):  RD designs exploit naturally occurring thresholds or cutoff points to estimate causal effects near the threshold. Participants just above and below the threshold are compared, assuming that they are similar except for their proximity to the threshold.
• Propensity Score Matching (PSM):  PSM matches individuals or units based on their propensity scores—the likelihood of receiving the treatment—creating comparable groups with similar observed characteristics. Matching reduces selection bias and allows for causal inference in observational studies.

Assessing Causality Strength

Assessing the strength of causality involves determining the magnitude and direction of causal effects between variables. While statistical significance indicates whether an observed relationship is unlikely to occur by chance, it does not necessarily imply a strong or meaningful effect.

• Effect Size:  Effect size measures the magnitude of the relationship between variables, providing information about the practical significance of the results. Standard effect size measures include Cohen's d for mean differences and odds ratios for categorical outcomes.
• Confidence Intervals:  Confidence intervals provide a range of values within which the actual effect size is likely to lie with a certain degree of certainty. Narrow confidence intervals indicate greater precision in estimating the true effect size.
• Practical Significance:  Practical significance considers whether the observed effect is meaningful or relevant in real-world terms. Researchers should interpret results in the context of their field and the implications for stakeholders.

Handling Confounding Variables

Confounding variables are extraneous factors that may distort the observed relationship between the independent and dependent variables, leading to spurious or biased conclusions. Addressing confounding variables is essential for establishing valid causal inferences.

• Statistical Control:  Statistical control involves including confounding variables as covariates in regression models to partially out their effects on the outcome variable. Controlling for confounders reduces bias and strengthens the validity of causal inferences.
• Matching:  Matching participants or units based on observed characteristics helps create comparable groups with similar distributions of confounding variables. Matching reduces selection bias and mimics the randomization process in experimental designs.
• Sensitivity Analysis:  Sensitivity analysis assesses the robustness of study findings to changes in model specifications or assumptions. By varying analytical choices and examining their impact on results, researchers can identify potential sources of bias and evaluate the stability of causal estimates.
• Subgroup Analysis:  Subgroup analysis explores whether the relationship between variables differs across subgroups defined by specific characteristics. Identifying effect modifiers helps understand the conditions under which causal effects may vary.

By employing rigorous causal inference methods, assessing the strength of causality, and addressing confounding variables, researchers can confidently draw valid conclusions about causal relationships in their studies, advancing scientific knowledge and informing evidence-based decision-making.

Causal Research Examples

Examples play a crucial role in understanding the application of causal research methods and their impact across various domains. Let's explore some detailed examples to illustrate how causal research is conducted and its real-world implications:

Example 1: Software as a Service (SaaS) User Retention Analysis

Suppose a SaaS company wants to understand the factors influencing user retention and engagement with their platform. The company conducts a longitudinal observational study, collecting data on user interactions, feature usage, and demographic information over several months.

• Design:  The company employs an observational cohort study design, tracking cohorts of users over time to observe changes in retention and engagement metrics. They use analytics tools to collect data on user behavior , such as logins, feature usage, session duration, and customer support interactions.
• Data Collection:  Data is collected from the company's platform logs, customer relationship management (CRM) system, and user surveys. Key metrics include user churn rates, active user counts, feature adoption rates, and Net Promoter Scores ( NPS ).
• Analysis:  Using statistical techniques like survival analysis and regression modeling, the company identifies factors associated with user retention, such as feature usage patterns, onboarding experiences, customer support interactions, and subscription plan types.
• Findings: The analysis reveals that users who engage with specific features early in their lifecycle have higher retention rates, while those who encounter usability issues or lack personalized onboarding experiences are more likely to churn. The company uses these insights to optimize product features, improve onboarding processes, and enhance customer support strategies to increase user retention and satisfaction.

Example 2: Business Impact of Digital Marketing Campaign

Consider a technology startup launching a digital marketing campaign to promote its new product offering. The company conducts an experimental study to evaluate the effectiveness of different marketing channels in driving website traffic, lead generation, and sales conversions.

• Design:  The company implements an A/B testing design, randomly assigning website visitors to different marketing treatment conditions, such as Google Ads, social media ads, email campaigns, or content marketing efforts. They track user interactions and conversion events using web analytics tools and marketing automation platforms.
• Data Collection:  Data is collected on website traffic, click-through rates, conversion rates, lead generation, and sales revenue. The company also gathers demographic information and user feedback through surveys and customer interviews to understand the impact of marketing messages and campaign creatives .
• Analysis:  Utilizing statistical methods like hypothesis testing and multivariate analysis, the company compares key performance metrics across different marketing channels to assess their effectiveness in driving user engagement and conversion outcomes. They calculate return on investment (ROI) metrics to evaluate the cost-effectiveness of each marketing channel.
• Findings:  The analysis reveals that social media ads outperform other marketing channels in generating website traffic and lead conversions, while email campaigns are more effective in nurturing leads and driving sales conversions. Armed with these insights, the company allocates marketing budgets strategically, focusing on channels that yield the highest ROI and adjusting messaging and targeting strategies to optimize campaign performance.

These examples demonstrate the diverse applications of causal research methods in addressing important questions, informing policy decisions, and improving outcomes in various fields. By carefully designing studies, collecting relevant data, employing appropriate analysis techniques, and interpreting findings rigorously, researchers can generate valuable insights into causal relationships and contribute to positive social change.

How to Interpret Causal Research Results?

Interpreting and reporting research findings is a crucial step in the scientific process, ensuring that results are accurately communicated and understood by stakeholders.

Interpreting Statistical Significance

Statistical significance indicates whether the observed results are unlikely to occur by chance alone, but it does not necessarily imply practical or substantive importance. Interpreting statistical significance involves understanding the meaning of p-values and confidence intervals and considering their implications for the research findings.

• P-values:  A p-value represents the probability of obtaining the observed results (or more extreme results) if the null hypothesis is true. A p-value below a predetermined threshold (typically 0.05) suggests that the observed results are statistically significant, indicating that the null hypothesis can be rejected in favor of the alternative hypothesis.
• Confidence Intervals:  Confidence intervals provide a range of values within which the true population parameter is likely to lie with a certain degree of confidence (e.g., 95%). If the confidence interval does not include the null value, it suggests that the observed effect is statistically significant at the specified confidence level.

Interpreting statistical significance requires considering factors such as sample size, effect size, and the practical relevance of the results rather than relying solely on p-values to draw conclusions.

Discussing Practical Significance

While statistical significance indicates whether an effect exists, practical significance evaluates the magnitude and meaningfulness of the effect in real-world terms. Discussing practical significance involves considering the relevance of the results to stakeholders and assessing their impact on decision-making and practice.

• Effect Size:  Effect size measures the magnitude of the observed effect, providing information about its practical importance. Researchers should interpret effect sizes in the context of their field and the scale of measurement (e.g., small, medium, or large effect sizes).
• Contextual Relevance:  Consider the implications of the results for stakeholders, policymakers, and practitioners. Are the observed effects meaningful in the context of existing knowledge, theory, or practical applications? How do the findings contribute to addressing real-world problems or informing decision-making?

Discussing practical significance helps contextualize research findings and guide their interpretation and application in practice, beyond statistical significance alone.

Addressing Limitations and Assumptions

No study is without limitations, and researchers should transparently acknowledge and address potential biases, constraints, and uncertainties in their research design and findings.

• Methodological Limitations:  Identify any limitations in study design, data collection, or analysis that may affect the validity or generalizability of the results. For example, sampling biases , measurement errors, or confounding variables.
• Assumptions:  Discuss any assumptions made in the research process and their implications for the interpretation of results. Assumptions may relate to statistical models, causal inference methods, or theoretical frameworks underlying the study.
• Alternative Explanations:  Consider alternative explanations for the observed results and discuss their potential impact on the validity of causal inferences. How robust are the findings to different interpretations or competing hypotheses?

Addressing limitations and assumptions demonstrates transparency and rigor in the research process, allowing readers to critically evaluate the validity and reliability of the findings.

Communicating Findings Clearly

Effectively communicating research findings is essential for disseminating knowledge, informing decision-making, and fostering collaboration and dialogue within the scientific community.

• Clarity and Accessibility:  Present findings in a clear, concise, and accessible manner, using plain language and avoiding jargon or technical terminology. Organize information logically and use visual aids (e.g., tables, charts, graphs) to enhance understanding.
• Contextualization:  Provide context for the results by summarizing key findings, highlighting their significance, and relating them to existing literature or theoretical frameworks. Discuss the implications of the findings for theory, practice, and future research directions.
• Transparency:  Be transparent about the research process, including data collection procedures, analytical methods, and any limitations or uncertainties associated with the findings. Clearly state any conflicts of interest or funding sources that may influence interpretation.

By communicating findings clearly and transparently, researchers can facilitate knowledge exchange, foster trust and credibility, and contribute to evidence-based decision-making.

Causal Research Tips

When conducting causal research, it's essential to approach your study with careful planning, attention to detail, and methodological rigor. Here are some tips to help you navigate the complexities of causal research effectively:

• Define Clear Research Questions:  Start by clearly defining your research questions and hypotheses. Articulate the causal relationship you aim to investigate and identify the variables involved.
• Consider Alternative Explanations:  Be mindful of potential confounding variables and alternative explanations for the observed relationships. Take steps to control for confounders and address alternative hypotheses in your analysis.
• Prioritize Internal Validity:  While external validity is important for generalizability, prioritize internal validity in your study design to ensure that observed effects can be attributed to the manipulation of the independent variable.
• Use Randomization When Possible:  If feasible, employ randomization in experimental designs to distribute potential confounders evenly across experimental conditions and enhance the validity of causal inferences.
• Be Transparent About Methods:  Provide detailed descriptions of your research methods, including data collection procedures, analytical techniques, and any assumptions or limitations associated with your study.
• Utilize Multiple Methods:  Consider using a combination of experimental and observational methods to triangulate findings and strengthen the validity of causal inferences.
• Be Mindful of Sample Size:  Ensure that your sample size is adequate to detect meaningful effects and minimize the risk of Type I and Type II errors. Conduct power analyses to determine the sample size needed to achieve sufficient statistical power.
• Validate Measurement Instruments:  Validate your measurement instruments to ensure that they are reliable and valid for assessing the variables of interest in your study. Pilot test your instruments if necessary.
• Seek Feedback from Peers:  Collaborate with colleagues or seek feedback from peer reviewers to solicit constructive criticism and improve the quality of your research design and analysis.

Conclusion for Causal Research

Mastering causal research empowers researchers to unlock the secrets of cause and effect, shedding light on the intricate relationships between variables in diverse fields. By employing rigorous methods such as experimental designs, causal inference techniques, and careful data analysis, you can uncover causal mechanisms, predict outcomes, and inform evidence-based practices. Through the lens of causal research, complex phenomena become more understandable, and interventions become more effective in addressing societal challenges and driving progress. In a world where understanding the reasons behind events is paramount, causal research serves as a beacon of clarity and insight. Armed with the knowledge and techniques outlined in this guide, you can navigate the complexities of causality with confidence, advancing scientific knowledge, guiding policy decisions, and ultimately making meaningful contributions to our understanding of the world.

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Causal Research: What it is, Tips & Examples

Causal research is classified as conclusive research since it attempts to build a cause-and-effect link between two variables. This research is mainly used to determine the cause of particular behavior. We can use this research to determine what changes occur in an independent variable due to a change in the dependent variable.

It can assist you in evaluating marketing activities, improving internal procedures, and developing more effective business plans. Understanding how one circumstance affects another may help you determine the most effective methods for satisfying your business needs.

LEARN ABOUT: Behavioral Research

This post will explain causal research, define its essential components, describe its benefits and limitations, and provide some important tips.

Content Index

What is causal research?

Temporal sequence, non-spurious association, concomitant variation, the advantages, the disadvantages, causal research examples, causal research tips.

Causal research is also known as explanatory research . It’s a type of research that examines if there’s a cause-and-effect relationship between two separate events. This would occur when there is a change in one of the independent variables, which is causing changes in the dependent variable.

You can use causal research to evaluate the effects of particular changes on existing norms, procedures, and so on. This type of research examines a condition or a research problem to explain the patterns of interactions between variables.

LEARN ABOUT: Research Process Steps

Components of causal research

Only specific causal information can demonstrate the existence of cause-and-effect linkages. The three key components of causal research are as follows:

Prior to the effect, the cause must occur. If the cause occurs before the appearance of the effect, the cause and effect can only be linked. For example, if the profit increase occurred before the advertisement aired, it cannot be linked to an increase in advertising spending.

Linked fluctuations between two variables are only allowed if there is no other variable that is related to both cause and effect. For example, a notebook manufacturer has discovered a correlation between notebooks and the autumn season. They see that during this season, more people buy notebooks because students are buying them for the upcoming semester.

During the summer, the company launched an advertisement campaign for notebooks. To test their assumption, they can look up the campaign data to see if the increase in notebook sales was due to the student’s natural rhythm of buying notebooks or the advertisement.

Concomitant variation is defined as a quantitative change in effect that happens solely as a result of a quantitative change in the cause. This means that there must be a steady change between the two variables. You can examine the validity of a cause-and-effect connection by seeing if the independent variable causes a change in the dependent variable.

For example, if any company does not make an attempt to enhance sales by acquiring skilled employees or offering training to them, then the hire of experienced employees cannot be credited for an increase in sales. Other factors may have contributed to the increase in sales.

Causal Research Advantages and Disadvantages

Causal or explanatory research has various advantages for both academics and businesses. As with any other research method, it has a few disadvantages that researchers should be aware of. Let’s look at some of the advantages and disadvantages of this research design .

• Helps in the identification of the causes of system processes. This allows the researcher to take the required steps to resolve issues or improve outcomes.
• It provides replication if it is required.
• Causal research assists in determining the effects of changing procedures and methods.
• Subjects are chosen in a methodical manner. As a result, it is beneficial for improving internal validity .
• The ability to analyze the effects of changes on existing events, processes, phenomena, and so on.
• Finds the sources of variable correlations, bridging the gap in correlational research .
• It is not always possible to monitor the effects of all external factors, so causal research is challenging to do.
• It is time-consuming and might be costly to execute.
• The effect of a large range of factors and variables existing in a particular setting makes it difficult to draw results.
• The most major error in this research is a coincidence. A coincidence between a cause and an effect can sometimes be interpreted as a direction of causality.
• To corroborate the findings of the explanatory research , you must undertake additional types of research. You can’t just make conclusions based on the findings of a causal study.
• It is sometimes simple for a researcher to see that two variables are related, but it can be difficult for a researcher to determine which variable is the cause and which variable is the effect.

Since different industries and fields can carry out causal comparative research , it can serve many different purposes. Let’s discuss 3 examples of causal research:

Advertising Research

Companies can use causal research to enact and study advertising campaigns. For example, six months after a business debuts a new ad in a region. They see a 5% increase in sales revenue.

To assess whether the ad has caused the lift, they run the same ad in randomly selected regions so they can compare sales data across regions over another six months. When sales pick up again in these regions, they can conclude that the ad and sales have a valuable cause-and-effect relationship.

LEARN ABOUT: Ad Testing

Customer Loyalty Research

Businesses can use causal research to determine the best customer retention strategies. They monitor interactions between associates and customers to identify patterns of cause and effect, such as a product demonstration technique leading to increased or decreased sales from the same customers.

For example, a company implements a new individual marketing strategy for a small group of customers and sees a measurable increase in monthly subscriptions. After receiving identical results from several groups, they concluded that the one-to-one marketing strategy has the causal relationship they intended.

Educational Research

Learning specialists, academics, and teachers use causal research to learn more about how politics affects students and identify possible student behavior trends. For example, a university administration notices that more science students drop out of their program in their third year, which is 7% higher than in any other year.

They interview a random group of science students and discover many factors that could lead to these circumstances, including non-university components. Through the in-depth statistical analysis, researchers uncover the top three factors, and management creates a committee to address them in the future.

Causal research is frequently the last type of research done during the research process and is considered definitive. As a result, it is critical to plan the research with specific parameters and goals in mind. Here are some tips for conducting causal research successfully:

1. Understand the parameters of your research

Identify any design strategies that change the way you understand your data. Determine how you acquired data and whether your conclusions are more applicable in practice in some cases than others.

2. Pick a random sampling strategy

Choosing a technique that works best for you when you have participants or subjects is critical. You can use a database to generate a random list, select random selections from sorted categories, or conduct a survey.

3. Determine all possible relations

Examine the different relationships between your independent and dependent variables to build more sophisticated insights and conclusions.

To summarize, causal or explanatory research helps organizations understand how their current activities and behaviors will impact them in the future. This is incredibly useful in a wide range of business scenarios. This research can ensure the outcome of various marketing activities, campaigns, and collaterals. Using the findings of this research program, you will be able to design more successful business strategies that take advantage of every business opportunity.

At QuestionPro, we offer all kinds of necessary tools for researchers to carry out their projects. It can help you get the most out of your data by guiding you through the process.

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An Introduction to Causal Inference *

Judea pearl.

* University of California, Los Angeles, ude.alcu.sc@aeduj

This paper summarizes recent advances in causal inference and underscores the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underlie all causal inferences, the languages used in formulating those assumptions, the conditional nature of all causal and counterfactual claims, and the methods that have been developed for the assessment of such claims. These advances are illustrated using a general theory of causation based on the Structural Causal Model (SCM) described in Pearl (2000a) , which subsumes and unifies other approaches to causation, and provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: those about (1) the effects of potential interventions, (2) probabilities of counterfactuals, and (3) direct and indirect effects (also known as "mediation"). Finally, the paper defines the formal and conceptual relationships between the structural and potential-outcome frameworks and presents tools for a symbiotic analysis that uses the strong features of both. The tools are demonstrated in the analyses of mediation, causes of effects, and probabilities of causation.

1. Introduction

Most studies in the health, social and behavioral sciences aim to answer causal rather than associative – questions. Such questions require some knowledge of the data-generating process, and cannot be computed from the data alone, nor from the distributions that govern the data. Remarkably, although much of the conceptual framework and algorithmic tools needed for tackling such problems are now well established, they are not known to many of the researchers who could put them into practical use. Solving causal problems systematically requires certain extensions in the standard mathematical language of statistics, and these extensions are not typically emphasized in the mainstream literature. As a result, many statistical researchers have not yet benefited from causal inference results in (i) counterfactual analysis, (ii) nonparametric structural equations, (iii) graphical models, and (iv) the symbiosis between counterfactual and graphical methods. This survey aims at making these contemporary advances more accessible by providing a gentle introduction to causal inference for a more in-depth treatment and its methodological principles (see ( Pearl, 2000a , 2009a , b )).

In Section 2, we discuss coping with untested assumptions and new mathematical notation which is required to move from associational to causal statistics. Section 3.1 introduces the fundamentals of the structural theory of causation and uses these modeling fundamentals to represent interventions and develop mathematical tools for estimating causal effects (Section 3.3) and counterfactual quantities (Section 3.4). Section 4 outlines a general methodology to guide problems of causal inference: Define, Assume, Identify and Estimate, with each step benefiting from the tools developed in Section 3.

Section 5 relates these tools to those used in the potential-outcome framework, and offers a formal mapping between the two frameworks and a symbiosis (Section 5.3) that exploits the best features of both. Finally, the benefit of this symbiosis is demonstrated in Section 6, in which the structure-based logic of counterfactuals is harnessed to estimate causal quantities that cannot be defined within the paradigm of controlled randomized experiments. These include direct and indirect effects, the effect of treatment on the treated, and questions of attribution, i.e., whether one event can be deemed “responsible” for another.

2. From Association to Causation

2.1. understanding the distinction and its implications.

The aim of standard statistical analysis is to assess parameters of a distribution from samples drawn of that distribution. With the help of such parameters, associations among variables can be inferred, which permits the researcher to estimate probabilities of past and future events and update those probabilities in light of new information. These tasks are managed well by standard statistical analysis so long as experimental conditions remain the same. Causal analysis goes one step further; its aim is to infer probabilities under conditions that are changing , for example, changes induced by treatments or external interventions.

This distinction implies that causal and associational concepts do not mix; there is nothing in a distribution function to tell us how that distribution would differ if external conditions were to change—say from observational to experimental setup—because the laws of probability theory do not dictate how one property of a distribution ought to change when another property is modified. This information must be provided by causal assumptions which identify relationships that remain invariant when external conditions change.

A useful demarcation line between associational and causal concepts crisp and easy to apply, can be formulated as follows. An associational concept is any relationship that can be defined in terms of a joint distribution of observed variables, and a causal concept is any relationship that cannot be defined from the distribution alone. Examples of associational concepts are: correlation, regression, dependence, conditional independence, likelihood, collapsibility, propensity score, risk ratio, odds ratio, marginalization, conditionalization, “controlling for,” and many more. Examples of causal concepts are: randomization, influence, effect, confounding, “holding constant,” disturbance, error terms, structural coefficients, spurious correlation, faithfulness/stability, instrumental variables, intervention, explanation, and attribution. The former can, while the latter cannot be defined in term of distribution functions.

This demarcation line is extremely useful in tracing the assumptions that are needed for substantiating various types of scientific claims. Every claim invoking causal concepts must rely on some premises that invoke such concepts; it cannot be inferred from, or even defined in terms statistical associations alone.

This distinction further implies that causal relations cannot be expressed in the language of probability and, hence, that any mathematical approach to causal analysis must acquire new notation – probability calculus is insufficient. To illustrate, the syntax of probability calculus does not permit us to express the simple fact that “symptoms do not cause diseases,” let alone draw mathematical conclusions from such facts. All we can say is that two events are dependent—meaning that if we find one, we can expect to encounter the other, but we cannot distinguish statistical dependence, quantified by the conditional probability P ( disease | symptom ) from causal dependence, for which we have no expression in standard probability calculus.

2.2. Untested assumptions and new notation

The preceding two requirements: (1) to commence causal analysis with untested, 1 theoretically or judgmentally based assumptions, and (2) to extend the syntax of probability calculus, constitute the two primary barriers to the acceptance of causal analysis among professionals with traditional training in statistics.

Associational assumptions, even untested, are testable in principle, given sufficiently large sample and sufficiently fine measurements. Causal assumptions, in contrast, cannot be verified even in principle, unless one resorts to experimental control. This difference stands out in Bayesian analysis. Though the priors that Bayesians commonly assign to statistical parameters are untested quantities, the sensitivity to these priors tends to diminish with increasing sample size. In contrast, sensitivity to prior causal assumptions, say that treatment does not change gender, remains substantial regardless of sample size.

This makes it doubly important that the notation we use for expressing causal assumptions be cognitively meaningful and unambiguous so that one can clearly judge the plausibility or inevitability of the assumptions articulated. Statisticians can no longer ignore the mental representation in which scientists store experiential knowledge, since it is this representation, and the language used to access it that determine the reliability of the judgments upon which the analysis so crucially depends.

Those versed in the potential-outcome notation ( Neyman, 1923 , Rubin, 1974 , Holland, 1988 ), can recognize causal expressions through the subscripts that are attached to counterfactual events and variables, e.g. Y x ( u ) or Z xy . (Some authors use parenthetical expressions, e.g. Y (0), Y (1), Y ( x , u ) or Z ( x , y ).) The expression Y x ( u ), for example, stands for the value that outcome Y would take in individual u , had treatment X been at level x . If u is chosen at random, Y x is a random variable, and one can talk about the probability that Y x would attain a value y in the population, written P ( Y x = y ) (see Section 5 for semantics). Alternatively, Pearl (1995) used expressions of the form P ( Y = y | set ( X = x )) or P ( Y = y | do ( X = x )) to denote the probability (or frequency) that event ( Y = y ) would occur if treatment condition X = x were enforced uniformly over the population. 2 Still a third notation that distinguishes causal expressions is provided by graphical models, where the arrows convey causal directionality.

However, few have taken seriously the textbook requirement that any introduction of new notation must entail a systematic definition of the syntax and semantics that governs the notation. Moreover, in the bulk of the statistical literature before 2000, causal claims rarely appear in the mathematics. They surface only in the verbal interpretation that investigators occasionally attach to certain associations, and in the verbal description with which investigators justify assumptions. For example, the assumption that a covariate not be affected by a treatment, a necessary assumption for the control of confounding ( Cox, 1958 , p. 48), is expressed in plain English, not in a mathematical expression.

The next section provides a conceptualization that overcomes these mental barriers by offering a friendly mathematical machinery for cause-effect analysis and a formal foundation for counterfactual analysis.

3. Structural Models, Diagrams, Causal Effects, and Counterfactuals

Any conception of causation worthy of the title “theory” must be able to (1) represent causal questions in some mathematical language, (2) provide a precise language for communicating assumptions under which the questions need to be answered, (3) provide a systematic way of answering at least some of these questions and labeling others “unanswerable,” and (4) provide a method of determining what assumptions or new measurements would be needed to answer the “unanswerable” questions.

A “general theory” should do more. In addition to embracing all questions judged to have causal character, a general theory must also subsume any other theory or method that scientists have found useful in exploring the various aspects of causation. In other words, any alternative theory needs to evolve as a special case of the “general theory” when restrictions are imposed on either the model, the type of assumptions admitted, or the language in which those assumptions are cast.

The structural theory that we use in this survey satisfies the criteria above. It is based on the Structural Causal Model (SCM) developed in ( Pearl, 1995 , 2000a ) which combines features of the structural equation models (SEM) used in economics and social science ( Goldberger, 1973 , Duncan, 1975 ), the potential-outcome framework of Neyman (1923) and Rubin (1974) , and the graphical models developed for probabilistic reasoning and causal analysis ( Pearl, 1988 , Lauritzen, 1996 , Spirtes, Glymour, and Scheines, 2000 , Pearl, 2000a ).

Although the basic elements of SCM were introduced in the mid 1990’s ( Pearl, 1995 ), and have been adapted widely by epidemiologists ( Greenland, Pearl, and Robins, 1999 , Glymour and Greenland, 2008 ), statisticians ( Cox and Wermuth, 2004 , Lauritzen, 2001 ), and social scientists ( Morgan and Winship, 2007 ), its potentials as a comprehensive theory of causation are yet to be fully utilized. Its ramifications thus far include:

• The unification of the graphical, potential outcome, structural equations, decision analytical ( Dawid, 2002 ), interventional ( Woodward, 2003 ), sufficient component ( Rothman, 1976 ) and probabilistic ( Suppes, 1970 ) approaches to causation; with each approach viewed as a restricted version of the SCM.
• The definition, axiomatization and algorithmization of counterfactuals and joint probabilities of counterfactuals
• Reducing the evaluation of “effects of causes,” “mediated effects,” and “causes of effects” to an algorithmic level of analysis.
• Solidifying the mathematical foundations of the potential-outcome model, and formulating the counterfactual foundations of structural equation models.
• Demystifying enigmatic notions such as “confounding,” “mediation,” “ignorability,” “comparability,” “exchangeability (of populations),” “superexogeneity” and others within a single and familiar conceptual framework.
• Weeding out myths and misconceptions from outdated traditions ( Meek and Glymour, 1994 , Greenland et al., 1999 , Cole and Hernán, 2002 , Arah, 2008 , Shrier, 2009 , Pearl, 2009c ).

This section provides a gentle introduction to the structural framework and uses it to present the main advances in causal inference that have emerged in the past two decades.

3.1. A brief introduction to structural equation models

How can one express mathematically the common understanding that symptoms do not cause diseases? The earliest attempt to formulate such relationship mathematically was made in the 1920’s by the geneticist Sewall Wright (1921) . Wright used a combination of equations and graphs to communicate causal relationships. For example, if X stands for a disease variable and Y stands for a certain symptom of the disease, Wright would write a linear equation: 3

where x stands for the level (or severity) of the disease, y stands for the level (or severity) of the symptom, and u Y stands for all factors, other than the disease in question, that could possibly affect Y when X is held constant. In interpreting this equation one should think of a physical process whereby Nature examines the values of x and u and, accordingly, assigns variable Y the value y = βx + u Y . Similarly, to “explain” the occurrence of disease X , one could write x = u X , where U X stands for all factors affecting X .

Equation (1) still does not properly express the causal relationship implied by this assignment process, because algebraic equations are symmetrical objects; if we re-write (1) as

it might be misinterpreted to mean that the symptom influences the disease. To express the directionality of the underlying process, Wright augmented the equation with a diagram, later called “path diagram,” in which arrows are drawn from (perceived) causes to their (perceived) effects, and more importantly, the absence of an arrow makes the empirical claim that Nature assigns values to one variable irrespective of another. In Fig. 1 , for example, the absence of arrow from Y to X represents the claim that symptom Y is not among the factors U X which affect disease X . Thus, in our example, the complete model of a symptom and a disease would be written as in Fig. 1 : The diagram encodes the possible existence of (direct) causal influence of X on Y , and the absence of causal influence of Y on X , while the equations encode the quantitative relationships among the variables involved, to be determined from the data. The parameter β in the equation is called a “path coefficient” and it quantifies the (direct) causal effect of X on Y ; given the numerical values of β and U Y , the equation claims that, a unit increase for X would result in β units increase of Y regardless of the values taken by other variables in the model, and regardless of whether the increase in X originates from external or internal influences.

A simple structural equation model, and its associated diagrams. Unobserved exogenous variables are connected by dashed arrows.

The variables U X and U Y are called “exogenous;” they represent observed or unobserved background factors that the modeler decides to keep unexplained, that is, factors that influence but are not influenced by the other variables (called “endogenous”) in the model. Unobserved exogenous variables are sometimes called “disturbances” or “errors”, they represent factors omitted from the model but judged to be relevant for explaining the behavior of variables in the model. Variable U X , for example, represents factors that contribute to the disease X , which may or may not be correlated with U Y (the factors that influence the symptom Y ). Thus, background factors in structural equations differ fundamentally from residual terms in regression equations. The latters are artifacts of analysis which, by definition, are uncorrelated with the regressors. The formers are part of physical reality (e.g., genetic factors, socio-economic conditions) which are responsible for variations observed in the data; they are treated as any other variable, though we often cannot measure their values precisely and must resign to merely acknowledging their existence and assessing qualitatively how they relate to other variables in the system.

If correlation is presumed possible, it is customary to connect the two variables, U Y and U X , by a dashed double arrow, as shown in Fig. 1(b) .

In reading path diagrams, it is common to use kinship relations such as parent, child, ancestor, and descendent, the interpretation of which is usually self evident. For example, an arrow X → Y designates X as a parent of Y and Y as a child of X . A “path” is any consecutive sequence of edges, solid or dashed. For example, there are two paths between X and Y in Fig. 1(b) , one consisting of the direct arrow X → Y while the other tracing the nodes X , U X , U Y and Y .

Wright’s major contribution to causal analysis, aside from introducing the language of path diagrams, has been the development of graphical rules for writing down the covariance of any pair of observed variables in terms of path coefficients and of covariances among the error terms. In our simple example, one can immediately write the relations

for Fig. 1(a) , and

for Fig. 1(b) (These can be derived of course from the equations, but, for large models, algebraic methods tend to obscure the origin of the derived quantities). Under certain conditions, (e.g. if Cov ( U Y , U X ) = 0), such relationships may allow one to solve for the path coefficients in term of observed covariance terms only, and this amounts to inferring the magnitude of (direct) causal effects from observed, nonexperimental associations, assuming of course that one is prepared to defend the causal assumptions encoded in the diagram.

It is important to note that, in path diagrams, causal assumptions are encoded not in the links but, rather, in the missing links. An arrow merely indicates the possibility of causal connection, the strength of which remains to be determined (from data); a missing arrow represents a claim of zero influence, while a missing double arrow represents a claim of zero covariance. In Fig. 1(a) , for example, the assumptions that permits us to identify the direct effect β are encoded by the missing double arrow between U X and U Y , indicating Cov ( U Y , U X )=0, together with the missing arrow from Y to X . Had any of these two links been added to the diagram, we would not have been able to identify the direct effect β . Such additions would amount to relaxing the assumption Cov ( U Y , U X ) = 0, or the assumption that Y does not effect X , respectively. Note also that both assumptions are causal, not associational, since none can be determined from the joint density of the observed variables, X and Y ; the association between the unobserved terms, U Y and U X , can only be uncovered in an experimental setting; or (in more intricate models, as in Fig. 5 ) from other causal assumptions.

Causal diagram representing the assignment ( Z ), treatment ( X ), and outcome ( Y ) in a clinical trial with imperfect compliance.

Although each causal assumption in isolation cannot be tested, the sum total of all causal assumptions in a model often has testable implications. The chain model of Fig. 2(a) , for example, encodes seven causal assumptions, each corresponding to a missing arrow or a missing double-arrow between a pair of variables. None of those assumptions is testable in isolation, yet the totality of all those assumptions implies that Z is unassociated with Y in every stratum of X . Such testable implications can be read off the diagrams using a graphical criterion known as d-separation ( Pearl, 1988 ).

(a) The diagram associated with the structural model of Eq. (5) . (b) The diagram associated with the modified model of Eq. (6) , representing the intervention do ( X = x 0 ).

Definition 1 ( d -separation) A set S of nodes is said to block a path p if either (i) p contains at least one arrow-emitting node that is in S, or (ii) p contains at least one collision node that is outside S and has no descendant in S. If S blocks all paths from X to Y, it is said to “d-separate X and Y,” and then, X and Y are independent given S, written X ⊥⊥ Y|S .

To illustrate, the path U Z → Z → X → Y is blocked by S = { Z } and by S = { X }, since each emits an arrow along that path. Consequently we can infer that the conditional independencies U Z ⊥⊥ Y | Z and U Z ⊥⊥ Y | X will be satisfied in any probability function that this model can generate, regardless of how we parametrize the arrows. Likewise, the path U Z → Z → X ← U X is blocked by the null set {∅︀} but is not blocked by S = { Y }, since Y is a descendant of the collision node X . Consequently, the marginal independence U Z ⊥⊥ U X will hold in the distribution, but U Z ⊥⊥ U X | Y may or may not hold. This special handling of collision nodes (or colliders, e.g., Z → X ← U X ) reflects a general phenomenon known as Berkson’s paradox ( Berkson, 1946 ), whereby observations on a common consequence of two independent causes render those causes dependent. For example, the outcomes of two independent coins are rendered dependent by the testimony that at least one of them is a tail.

The conditional independencies entailed by d -separation constitute the main opening through which the assumptions embodied in structural equation models can confront the scrutiny of nonexperimental data. In other words, almost all statistical tests capable of invalidating the model are entailed by those implications. 4

3.2. From linear to nonparametric models and graphs

Structural equation modeling (SEM) has been the main vehicle for effect analysis in economics and the behavioral and social sciences ( Goldberger, 1972 , Duncan, 1975 , Bollen, 1989 ). However, the bulk of SEM methodology was developed for linear analysis and, until recently, no comparable methodology has been devised to extend its capabilities to models involving dichotomous variables or nonlinear dependencies. A central requirement for any such extension is to detach the notion of “effect” from its algebraic representation as a coefficient in an equation, and redefine “effect” as a general capacity to transmit changes among variables. Such an extension, based on simulating hypothetical interventions in the model, was proposed in ( Haavelmo, 1943 , Strotz and Wold, 1960 , Spirtes, Glymour, and Scheines, 1993 , Pearl, 1993a , 2000a , Lindley, 2002 ) and has led to new ways of defining and estimating causal effects in nonlinear and nonparametric models (that is, models in which the functional form of the equations is unknown).

The central idea is to exploit the invariant characteristics of structural equations without committing to a specific functional form. For example, the nonparametric interpretation of the diagram of Fig. 2(a) corresponds to a set of three functions, each corresponding to one of the observed variables:

where in this particular example U Z , U X and U Y are assumed to be jointly independent but, otherwise, arbitrarily distributed. Each of these functions represents a causal process (or mechanism) that determines the value of the left variable (output) from those on the right variables (inputs). The absence of a variable from the right hand side of an equation encodes the assumption that Nature ignores that variable in the process of determining the value of the output variable. For example, the absence of variable Z from the arguments of f Y conveys the empirical claim that variations in Z will leave Y unchanged, as long as variables U Y , and X remain constant. A system of such functions are said to be structural if they are assumed to be autonomous, that is, each function is invariant to possible changes in the form of the other functions ( Simon, 1953 , Koopmans, 1953 ).

3.2.1. Representing interventions

This feature of invariance permits us to use structural equations as a basis for modeling causal effects and counterfactuals. This is done through a mathematical operator called do ( x ) which simulates physical interventions by deleting certain functions from the model, replacing them by a constant X = x , while keeping the rest of the model unchanged. For example, to emulate an intervention do ( x 0 ) that holds X constant (at X = x 0 ) in model M of Fig. 2(a) , we replace the equation for x in Eq. (5) with x = x 0 , and obtain a new model, M x 0 ,

the graphical description of which is shown in Fig. 2(b) .

The joint distribution associated with the modified model, denoted P ( z , y | do ( x 0 )) describes the post-intervention distribution of variables Y and Z (also called “controlled” or “experimental” distribution), to be distinguished from the pre-intervention distribution, P ( x , y , z ), associated with the original model of Eq. (5) . For example, if X represents a treatment variable, Y a response variable, and Z some covariate that affects the amount of treatment received, then the distribution P ( z , y | do ( x 0 )) gives the proportion of individuals that would attain response level Y = y and covariate level Z = z under the hypothetical situation in which treatment X = x 0 is administered uniformly to the population.

In general, we can formally define the post-intervention distribution by the equation:

In words: In the framework of model M , the post-intervention distribution of outcome Y is defined as the probability that model M x assigns to each outcome level Y = y .

From this distribution, one is able to assess treatment efficacy by comparing aspects of this distribution at different levels of x 0 . A common measure of treatment efficacy is the average difference

where x ′ 0 and x 0 are two levels (or types) of treatment selected for comparison. Another measure is the experimental Risk Ratio

The variance Var ( Y | do ( x 0 )), or any other distributional parameter, may also enter the comparison; all these measures can be obtained from the controlled distribution function P ( Y = y | do ( x )) = ∑ z P ( z , y | do ( x )) which was called “causal effect” in Pearl (2000a , 1995) (see footnote 2 ). The central question in the analysis of causal effects is the question of identification : Can the controlled (post-intervention) distribution, P ( Y = y | do ( x )), be estimated from data governed by the pre-intervention distribution, P ( z , x , y )?

The problem of identification has received considerable attention in econometrics ( Hurwicz, 1950 , Marschak, 1950 , Koopmans, 1953 ) and social science ( Duncan, 1975 , Bollen, 1989 ), usually in linear parametric settings, where it reduces to asking whether some model parameter, β , has a unique solution in terms of the parameters of P (the distribution of the observed variables). In the nonparametric formulation, identification is more involved, since the notion of “has a unique solution” does not directly apply to causal quantities such as Q ( M ) = P ( y | do ( x )) which have no distinct parametric signature, and are defined procedurally by simulating an intervention in a causal model M (as in (6)). The following definition overcomes these difficulties:

Definition 2 (Identifiability ( Pearl, 2000a , p. 77)) A quantity Q(M) is identifiable, given a set of assumptions A, if for any two models M 1 and M 2 that satisfy A, we have

In words, the details of M 1 and M 2 do not matter; what matters is that the assumptions in A (e.g., those encoded in the diagram) would constrain the variability of those details in such a way that equality of P ’s would entail equality of Q ’s. When this happens, Q depends on P only, and should therefore be expressible in terms of the parameters of P . The next subsections exemplify and operationalize this notion.

3.2.2. Estimating the effect of interventions

To understand how hypothetical quantities such as P ( y | do ( x )) or E ( Y | do ( x 0 )) can be estimated from actual data and a partially specified model let us begin with a simple demonstration on the model of Fig. 2(a) . We will see that, despite our ignorance of f X , f Y , f Z and P ( u ), E ( Y | do ( x 0 )) is nevertheless identifiable and is given by the conditional expectation E ( Y | X = x 0 ). We do this by deriving and comparing the expressions for these two quantities, as defined by (5) and (6), respectively. The mutilated model in Eq. (6) dictates:

whereas the pre-intervention model of Eq. (5) gives

which is identical to (11). Therefore,

Using a similar derivation, though somewhat more involved, we can show that P ( y | do ( x )) is identifiable and given by the conditional probability P ( y | x ).

We see that the derivation of (13) was enabled by two assumptions; first, Y is a function of X and U Y only, and, second, U Y is independent of { U Z , U X }, hence of X . The latter assumption parallels the celebrated “orthogonality” condition in linear models, Cov ( X , U Y ) = 0, which has been used routinely, often thoughtlessly, to justify the estimation of structural coefficients by regression techniques.

Naturally, if we were to apply this derivation to the linear models of Fig. 1(a) or 1(b) , we would get the expected dependence between Y and the intervention do ( x 0 ):

This equality endows β with its causal meaning as “effect coefficient.” It is extremely important to keep in mind that in structural (as opposed to regressional) models, β is not “interpreted” as an effect coefficient but is “proven” to be one by the derivation above. β will retain this causal interpretation regardless of how X is actually selected (through the function f X , Fig. 2(a) ) and regardless of whether U X and U Y are correlated (as in Fig. 1(b) ) or uncorrelated (as in Fig. 1(a) ). Correlations may only impede our ability to estimate β from nonexperimental data, but will not change its definition as given in (14). Accordingly, and contrary to endless confusions in the literature (see footnote 12 ) structural equations say absolutely nothing about the conditional expectation E ( Y | X = x ). Such connection may exist under special circumstances, e.g., if cov ( X , U Y ) = 0, as in Eq. (13) , but is otherwise irrelevant to the definition or interpretation of β as effect coefficient, or to the empirical claims of Eq. (1) .

The next subsection will circumvent these derivations altogether by reducing the identification problem to a graphical procedure. Indeed, since graphs encode all the information that non-parametric structural equations represent, they should permit us to solve the identification problem without resorting to algebraic analysis.

3.2.3. Causal effects from data and graphs

Causal analysis in graphical models begins with the realization that all causal effects are identifiable whenever the model is Markovian , that is, the graph is acyclic (i.e., containing no directed cycles) and all the error terms are jointly independent. Non-Markovian models, such as those involving correlated errors (resulting from unmeasured confounders), permit identification only under certain conditions, and these conditions too can be determined from the graph structure (Section 3.3). The key to these results rests with the following basic theorem.

Theorem 1 (The Causal Markov Condition) Any distribution generated by a Markovian model M can be factorized as:

where V 1 , V 2 , . . ., V n are the endogenous variables in M, and pa i are (values of) the endogenous “parents” of V i in the causal diagram associated with M.

For example, the distribution associated with the model in Fig. 2(a) can be factorized as

since X is the (endogenous) parent of Y , Z is the parent of X , and Z has no parents.

Corollary 1 (Truncated factorization) For any Markovian model, the distribution generated by an intervention do ( X = x 0 ) on a set X of endogenous variables is given by the truncated factorization

where P(v i | pa i ) are the pre-intervention conditional probabilities. 5

Corollary 1 instructs us to remove from the product of Eq. (15) those factors that quantify how the intervened variables (members of set X ) are influenced by their pre-intervention parents. This removal follows from the fact that the post-intervention model is Markovian as well, hence, following Theorem 1, it must generate a distribution that is factorized according to the modified graph, yielding the truncated product of Corollary 1. In our example of Fig. 2(b) , the distribution P ( z , y | do ( x 0 )) associated with the modified model is given by

where P ( z ) and P ( y | x 0 ) are identical to those associated with the pre-intervention distribution of Eq. (16) . As expected, the distribution of Z is not affected by the intervention, since

while that of Y is sensitive to x 0 , and is given by

This example demonstrates how the (causal) assumptions embedded in the model M permit us to predict the post-intervention distribution from the pre-intervention distribution, which further permits us to estimate the causal effect of X on Y from nonexperimental data, since P ( y | x 0 ) is estimable from such data. Note that we have made no assumption whatsoever on the form of the equations or the distribution of the error terms; it is the structure of the graph alone (specifically, the identity of X ’s parents) that permits the derivation to go through.

The truncated factorization formula enables us to derive causal quantities directly, without dealing with equations or equation modification as in Eqs. (11) – (13) . Consider, for example, the model shown in Fig. 3 , in which the error variables are kept implicit. Instead of writing down the corresponding five nonparametric equations, we can write the joint distribution directly as

where each marginal or conditional probability on the right hand side is directly estimable from the data. Now suppose we intervene and set variable X to x 0 . The post-intervention distribution can readily be written (using the truncated factorization formula (17) ) as

and the causal effect of X on Y can be obtained immediately by marginalizing over the Z variables, giving

Note that this formula corresponds precisely to what is commonly called “adjusting for Z 1 , Z 2 and Z 3 ” and, moreover, we can write down this formula by inspection, without thinking on whether Z 1 , Z 2 and Z 3 are confounders, whether they lie on the causal pathways, and so on. Though such questions can be answered explicitly from the topology of the graph, they are dealt with automatically when we write down the truncated factorization formula and marginalize.

Markovian model illustrating the derivation of the causal effect of X on Y , Eq. (20) . Error terms are not shown explicitly.

Note also that the truncated factorization formula is not restricted to interventions on a single variable; it is applicable to simultaneous or sequential interventions such as those invoked in the analysis of time varying treatment with time varying confounders ( Robins, 1986 , Arjas and Parner, 2004 ). For example, if X and Z 2 are both treatment variables, and Z 1 and Z 3 are measured covariates, then the post-intervention distribution would be

and the causal effect of the treatment sequence do ( X = x ), do ( Z 2 = z 2 ) 6 would be

This expression coincides with Robins’ (1987) G -computation formula, which was derived from a more complicated set of (counterfactual) assumptions. As noted by Robins, the formula dictates an adjustment for covariates (e.g., Z 3 ) that might be affected by previous treatments (e.g., Z 2 ).

3.3. Coping with unmeasured confounders

Things are more complicated when we face unmeasured confounders. For example, it is not immediately clear whether the formula in Eq. (20) can be estimated if any of Z 1 , Z 2 and Z 3 is not measured. A few but challenging algebraic steps would reveal that one can perform the summation over Z 2 to obtain

which means that we need only adjust for Z 1 and Z 3 without ever measuring Z 2 . In general, it can be shown ( Pearl, 2000a , p. 73) that, whenever the graph is Markovian the post-interventional distribution P ( Y = y | do ( X = x )) is given by the following expression:

where T is the set of direct causes of X (also called “parents”) in the graph. This allows us to write (23) directly from the graph, thus skipping the algebra that led to (23). It further implies that, no matter how complicated the model, the parents of X are the only variables that need to be measured to estimate the causal effects of X .

It is not immediately clear however whether other sets of variables beside X ’s parents suffice for estimating the effect of X , whether some algebraic manipulation can further reduce Eq. (23) , or that measurement of Z 3 (unlike Z 1 , or Z 2 ) is necessary in any estimation of P ( y | do ( x 0 )). Such considerations become transparent from a graphical criterion to be discussed next.

3.3.1. Covariate selection – the back-door criterion

Consider an observational study where we wish to find the effect of X on Y , for example, treatment on response, and assume that the factors deemed relevant to the problem are structured as in Fig. 4 ; some are affecting the response, some are affecting the treatment and some are affecting both treatment and response. Some of these factors may be unmeasurable, such as genetic trait or life style, others are measurable, such as gender, age, and salary level. Our problem is to select a subset of these factors for measurement and adjustment, namely, that if we compare treated vs. untreated subjects having the same values of the selected factors, we get the correct treatment effect in that subpopulation of subjects. Such a set of factors is called a “sufficient set” or “admissible set” for adjustment. The problem of defining an admissible set, let alone finding one, has baffled epidemiologists and social scientists for decades (see ( Greenland et al., 1999 , Pearl, 1998 ) for review).

Markovian model illustrating the back-door criterion. Error terms are not shown explicitly.

The following criterion, named “back-door” in ( Pearl, 1993a ), settles this problem by providing a graphical method of selecting admissible sets of factors for adjustment.

Definition 3 (Admissible sets – the back-door criterion) A set S is admissible (or “sufficient”) for adjustment if two conditions hold:

• No element of S is a descendant of X
• The elements of S “block” all “back-door” paths from X to Y, namely all paths that end with an arrow pointing to X.

In this criterion, “blocking” is interpreted as in Definition 1. For example, the set S = { Z 3 } blocks the path X ← W 1 ← Z 1 → Z 3 → Y , because the arrow-emitting node Z 3 is in S . However, the set S = { Z 3 } does not block the path X ← W 1 ← Z 1 → Z 3 ← Z 2 → W 2 → Y , because none of the arrow-emitting nodes, Z 1 and Z 2 , is in S , and the collision node Z 3 is not outside S .

Based on this criterion we see, for example, that the sets { Z 1 , Z 2 , Z 3 }, { Z 1 , Z 3 }, { W 1 , Z 3 }, and { W 2 , Z 3 }, each is sufficient for adjustment, because each blocks all back-door paths between X and Y . The set { Z 3 }, however, is not sufficient for adjustment because, as explained above, it does not block the path X ← W 1 ← Z 1 → Z 3 ← Z 2 → W 2 → Y .

The intuition behind the back-door criterion is as follows. The back-door paths in the diagram carry spurious associations from X to Y , while the paths directed along the arrows from X to Y carry causative associations. Blocking the former paths (by conditioning on S ) ensures that the measured association between X and Y is purely causative, namely, it correctly represents the target quantity: the causal effect of X on Y . The reason for excluding descendants of X (e.g., W 3 or any of its descendants) is given in ( Pearl, 2009b , pp. 338–41).

Formally, the implication of finding an admissible set S is that, stratifying on S is guaranteed to remove all confounding bias relative the causal effect of X on Y . In other words, the risk difference in each stratum of S gives the correct causal effect in that stratum. In the binary case, for example, the risk difference in stratum s of S is given by

while the causal effect (of X on Y ) at that stratum is given by

These two expressions are guaranteed to be equal whenever S is a sufficient set, such as { Z 1 , Z 3 } or { Z 2 , Z 3 } in Fig. 4 . Likewise, the average stratified risk difference, taken over all strata,

gives the correct causal effect of X on Y in the entire population

In general, for multi-valued variables X and Y , finding a sufficient set S permits us to write

Since all factors on the right hand side of the equation are estimable (e.g., by regression) from the pre-interventional data, the causal effect can likewise be estimated from such data without bias.

An equivalent expression for the causal effect (25) can be obtained by multiplying and dividing by the conditional probability P ( X = x | S = s ), giving

from which the name “Inverse Probability Weighting” has evolved ( Pearl, 2000a , pp. 73, 95).

Interestingly, it can be shown that any irreducible sufficient set, S , taken as a unit, satisfies the associational criterion that epidemiologists have been using to define “confounders”. In other words, S must be associated with X and, simultaneously, associated with Y , given X . This need not hold for any specific members of S . For example, the variable Z 3 in Fig. 4 , though it is a member of every sufficient set and hence a confounder, can be unassociated with both Y and X ( Pearl, 2000a , p. 195). Conversely, a pre-treatment variable Z that is associated with both Y and X may need to be excluded from entering a sufficient set.

The back-door criterion allows us to write Eq. (25) directly, by selecting a sufficient set S directly from the diagram, without manipulating the truncated factorization formula. The selection criterion can be applied systematically to diagrams of any size and shape, thus freeing analysts from judging whether “ X is conditionally ignorable given S ,” a formidable mental task required in the potential-response framework ( Rosenbaum and Rubin, 1983 ). The criterion also enables the analyst to search for an optimal set of covariate—namely, a set S that minimizes measurement cost or sampling variability ( Tian, Paz, and Pearl, 1998 ).

All in all, one can safely state that, armed with the back-door criterion, causality has removed “confounding” from its store of enigmatic and controversial concepts.

3.3.2. Confounding equivalence – a graphical test

Another problem that has been given graphical solution recently is that of determining whether adjustment for two sets of covariates would result in the same confounding bias ( Pearl and Paz, 2009 ). The reasons for posing this question are several. First, an investigator may wish to assess, prior to taking any measurement, whether two candidate sets of covariates, differing substantially in dimensionality, measurement error, cost, or sample variability are equally valuable in their bias-reduction potential. Second, assuming that the structure of the underlying DAG is only partially known, one may wish to test, using adjustment, which of two hypothesized structures is compatible with the data. Structures that predict equal response to adjustment for two sets of variables must be rejected if, after adjustment, such equality is not found in the data.

Definition 4 (( c -equivalence)) Define two sets, T and Z of covariates as c-equivalent, (c connotes “confounding”) if the following equality holds:

Definition 5 ((Markov boundary)) For any set of variables S in a DAG G, the Markov boundary S m of S is the minimal subset of S that d-separates X from all other members of S.

In Fig. 4 , for example, the Markov boundary of S = { W 1 , Z 1 , Z 2 , Z 3 } is S m = { W 1 , Z 3 }.

Theorem 2 ( Pearl and Paz, 2009 )

Let Z and T be two sets of variables in G, containing no descendant of X. A necessary and sufficient conditions for Z and T to be c-equivalent is that at least one of the following conditions holds:

• Z m = T m , (i.e., the Markov boundary of Z coincides with that of T)
• Z and T are admissible (i.e., satisfy the back-door condition)

For example, the sets T = { W 1 , Z 3 } and Z = { Z 3 , W 2 } in Fig. 4 are c -equivalent, because each blocks all back-door paths from X to Y . Similarly, the non-admissible sets T = { Z 2 } and Z = { W 2 , Z 2 } are c -equivalent, since their Markov boundaries are the same ( T m = Z m = { Z 2 }). In contrast, the sets { W 1 } and { Z 1 }, although they block the same set of paths in the graph, are not c -equivalent; they fail both conditions of Theorem 2.

Tests for c -equivalence (27) are fairly easy to perform, and they can also be assisted by propensity scores methods. The information that such tests provide can be as powerful as conditional independence tests. The statistical ramification of such tests are explicated in ( Pearl and Paz, 2009 ).

3.3.3. General control of confounding

Adjusting for covariates is only one of many methods that permits us to estimate causal effects in nonexperimental studies. Pearl (1995) has presented examples in which there exists no set of variables that is sufficient for adjustment and where the causal effect can nevertheless be estimated consistently. The estimation, in such cases, employs multi-stage adjustments. For example, if W 3 is the only observed covariate in the model of Fig. 4 , then there exists no sufficient set for adjustment (because no set of observed covariates can block the paths from X to Y through Z 3 ), yet P ( y | do ( x )) can be estimated in two steps; first we estimate P ( w 3 | do ( x )) = P ( w 3 | x ) (by virtue of the fact that there exists no unblocked back-door path from X to W 3 ), second we estimate P ( y | do ( w 3 )) (since X constitutes a sufficient set for the effect of W 3 on Y ) and, finally, we combine the two effects together and obtain

In this example, the variable W 3 acts as a “mediating instrumental variable” ( Pearl, 1993b , Chalak and White, 2006 ).

The analysis used in the derivation and validation of such results invokes mathematical rules of transforming causal quantities, represented by expressions such as P ( Y = y | do ( x )), into do -free expressions derivable from P ( z , x , y ), since only do -free expressions are estimable from non-experimental data. When such a transformation is feasible, we are ensured that the causal quantity is identifiable.

Applications of this calculus to problems involving multiple interventions (e.g., time varying treatments), conditional policies, and surrogate experiments were developed in Pearl and Robins (1995) , Kuroki and Miyakawa (1999) , and Pearl (2000a , Chapters 3–4).

A more recent analysis ( Tian and Pearl, 2002 ) shows that the key to identifiability lies not in blocking paths between X and Y but, rather, in blocking paths between X and its immediate successors on the pathways to Y . All existing criteria for identification are special cases of the one defined in the following theorem:

Theorem 3 ( Tian and Pearl, 2002 ) A sufficient condition for identifying the causal effect P ( y|do ( x )) is that every path between X and any of its children traces at least one arrow emanating from a measured variable. 7

For example, if W 3 is the only observed covariate in the model of Fig. 4 , P ( y | do ( x )) can be estimated since every path from X to W 3 (the only child of X ) traces either the arrow X → W 3 , or the arrow W 3 → Y , both emanating from a measured variable ( W 3 ).

Shpitser and Pearl (2006) have further extended this theorem by (1) presenting a necessary and sufficient condition for identification, and (2) extending the condition from causal effects to any counterfactual expression. The corresponding unbiased estimands for these causal quantities are readable directly from the diagram.

Graph-based methods for effect identification under measurement errors are discussed in ( Pearl, 2009f , Hernán and Cole, 2009 , Cai and Kuroki, 2008 ).

3.3.4. From identification to estimation

The mathematical derivation of causal effect estimands, like Eqs. (25) and (28) is merely a first step toward computing quantitative estimates of those effects from finite samples, using the rich traditions of statistical estimation and machine learning Bayesian as well as non-Bayesian. Although the estimands derived in (25) and (28) are non-parametric, this does not mean that one should refrain from using parametric forms in the estimation phase of the study. Parameterization is in fact necessary when the dimensionality of a problem is high. For example, if the assumptions of Gaussian, zero-mean disturbances and additive interactions are deemed reasonable, then the estimand given in (28) can be converted to the product E ( Y | do ( x )) = r W 3 X r YW 3·X x , where r YZ·X is the (standardized) coefficient of Z in the regression of Y on Z and X . More sophisticated estimation techniques are the “marginal structural models” of ( Robins, 1999 ), and the “propensity score” method of ( Rosenbaum and Rubin, 1983 ) which were found to be particularly useful when dimensionality is high and data are sparse (see Pearl (2009b , pp. 348–52)).

It should be emphasized, however, that contrary to conventional wisdom (e.g., ( Rubin, 2007 , 2009 )), propensity score methods are merely efficient estimators of the right hand side of (25); they entail the same asymptotic bias, and cannot be expected to reduce bias in case the set S does not satisfy the back-door criterion ( Pearl, 2000a , 2009c , d ). Consequently, the prevailing practice of conditioning on as many pre-treatment measurements as possible should be approached with great caution; some covariates (e.g., Z 3 in Fig. 3 ) may actually increase bias if included in the analysis (see footnote 16 ). Using simulation and parametric analysis, Heckman and Navarro-Lozano (2004) and Wooldridge (2009) indeed confirmed the bias-raising potential of certain covariates in propensity-score methods. The graphical tools presented in this section unveil the character of these covariates and show precisely what covariates should, and should not be included in the conditioning set for propensity-score matching (see also ( Pearl and Paz, 2009 , Pearl, 2009e )).

3.4. Counterfactual analysis in structural models

Not all questions of causal character can be encoded in P ( y | do ( x )) type expressions, thus implying that not all causal questions can be answered from experimental studies. For example, questions of attribution (e.g., what fraction of death cases are due to specific exposure?) or of susceptibility (what fraction of the healthy unexposed population would have gotten the disease had they been exposed?) cannot be answered from experimental studies, and naturally, this kind of questions cannot be expressed in P ( y | do ( x )) notation. 8 To answer such questions, a probabilistic analysis of counterfactuals is required, one dedicated to the relation “ Y would be y had X been x in situation U = u ,” denoted Y x ( u ) = y . Remarkably, unknown to most economists and philosophers, structural equation models provide the formal interpretation and symbolic machinery for analyzing such counterfactual relationships. 9

The key idea is to interpret the phrase “had X been x ” as an instruction to make a minimal modification in the current model, which may have assigned X a different value, say X = x ′ , so as to ensure the specified condition X = x . Such a minimal modification amounts to replacing the equation for X by a constant x , as we have done in Eq. (6) . This replacement permits the constant x to differ from the actual value of X (namely f X ( z , u X )) without rendering the system of equations inconsistent, thus yielding a formal interpretation of counterfactuals in multi-stage models, where the dependent variable in one equation may be an independent variable in another.

Definition 6 (Unit-level Counterfactuals – “surgical” definition, Pearl (2000a , p. 98)) Let M be a structural model and M x a modified version of M, with the equation(s) of X replaced by X = x. Denote the solution for Y in the equations of M x by the symbol Y M x (u). The counterfactual Y x (u) (Read: “The value of Y in unit u, had X been x”) is given by:

In words: The counterfactual Y x ( u ) in model M is defined as the solution for Y in the “surgically modified” submodel M x .

We see that the unit-level counterfactual Y x ( u ), which in the Neyman-Rubin approach is treated as a primitive, undefined quantity, is actually a derived quantity in the structural framework. The fact that we equate the experimental unit u with a vector of background conditions, U = u , in M , reflects the understanding that the name of a unit or its identity do not matter; it is only the vector U = u of attributes characterizing a unit which determines its behavior or response. As we go from one unit to another, the laws of nature, as they are reflected in the functions f X , f Y , etc. remain invariant; only the attributes U = u vary from individual to individual. 10

To illustrate, consider the solution of Y in the modified model M x 0 of Eq. (6) , which Definition 6 endows with the symbol Y x 0 ( u X , u Y , u Z ). This entity has a clear counterfactual interpretation, for it stands for the way an individual with characteristics ( u X , u Y , u Z ) would respond, had the treatment been x 0 , rather than the treatment x = f X ( z , u X ) actually received by that individual. In our example, since Y does not depend on u X and u Z , we can write:

In a similar fashion, we can derive

and so on. These examples reveal the counterfactual reading of each individual structural equation in the model of Eq. (5) . The equation x = f X ( z , u X ), for example, advertises the empirical claim that, regardless of the values taken by other variables in the system, had Z been z 0 , X would take on no other value but x = f X ( z 0 , u X ).

Clearly, the distribution P ( u Y , u X , u Z ) induces a well defined probability on the counterfactual event Y x 0 = y , as well as on joint counterfactual events, such as ‘ Y x 0 = y AND Y x 1 = y ′ ,’ which are, in principle, unobservable if x 0 ≠ x 1 . Thus, to answer attributional questions, such as whether Y would be y 1 if X were x 1 , given that in fact Y is y 0 and X is x 0 , we need to compute the conditional probability P ( Y x 1 = y 1 | Y = y 0 , X = x 0 ) which is well defined once we know the forms of the structural equations and the distribution of the exogenous variables in the model. For example, assuming linear equations (as in Fig. 1 ),

the conditioning events Y = y 0 and X = x 0 yield U X = x 0 and U Y = y 0 − βx 0 , and we can conclude that, with probability one, Y x 1 must take on the value: Y x 1 = βx 1 + U Y = β ( x 1 − x 0 ) + y 0 . In other words, if X were x 1 instead of x 0 , Y would increase by β times the difference ( x 1 − x 0 ). In nonlinear systems, the result would also depend on the distribution of { U X , U Y } and, for that reason, attributional queries are generally not identifiable in nonparametric models (see Section 6.3 and 2000a, Chapter 9).

In general, if x and x ′ are incompatible then Y x and Y x ′ cannot be measured simultaneously, and it may seem meaningless to attribute probability to the joint statement “ Y would be y if X = x and Y would be y ′ if X = x ′ .” 11 Such concerns have been a source of objections to treating counterfactuals as jointly distributed random variables ( Dawid, 2000 ). The definition of Y x and Y x ′ in terms of two distinct submodels neutralizes these objections ( Pearl, 2000b ), since the contradictory joint statement is mapped into an ordinary event, one where the background variables satisfy both statements simultaneously, each in its own distinct submodel; such events have well defined probabilities.

The surgical definition of counterfactuals given by (29), provides the conceptual and formal basis for the Neyman-Rubin potential-outcome framework, an approach to causation that takes a controlled randomized trial (CRT) as its ruling paradigm, assuming that nothing is known to the experimenter about the science behind the data. This “black-box” approach, which has thus far been denied the benefits of graphical or structural analyses, was developed by statisticians who found it difficult to cross the two mental barriers discussed in Section 2.2. Section 5 establishes the precise relationship between the structural and potential-outcome paradigms, and outlines how the latter can benefit from the richer representational power of the former.

4. Methodological Principles of Causal Inference

The structural theory described in the previous sections dictates a principled methodology that eliminates much of the confusion concerning the interpretations of study results as well as the ethical dilemmas that this confusion tends to spawn. The methodology dictates that every investigation involving causal relationships (and this entails the vast majority of empirical studies in the health, social, and behavioral sciences) should be structured along the following four-step process:

• Define: Express the target quantity Q as a function Q ( M ) that can be computed from any model M .
• Assume: Formulate causal assumptions using ordinary scientific language and represent their structural part in graphical form.
• Identify: Determine if the target quantity is identifiable (i.e., expressible in terms of estimable parameters).
• Estimate: Estimate the target quantity if it is identifiable, or approximate it, if it is not. Test the statistical implications of the model, if any, and modify the model when failure occurs.

4.1. Defining the target quantity

The definitional phase is the most neglected step in current practice of quantitative analysis. The structural modeling approach insists on defining the target quantity, be it “causal effect,” “mediated effect,” “effect on the treated,” or “probability of causation” before specifying any aspect of the model, without making functional or distributional assumptions and prior to choosing a method of estimation.

The investigator should view this definition as an algorithm that receives a model M as an input and delivers the desired quantity Q ( M ) as the output. Surely, such algorithm should not be tailored to any aspect of the input M ; it should be general, and ready to accommodate any conceivable model M whatsoever. Moreover, the investigator should imagine that the input M is a completely specified model, with all the functions f X , f Y , . . . and all the U variables (or their associated probabilities) given precisely. This is the hardest step for statistically trained investigators to make; knowing in advance that such model details will never be estimable from the data, the definition of Q ( M ) appears like a futile exercise in fantasy land – it is not.

For example, the formal definition of the causal effect P ( y | do ( x )), as given in Eq. (7) , is universally applicable to all models, parametric as well as nonparametric, through the formation of a submodel M x . By defining causal effect procedurally, thus divorcing it from its traditional parametric representation, the structural theory avoids the many pitfalls and confusions that have plagued the interpretation of structural and regressional parameters for the past half century. 12

4.2. Explicating causal assumptions

This is the second most neglected step in causal analysis. In the past, the difficulty has been the lack of a language suitable for articulating causal assumptions which, aside from impeding investigators from explicating assumptions, also inhibited them from giving causal interpretations to their findings.

Structural equation models, in their counterfactual reading, have removed this lingering difficulty by providing the needed language for causal analysis. Figures 3 and ​ and4 4 illustrate the graphical component of this language, where assumptions are conveyed through the missing arrows in the diagram. If numerical or functional knowledge is available, for example, linearity or monotonicity of the functions f X , f Y , . . ., those are stated separately, and applied in the identification and estimation phases of the study. Today we understand that the longevity and natural appeal of structural equations stem from the fact that they permit investigators to communicate causal assumptions formally and in the very same vocabulary in which scientific knowledge is stored.

Unfortunately, however, this understanding is not shared by all causal analysts; some analysts vehemently oppose the re-emergence of structure-based causation and insist, instead, on articulating causal assumptions exclusively in the unnatural (though formally equivalent) language of “potential outcomes,” “ignorability,” “missing data,” “treatment assignment,” and other metaphors borrowed from clinical trials. This modern assault on structural models is perhaps more dangerous than the regressional invasion that distorted the causal readings of these models in the late 1970s ( Richard, 1980 ). While sanctioning causal inference in one idiosyncratic style of analysis, the modern assault denies validity to any other style, including structural equations, thus discouraging investigators from subjecting models to the scrutiny of scientific knowledge.

This exclusivist attitude is manifested in passages such as: “The crucial idea is to set up the causal inference problem as one of missing data” or “If a problem of causal inference cannot be formulated in this manner (as the comparison of potential outcomes under different treatment assignments), it is not a problem of inference for causal effects, and the use of “causal” should be avoided,” or, even more bluntly, “the underlying assumptions needed to justify any causal conclusions should be carefully and explicitly argued, not in terms of technical properties like “uncorrelated error terms,” but in terms of real world properties, such as how the units received the different treatments” ( Wilkinson, the Task Force on Statistical Inference, and APA Board of Scientific Affairs , 1999 ).

The methodology expounded in this paper testifies against such restrictions. It demonstrates the viability and scientific soundness of the traditional structural equations paradigm, which stands diametrically opposed to the “missing data” paradigm. It renders the vocabulary of “treatment assignment” stifling and irrelevant (e.g., there is no “treatment assignment” in sex discrimination cases). Most importantly, it strongly prefers the use of “uncorrelated error terms,” (or “omitted factors”) over its “strong ignorability” alternative, as the proper way of articulating causal assumptions. Even the most devout advocates of the “strong ignorability” language use “omitted factors” when the need arises to defend assumptions (e.g., ( Sobel, 2008 ))

4.3. Identification, estimation, and approximation

Having unburden itself from parametric representations, the identification process in the structural framework proceeds either in the space of assumptions (i.e., the diagram) or in the space of mathematical expressions, after translating the graphical assumptions into a counterfactual language, as demonstrated in Section 5.3. Graphical criteria such as those of Definition 3 and Theorem 3 permit the identification of causal effects to be decided entirely within the graphical domain, where it can benefit from the guidance of scientific understanding. Identification of counterfactual queries, on the other hand, often require a symbiosis of both algebraic and graphical techniques. The nonparametric nature of the identification task (Definition 1) makes it clear that contrary to traditional folklore in linear analysis, it is not the model that need be identified but the query Q – the target of investigation. It also provides a simple way of proving non-identifiability: the construction of two parameterization of M , agreeing in P and disagreeing in Q , is sufficient to rule out identifiability.

When Q is identifiable, the structural framework also delivers an algebraic expression for the estimand EST ( Q ) of the target quantity Q , examples of which are given in Eqs. (24) and (25) , and estimation techniques are then unleashed as discussed in Section 3.3.4. An integral part of this estimation phase is a test for the testable implications, if any, of those assumptions in M that render Q identifiable – there is no point in estimating EST ( Q ) if the data proves those assumptions false and EST ( Q ) turns out to be a misrepresentation of Q . Investigators should be reminded, however, that only a fraction, called “kernel,” of the assumptions embodied in M are needed for identifying Q ( Pearl, 2004 ), the rest may be violated in the data with no effect on Q . In Fig. 2 , for example, the assumption { U Z ⊥⊥ U X } is not necessary for identifying Q = P ( y | do ( x )); the kernel { U Y ⊥⊥ U Z , U Y ⊥⊥ U X } (together with the missing arrows) is sufficient. Therefore, the testable implication of this kernel, Z ⊥⊥ Y | X , is all we need to test when our target quantity is Q ; the assumption { U Z ⊥⊥ U X } need not concern us.

More importantly, investigators must keep in mind that only a tiny fraction of any kernel lends itself to statistical tests, the bulk of it must remain untestable, at the mercy of scientific judgment. In Fig. 2 , for example, the assumption set { U X ⊥⊥ U Z , U Y ⊥⊥ U X } constitutes a sufficient kernel for Q = P ( y | do ( x )) (see Eq. (28) ) yet it has no testable implications whatsoever. The prevailing practice of submitting an entire structural equation model to a “goodness of fit” test ( Bollen, 1989 ) in support of causal claims is at odd with the logic of SCM (see ( Pearl, 2000a , pp. 144–5)). Alternative causal models usually exist that make contradictory claims and, yet, possess identical statistical implications. Statistical test can be used for rejecting certain kernels, in the rare cases where such kernels have testable implications, but the lion’s share of supporting causal claims falls on the shoulders of untested causal assumptions.

When conditions for identification are not met, the best one can do is derive bounds for the quantities of interest—namely, a range of possible values of Q that represents our ignorance about the details of the data-generating process M and that cannot be improved with increasing sample size. A classical example of non identifiable model that has been approximated by bounds, is the problem of estimating causal effect in experimental studies marred by non compliance, the structure of which is given in Fig. 5 .

Our task in this example is to find the highest and lowest values of Q

subject to the equality constraints imposed by the observed probabilities P ( x , y , | z ), where the maximization ranges over all possible functions P ( u Y , u X ), P ( y | x , u X ) and P ( x | z , u Y ) that satisfy those constraints.

Realizing that units in this example fall into 16 equivalence classes, each representing a binary function X = f ( z ) paired with a binary function y = g ( x ), Balke and Pearl (1997) were able to derive closed-form solutions for these bounds. 13 They showed that, in certain cases, the derived bounds can yield significant information on the treatment efficacy. Chickering and Pearl (1997) further used Bayesian techniques (with Gibbs sampling) to investigate the sharpness of these bounds as a function of sample size. Kaufman, Kaufman, and MacLenose (2009) used this technique to bound direct and indirect effects (see Section 6.1).

5. The Potential Outcome Framework

This section compares the structural theory presented in Sections 1–3 to the potential-outcome framework, usually associated with the names of Neyman (1923) and Rubin (1974) , which takes the randomized experiment as its ruling paradigm and has appealed therefore to researchers who do not find that paradigm overly constraining. This framework is not a contender for a comprehensive theory of causation for it is subsumed by the structural theory and excludes ordinary cause-effect relationships from its assumption vocabulary. We here explicate the logical foundation of the Neyman-Rubin framework, its formal subsumption by the structural causal model, and how it can benefit from the insights provided by the broader perspective of the structural theory.

The primitive object of analysis in the potential-outcome framework is the unit-based response variable, denoted Y x ( u ), read: “the value that outcome Y would obtain in experimental unit u , had treatment X been x .” Here, unit may stand for an individual patient, an experimental subject, or an agricultural plot. In Section 3.4 ( Eq. (29) we saw that this counterfactual entity has a natural interpretation in the SCM; it is the solution for Y in a modified system of equations, where unit is interpreted a vector u of background factors that characterize an experimental unit. Each structural equation model thus carries a collection of assumptions about the behavior of hypothetical units, and these assumptions permit us to derive the counterfactual quantities of interest. In the potential-outcome framework, however, no equations are available for guidance and Y x ( u ) is taken as primitive, that is, an undefined quantity in terms of which other quantities are defined; not a quantity that can be derived from the model. In this sense the structural interpretation of Y x ( u ) given in (29) provides the formal basis for the potential-outcome approach; the formation of the submodel M x explicates mathematically how the hypothetical condition “had X been x ” is realized, and what the logical consequences are of such a condition.

5.1. The “black-box” missing-data paradigm

The distinct characteristic of the potential-outcome approach is that, although investigators must think and communicate in terms of undefined, hypothetical quantities such as Y x ( u ), the analysis itself is conducted almost entirely within the axiomatic framework of probability theory. This is accomplished, by postulating a “super” probability function on both hypothetical and real events. If U is treated as a random variable then the value of the counterfactual Y x ( u ) becomes a random variable as well, denoted as Y x . The potential-outcome analysis proceeds by treating the observed distribution P ( x 1 , . . ., x n ) as the marginal distribution of an augmented probability function P* defined over both observed and counterfactual variables. Queries about causal effects (written P ( y | do ( x )) in the structural analysis) are phrased as queries about the marginal distribution of the counterfactual variable of interest, written P *( Y x = y ). The new hypothetical entities Y x are treated as ordinary random variables; for example, they are assumed to obey the axioms of probability calculus, the laws of conditioning, and the axioms of conditional independence.

Naturally, these hypothetical entities are not entirely whimsy. They are assumed to be connected to observed variables via consistency constraints ( Robins, 1986 ) such as

which states that, for every u , if the actual value of X turns out to be x , then the value that Y would take on if ‘ X were x ’ is equal to the actual value of Y . For example, a person who chose treatment x and recovered, would also have recovered if given treatment x by design. When X is binary, it is sometimes more convenient to write (32) as:

Whether additional constraints should tie the observables to the unobservables is not a question that can be answered in the potential-outcome framework; for it lacks an underlying model to define its axioms.

The main conceptual difference between the two approaches is that, whereas the structural approach views the intervention do ( x ) as an operation that changes a distribution but keeps the variables the same, the potential-outcome approach views the variable Y under do ( x ) to be a different variable, Y x , loosely connected to Y through relations such as (32), but remaining unobserved whenever X ≠ x . The problem of inferring probabilistic properties of Y x , then becomes one of “missing-data” for which estimation techniques have been developed in the statistical literature.

Pearl (2000a , Chapter 7) shows, using the structural interpretation of Y x ( u ), that it is indeed legitimate to treat counterfactuals as jointly distributed random variables in all respects, that consistency constraints like (32) are automatically satisfied in the structural interpretation and, moreover, that investigators need not be concerned about any additional constraints except the following two

Equation (33) ensures that the interventions do ( Y = y ) results in the condition Y = y , regardless of concurrent interventions, say do ( Z = z ), that may be applied to variables other than Y . Equation (34) generalizes (32) to cases where Z is held fixed, at z . (See ( Halpern, 1998 ) for proof of completeness.)

5.2. Problem formulation and the demystification of “ignorability”

The main drawback of this black-box approach surfaces in problem formulation, namely, the phase where a researcher begins to articulate the “science” or “causal assumptions” behind the problem of interest. Such knowledge, as we have seen in Section 1, must be articulated at the onset of every problem in causal analysis – causal conclusions are only as valid as the causal assumptions upon which they rest.

To communicate scientific knowledge, the potential-outcome analyst must express assumptions as constraints on P* , usually in the form of conditional independence assertions involving counterfactual variables. For instance, in our example of Fig. 5 , to communicate the understanding that Z is randomized (hence independent of U X and U Y ), the potential-outcome analyst would use the independence constraint Z ⊥⊥{ Y z 1 , Y z 2 , . . ., Y z k }. 14 To further formulate the understanding that Z does not affect Y directly, except through X , the analyst would write a, so called, “exclusion restriction”: Y xz = Y x .

A collection of constraints of this type might sometimes be sufficient to permit a unique solution to the query of interest. For example, if one can plausibly assume that, in Fig. 4 , a set Z of covariates satisfies the conditional independence

(an assumption termed “conditional ignorability” by Rosenbaum and Rubin (1983) ,) then the causal effect P ( y | do ( x )) = P* ( Y x = y ) can readily be evaluated to yield

The last expression contains no counterfactual quantities (thus permitting us to drop the asterisk from P* ) and coincides precisely with the standard covariate-adjustment formula of Eq. (25) .

We see that the assumption of conditional ignorability (35) qualifies Z as an admissible covariate for adjustment; it mirrors therefore the “back-door” criterion of Definition 3, which bases the admissibility of Z on an explicit causal structure encoded in the diagram.

The derivation above may explain why the potential-outcome approach appeals to mathematical statisticians; instead of constructing new vocabulary (e.g., arrows), new operators ( do ( x )) and new logic for causal analysis, almost all mathematical operations in this framework are conducted within the safe confines of probability calculus. Save for an occasional application of rule (34) or (32)), the analyst may forget that Y x stands for a counterfactual quantity—it is treated as any other random variable, and the entire derivation follows the course of routine probability exercises.

This orthodoxy exacts a high cost: Instead of bringing the theory to the problem, the problem must be reformulated to fit the theory; all background knowledge pertaining to a given problem must first be translated into the language of counterfactuals (e.g., ignorability conditions) before analysis can commence. This translation may in fact be the hardest part of the problem. The reader may appreciate this aspect by attempting to judge whether the assumption of conditional ignorability (35), the key to the derivation of (36), holds in any familiar situation, say in the experimental setup of Fig. 2(a) . This assumption reads: “the value that Y would obtain had X been x , is independent of X , given Z ”. Even the most experienced potential-outcome expert would be unable to discern whether any subset Z of covariates in Fig. 4 would satisfy this conditional independence condition. 15 Likewise, to derive Eq. (35) in the language of potential-outcome (see ( Pearl, 2000a , p. 223)), one would need to convey the structure of the chain X → W 3 → Y using the cryptic expression: W 3 x ⊥⊥{ Y w 3 , X }, read: “the value that W 3 would obtain had X been x is independent of the value that Y would obtain had W 3 been w 3 jointly with the value of X .” Such assumptions are cast in a language so far removed from ordinary understanding of scientific theories that, for all practical purposes, they cannot be comprehended or ascertained by ordinary mortals. As a result, researchers in the graph-less potential-outcome camp rarely use “conditional ignorability” (35) to guide the choice of covariates; they view this condition as a hoped-for miracle of nature rather than a target to be achieved by reasoned design. 16

Replacing “ignorability” with a conceptually meaningful condition (i.e., back-door) in a graphical model permits researchers to understand what conditions covariates must fulfill before they eliminate bias, what to watch for and what to think about when covariates are selected, and what experiments we can do to test, at least partially, if we have the knowledge needed for covariate selection.

Aside from offering no guidance in covariate selection, formulating a problem in the potential-outcome language encounters three additional hurdles. When counterfactual variables are not viewed as byproducts of a deeper, process-based model, it is hard to ascertain whether all relevant judgments have been articulated, whether the judgments articulated are redundant , or whether those judgments are self-consistent. The need to express, defend, and manage formidable counterfactual relationships of this type explain the slow acceptance of causal analysis among health scientists and statisticians, and why most economists and social scientists continue to use structural equation models ( Wooldridge, 2002 , Stock and Watson, 2003 , Heckman, 2008 ) instead of the potential-outcome alternatives advocated in Angrist, Imbens, and Rubin (1996) , Holland (1988) , Sobel (1998 , 2008) .

On the other hand, the algebraic machinery offered by the counterfactual notation, Y x ( u ), once a problem is properly formalized, can be extremely powerful in refining assumptions ( Angrist et al., 1996 , Heckman and Vytlacil, 2005 ), deriving consistent estimands ( Robins, 1986 ), bounding probabilities of necessary and sufficient causation ( Tian and Pearl, 2000 ), and combining data from experimental and nonexperimental studies ( Pearl, 2000a ). The next subsection (5.3) presents a way of combining the best features of the two approaches. It is based on encoding causal assumptions in the language of diagrams, translating these assumptions into counterfactual notation, performing the mathematics in the algebraic language of counterfactuals (using (32), (33), and (34)) and, finally, interpreting the result in graphical terms or plain causal language. The mediation problem of Section 6.1 illustrates how such symbiosis clarifies the definition and identification of direct and indirect effects, 17 and how it overcomes difficulties that were deemed insurmountable in the exclusivist potential-outcome framework ( Rubin, 2004 , 2005 ).

5.3. Combining graphs and potential outcomes

The formulation of causal assumptions using graphs was discussed in Section 3. In this subsection we will systematize the translation of these assumptions from graphs to counterfactual notation.

Structural equation models embody causal information in both the equations and the probability function P ( u ) assigned to the exogenous variables; the former is encoded as missing arrows in the diagrams the latter as missing (double arrows) dashed arcs. Each parent-child family ( PA i , X i ) in a causal diagram G corresponds to an equation in the model M . Hence, missing arrows encode exclusion assumptions, that is, claims that manipulating variables that are excluded from an equation will not change the outcome of the hypothetical experiment described by that equation. Missing dashed arcs encode independencies among error terms in two or more equations. For example, the absence of dashed arcs between a node Y and a set of nodes { Z 1 , . . ., Z k } implies that the corresponding background variables, U Y and { U Z 1 , . . ., U Z k }, are independent in P ( u ).

These assumptions can be translated into the potential-outcome notation using two simple rules ( Pearl, 2000a , p. 232); the first interprets the missing arrows in the graph, the second, the missing dashed arcs.

• Exclusion restrictions: For every variable Y having parents PA Y and for every set of endogenous variables S disjoint of PA Y , we have Y p a Y =  Y p a Y , s . (37)
• Independence restrictions: If Z 1 , . . ., Z k is any set of nodes not connected to Y via dashed arcs, and PA 1 , . . ., PA k their respective sets of parents, we have Y p a Y  ⊥  ⊥ { Z 1   p a 1 , …,  Z k    p a k }. (38)

The exclusion restrictions expresses the fact that each parent set includes all direct causes of the child variable, hence, fixing the parents of Y , determines the value of Y uniquely, and intervention on any other set S of (endogenous) variables can no longer affect Y . The independence restriction translates the independence between U Y and { U Z 1 , . . ., U Z k } into independence between the corresponding potential-outcome variables. This follows from the observation that, once we set their parents, the variables in { Y , Z 1 , . . ., Z k } stand in functional relationships to the U terms in their corresponding equations.

As an example, consider the model shown in Fig. 5 , which serves as the canonical representation for the analysis of instrumental variables ( Angrist et al., 1996 , Balke and Pearl, 1997 ). This model displays the following parent sets:

Consequently, the exclusion restrictions translate into:

the absence of any dashed arc between Z and { Y , X } translates into the independence restriction

This is precisely the condition of randomization; Z is independent of all its non-descendants, namely independent of U X and U Y which are the exogenous parents of Y and X , respectively. (Recall that the exogenous parents of any variable, say Y , may be replaced by the counterfactual variable Y pa Y , because holding PA Y constant renders Y a deterministic function of its exogenous parent U Y .)

The role of graphs is not ended with the formulation of causal assumptions. Throughout an algebraic derivation, like the one shown in Eq. (36) , the analyst may need to employ additional assumptions that are entailed by the original exclusion and independence assumptions, yet are not shown explicitly in their respective algebraic expressions. For example, it is hardly straightforward to show that the assumptions of Eqs. (40) – (41) imply the conditional independence ( Y x ⊥⊥ Z |{ X z , X }) but do not imply the conditional independence ( Y x ⊥⊥ Z | X ). These are not easily derived by algebraic means alone. Such implications can, however, easily be tested in the graph of Fig. 5 using the graphical reading for conditional independence (Definition 1). (See ( Pearl, 2000a , pp. 16–17, 213–215).) Thus, when the need arises to employ independencies in the course of a derivation, the graph may assist the procedure by vividly displaying the independencies that logically follow from our assumptions.

6. Counterfactuals at Work

6.1. mediation: direct and indirect effects, 6.1.1. direct versus total effects.

The causal effect we have analyzed so far, P ( y | do ( x )), measures the total effect of a variable (or a set of variables) X on a response variable Y . In many cases, this quantity does not adequately represent the target of investigation and attention is focused instead on the direct effect of X on Y . The term “direct effect” is meant to quantify an effect that is not mediated by other variables in the model or, more accurately, the sensitivity of Y to changes in X while all other factors in the analysis are held fixed. Naturally, holding those factors fixed would sever all causal paths from X to Y with the exception of the direct link X → Y , which is not intercepted by any intermediaries.

A classical example of the ubiquity of direct effects involves legal disputes over race or sex discrimination in hiring. Here, neither the effect of sex or race on applicants’ qualification nor the effect of qualification on hiring are targets of litigation. Rather, defendants must prove that sex and race do not directly influence hiring decisions, whatever indirect effects they might have on hiring by way of applicant qualification.

From a policy making viewpoint, an investigator may be interested in decomposing effects to quantify the extent to which racial salary disparity is due to educational disparity, or, taking a health-care example, the extent to which sensitivity to a given exposure can be reduced by eliminating sensitivity to an intermediate factor, standing between exposure and outcome. Another example concerns the identification of neural pathways in the brain or the structural features of protein-signaling networks in molecular biology ( Brent and Lok, 2005 ). Here, the decomposition of effects into their direct and indirect components carries theoretical scientific importance, for it tells us “how nature works” and, therefore, enables us to predict behavior under a rich variety of conditions.

Yet despite its ubiquity, the analysis of mediation has long been a thorny issue in the social and behavioral sciences ( Judd and Kenny, 1981 , Baron and Kenny, 1986 , Muller, Judd, and Yzerbyt, 2005 , Shrout and Bolger, 2002 , MacKinnon, Fairchild, and Fritz, 2007a ) primarily because structural equation modeling in those sciences were deeply entrenched in linear analysis, where the distinction between causal parameters and their regressional interpretations can easily be conflated. 18 As demands grew to tackle problems involving binary and categorical variables, researchers could no longer define direct and indirect effects in terms of structural or regressional coefficients, and all attempts to extend the linear paradigms of effect decomposition to non-linear systems produced distorted results ( MacKinnon, Lockwood, Brown, Wang, and Hoffman, 2007b ). These difficulties have accentuated the need to redefine and derive causal effects from first principles, uncommitted to distributional assumptions or a particular parametric form of the equations. The structural methodology presented in this paper adheres to this philosophy and it has produced indeed a principled solution to the mediation problem, based on the counterfactual reading of structural equations (29) . The following subsections summarize the method and its solution.

6.1.2. Controlled direct-effects

A major impediment to progress in mediation analysis has been the lack of notational facility for expressing the key notion of “holding the mediating variables fixed” in the definition of direct effect. Clearly, this notion must be interpreted as (hypothetically) setting the intermediate variables to constants by physical intervention, not by analytical means such as selection, regression, conditioning, matching or adjustment. For example, consider the simple mediation models of Fig. 6 , where the error terms (not shown explicitly) are assumed to be independent. It will not be sufficient to measure the association between gender ( X ) and hiring ( Y ) for a given level of qualification ( Z ), (see Fig. 6(b) ) because, by conditioning on the mediator Z , we create spurious associations between X and Y through W 2 , even when there is no direct effect of X on Y ( Pearl, 1998 , Cole and Hernán, 2002 ).

(a) A generic model depicting mediation through Z with no confounders, and (b) with two confounders, W 1 and W 2 .

Using the do ( x ) notation, enables us to correctly express the notion of “holding Z fixed” and obtain a simple definition of the controlled direct effect of the transition from X = x to X = x ′ :

or, equivalently, using counterfactual notation:

where Z is the set of all mediating variables. The readers can easily verify that, in linear systems, the controlled direct effect reduces to the path coefficient of the link X → Y (see footnote 12 ) regardless of whether confounders are present (as in Fig. 6(b) ) and regardless of whether the error terms are correlated or not.

This separates the task of definition from that of identification, as demanded by Section 4.1. The identification of CDE would depend, of course, on whether confounders are present and whether they can be neutralized by adjustment, but these do not alter its definition. Nor should trepidation about infeasibility of the action do ( gender = male ) enter the definitional phase of the study, Definitions apply to symbolic models, not to human biology. Graphical identification conditions for expressions of the type E ( Y | do ( x ), do ( z 1 ), do ( z 2 ), . . ., do ( z k )) in the presence of unmeasured confounders were derived by Pearl and Robins (1995) (see Pearl (2000a , Chapter 4) and invoke sequential application of the back-door conditions discussed in Section 3.2.

6.1.3. Natural direct effects

In linear systems, the direct effect is fully specified by the path coefficient attached to the link from X to Y ; therefore, the direct effect is independent of the values at which we hold Z . In nonlinear systems, those values would, in general, modify the effect of X on Y and thus should be chosen carefully to represent the target policy under analysis. For example, it is not uncommon to find employers who prefer males for the high-paying jobs (i.e., high z ) and females for low-paying jobs (low z ).

When the direct effect is sensitive to the levels at which we hold Z , it is often more meaningful to define the direct effect relative to some “natural” base-line level that may vary from individual to individual, and represents the level of Z just before the change in X . Conceptually, we can define the natural direct effect DE x,x ′ ( Y ) as the expected change in Y induced by changing X from x to x ′ while keeping all mediating factors constant at whatever value they would have obtained under do ( x ). This hypothetical change, which Robins and Greenland (1992) conceived and called “pure” and Pearl (2001) formalized and analyzed under the rubric “natural,” mirrors what lawmakers instruct us to consider in race or sex discrimination cases: “The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of a different race (age, sex, religion, national origin etc.) and everything else had been the same.” (In Carson versus Bethlehem Steel Corp. , 70 FEP Cases 921, 7th Cir. (1996)).

Extending the subscript notation to express nested counterfactuals, Pearl (2001) gave a formal definition for the “natural direct effect”:

Here, Y x ′ , Z x represents the value that Y would attain under the operation of setting X to x ′ and, simultaneously, setting Z to whatever value it would have obtained under the setting X = x . We see that DE x,x′ ( Y ), the natural direct effect of the transition from x to x ′ , involves probabilities of nested counterfactuals and cannot be written in terms of the do ( x ) operator. Therefore, the natural direct effect cannot in general be identified, even with the help of ideal, controlled experiments (see footnote 8 for intuitive explanation). However, aided by the surgical definition of Eq. (29) and the notational power of nested counterfactuals, Pearl (2001) was nevertheless able to show that, if certain assumptions of “no confounding” are deemed valid, the natural direct effect can be reduced to

The intuition is simple; the natural direct effect is the weighted average of the controlled direct effect, using the causal effect P ( z | do ( x )) as a weighing function.

One condition for the validity of (43) is that Z x ⊥⊥ Y x′,z | W holds for some set W of measured covariates. This technical condition in itself, like the ignorability condition of (35), is close to meaningless for most investigators, as it is not phrased in terms of realized variables. The surgical interpretation of counterfactuals (29) can be invoked at this point to unveil the graphical interpretation of this condition. It states that W should be admissible (i.e., satisfy the back-door condition) relative the path(s) from Z to Y . This condition, satisfied by W 2 in Fig. 6(b) , is readily comprehended by empirical researchers, and the task of selecting such measurements, W , can then be guided by the available scientific knowledge. Additional graphical and counterfactual conditions for identification are derived in Pearl (2001) Petersen et al. (2006) and Imai, Keele, and Yamamoto (2008) .

In particular, it can be shown ( Pearl, 2001 ) that expression (43) is both valid and identifiable in Markovian models (i.e., no unobserved confounders) where each term on the right can be reduced to a “ do -free” expression using Eq. (24) or (25) and then estimated by regression.

For example, for the model in Fig. 6(b) , Eq. (43) reads:

while for the confounding-free model of Fig. 6(a) we have:

Both (44) and (45) can easily be estimated by a two-step regression.

6.1.4. Natural indirect effects

Remarkably, the definition of the natural direct effect (42) can be turned around and provide an operational definition for the indirect effect – a concept shrouded in mystery and controversy, because it is impossible, using the do ( x ) operator, to disable the direct link from X to Y so as to let X influence Y solely via indirect paths.

The natural indirect effect , IE , of the transition from x to x ′ is defined as the expected change in Y affected by holding X constant, at X = x , and changing Z to whatever value it would have attained had X been set to X = x ′ . Formally, this reads ( Pearl, 2001 ):

which is almost identical to the direct effect ( Eq. (42) ) save for exchanging x and x ′ in the first term.

Indeed, it can be shown that, in general, the total effect TE of a transition is equal to the difference between the direct effect of that transition and the indirect effect of the reverse transition. Formally,

In linear systems, where reversal of transitions amounts to negating the signs of their effects, we have the standard additive formula

Since each term above is based on an independent operational definition, this equality constitutes a formal justification for the additive formula used routinely in linear systems.

Note that, although it cannot be expressed in do -notation, the indirect effect has clear policy-making implications. For example: in the hiring discrimination context, a policy maker may be interested in predicting the gender mix in the work force if gender bias is eliminated and all applicants are treated equally—say, the same way that males are currently treated. This quantity will be given by the indirect effect of gender on hiring, mediated by factors such as education and aptitude, which may be gender-dependent.

More generally, a policy maker may be interested in the effect of issuing a directive to a select set of subordinate employees, or in carefully controlling the routing of messages in a network of interacting agents. Such applications motivate the analysis of path-specific effects , that is, the effect of X on Y through a selected set of paths ( Avin, Shpitser, and Pearl, 2005 ).

In all these cases, the policy intervention invokes the selection of signals to be sensed, rather than variables to be fixed. Pearl (2001) has suggested therefore that signal sensing is more fundamental to the notion of causation than manipulation ; the latter being but a crude way of stimulating the former in experimental setup. The mantra “No causation without manipulation” must be rejected. (See ( Pearl, 2009b , Section 11.4.5).)

It is remarkable that counterfactual quantities like DE and IE that could not be expressed in terms of do ( x ) operators, and appear therefore void of empirical content, can, under certain conditions be estimated from empirical studies, and serve to guide policies. Awareness of this potential should embolden researchers to go through the definitional step of the study and freely articulate the target quantity Q ( M ) in the language of science, i.e., counterfactuals, despite the seemingly speculative nature of each assumption in the model ( Pearl, 2000b ).

6.2. The Mediation Formula: a simple solution to a thorny problem

This subsection demonstrates how the solution provided in equations (45) and (48) can be applied to practical problems of assessing mediation effects in non-linear models. We will use the simple mediation model of Fig. 6(a) , where all error terms (not shown explicitly) are assumed to be mutually independent, with the understanding that adjustment for appropriate sets of covariates W may be necessary to achieve this independence and that integrals should replace summations when dealing with continuous variables ( Imai et al., 2008 ).

Combining (45) and (48), the expression for the indirect effect, IE , becomes:

which provides a general formula for mediation effects, applicable to any nonlinear system, any distribution (of U ), and any type of variables. Moreover, the formula is readily estimable by regression. Owed to its generality and ubiquity, I will refer to this expression as the “Mediation Formula.”

The Mediation Formula represents the average increase in the outcome Y that the transition from X = x to X = x ′ is expected to produce absent any direct effect of X on Y . Though based on solid causal principles, it embodies no causal assumption other than the generic mediation structure of Fig. 6(a) . When the outcome Y is binary (e.g., recovery, or hiring) the ratio (1 − IE / TE ) represents the fraction of responding individuals who owe their response to direct paths, while (1 − DE / TE ) represents the fraction who owe their response to Z -mediated paths.

The Mediation Formula tells us that IE depends only on the expectation of the counterfactual Y xz , not on its functional form f Y ( x , z , u Y ) or its distribution P ( Y xz = y ). It calls therefore for a two-step regression which, in principle, can be performed non-parametrically. In the first step we regress Y on X and Z , and obtain the estimate

for every ( x , z ) cell. In the second step we estimate the expectation of g ( x , z ) conditional on X = x ′ and X = x , respectively, and take the difference:

Nonparametric estimation is not always practical. When Z consists of a vector of several mediators, the dimensionality of the problem would prohibit the estimation of E ( Y | x , z ) for every ( x , z ) cell, and the need arises to use parametric approximation. We can then choose any convenient parametric form for E ( Y | x , z ) (e.g., linear, logit, probit), estimate the parameters separately (e.g., by regression or maximum likelihood methods), insert the parametric approximation into (49) and estimate its two conditional expectations (over z ) to get the mediated effect ( VanderWeele, 2009 ).

Let us examine what the Mediation Formula yields when applied to both linear and non-linear versions of model 6(a). In the linear case, the structural model reads:

Computing the conditional expectation in (49) gives

where b is the total effect coefficient, b = ( E ( Y | x ′ ) − E ( Y | x ))/( x ′ − x ) = c x + c z b x .

We thus obtained the standard expressions for indirect effects in linear systems, which can be estimated either as a difference in two regression coefficients ( Eq. 53 ) or a product of two regression coefficients ( Eq. 52 ), with Y regressed on both X and Z . (see ( MacKinnon et al., 2007b )). These two strategies do not generalize to non-linear system as we shall see next.

Suppose we apply (49) to a non-linear process ( Fig. 7 ) in which X, Y , and Z are binary variables, and Y and Z are given by the Boolean formula

Such disjunctive interaction would describe, for example, a disease Y that would be triggered either by X directly, if enabled by e x , or by Z , if enabled by e z . Let us further assume that e x , e z and e xz are three independent Bernoulli variables with probabilities p x , p z , and p xz , respectively.

Stochastic non-linear model of mediation. All variables are binary.

As investigators, we are not aware, of course, of these underlying mechanisms; all we know is that X , Y , and Z are binary, that Z is hypothesized to be a mediator, and that the assumption of nonconfoundedness permits us to use the Mediation Formula (49) for estimating the Z -mediated effect of X on Y . Assume that our plan is to conduct a nonparametric estimation of the terms in (49) over a very large sample drawn from P ( x , y.z ); it is interesting to ask what the asymptotic value of the Mediation Formula would be, as a function of the model parameters: p x , p z , and p xz .

From knowledge of the underlying mechanism, we have:

Taking x = 0, x ′ = 1 and substituting these expressions in (45), (48), and (49) yields

Two observations are worth noting. First, we see that, despite the non-linear interaction between the two causal paths, the parameters of one do not influence on the causal effect mediated by the other. Second, the total effect is not the sum of the direct and indirect effects. Instead, we have:

which means that a fraction DE · IE / TE of outcome cases triggered by the transition from X = 0 to X = 1 are triggered simultaneously, through both causal paths, and would have been triggered even if one of the paths was disabled.

Now assume that we choose to approximate E ( Y | x , z ) by the linear expression

After fitting the a ’s parameters to the data (e.g., by OLS) and substituting in (49) one would obtain

which holds whenever we use the approximation in (57), regardless of the underlying mechanism.

If the correct data-generating process was the linear model of (50), we would obtain the expected estimates a 2 = c z , E ( z | x ′ ) − E ( z | x ′ ) = b x ( x ′ − x ) and

If however we were to apply the approximation in (57) to data generated by the nonlinear model of Fig. 7 , a distorted solution would ensue; a 2 would evaluate to

E ( z | x ′ ) − E ( z | x ) would evaluate to p xz ( x ′ − x ), and (58) would yield the approximation

We see immediately that the result differs from the correct value p z p xz derived in (54). Whereas the approximate value depends on P ( x = 1), the correct value shows no such dependence, and rightly so; no causal effect should depend on the probability of the causal variable.

Fortunately, the analysis permits us to examine under what condition the distortion would be significant. Comparing (59) and (54) reveals that the approximate method always underestimates the indirect effect and the distortion is minimal for high values of P ( x = 1) and (1− p x ).

Had we chosen to include an interaction term in the approximation of E ( Y | x , z ), the correct result would obtain. To witness, writing

a 2 would evaluate to p z , a 3 to p x p z , and the correct result obtains through:

We see that, in addition to providing causally-sound estimates for mediation effects, the Mediation Formula also enables researchers to evaluate analytically the effectiveness of various parametric specifications relative to any assumed model. This type of analytical “sensitivity analysis” has been used extensively in statistics for parameter estimation, but could not be applied to mediation analysis, owed to the absence of an objective target quantity that captures the notion of indirect effect in both linear and non-linear systems, free of parametric assumptions. The Mediation Formula of Eq. (49) explicates this target quantity formally, and casts it in terms of estimable quantities.

The derivation of the Mediation Formula was facilitated by taking seriously the four steps of the structural methodology (Section 4) together with the graphical-counterfactual-structural symbiosis spawned by the surgical interpretation of counterfactuals ( Eq. (29) ).

In contrast, when the mediation problem is approached from an exclusivist potential-outcome viewpoint, void of the structural guidance of Eq. (29) , counterintuitive definitions ensue, carrying the label “principal stratification” ( Rubin, 2004 , 2005 ), which are at variance with common understanding of direct and indirect effects. For example, the direct effect is definable only in units absent of indirect effects. This means that a grandfather would be deemed to have no direct effect on his grandson’s behavior in families where he has had some effect on the father. This precludes from the analysis all typical families, in which a father and a grandfather have simultaneous, complementary influences on children’s upbringing. In linear systems, to take an even sharper example, the direct effect would be undefined whenever indirect paths exist from the cause to its effect. The emergence of such paradoxical conclusions underscores the wisdom, if not necessity of a symbiotic analysis, in which the counterfactual notation Y x ( u ) is governed by its structural definition, Eq. (29) . 19

6.3. Causes of effects and probabilities of causation

The likelihood that one event was the cause of another guides much of what we understand about the world (and how we act in it). For example, knowing whether it was the aspirin that cured my headache or the TV program I was watching would surely affect my future use of aspirin. Likewise, to take an example from common judicial standard, judgment in favor of a plaintiff should be made if and only if it is “more probable than not” that the damage would not have occurred but for the defendant’s action ( Robertson, 1997 ).

These two examples fall under the category of “causes of effects” because they concern situations in which we observe both the effect, Y = y , and the putative cause X = x and we are asked to assess, counterfactually, whether the former would have occurred absent the latter.

We have remarked earlier ( footnote 8 ) that counterfactual probabilities conditioned on the outcome cannot in general be identified from observational or even experimental studies. This does not mean however that such probabilities are useless or void of empirical content; the structural perspective may guide us in fact toward discovering the conditions under which they can be assessed from data, thus defining the empirical content of these counterfactuals.

Following the 4-step process of structural methodology – define, assume, identify, and estimate – our first step is to express the target quantity in counterfactual notation and verify that it is well defined, namely, that it can be computed unambiguously from any fully-specified causal model.

In our case, this step is simple. Assuming binary events, with X = x and Y = y representing treatment and outcome, respectively, and X = x ′ , Y = y ′ their negations, our target quantity can be formulated directly from the English sentence:

“Find the probability that Y would be y ′ had X been x ′ , given that, in reality, Y is actually y and X is x ,”

This counterfactual quantity, which Robins and Greenland (1989b) named “probability of causation” and Pearl (2000a , p. 296) named “probability of necessity” (PN), to be distinguished from two other nuances of “causation,” is certainly computable from any fully specified structural model, i.e., one in which P ( u ) and all functional relationships are given. This follows from the fact that every structural model defines a joint distribution of counterfactuals, through Eq. (29) .

Having written a formal expression for PN, Eq. (60) , we can move on to the formulation and identification phases and ask what assumptions would permit us to identify PN from empirical studies, be they observational, experimental or a combination thereof.

This problem was analyzed in Pearl (2000a , Chapter 9) and yielded the following results:

Theorem 4 If Y is monotonic relative to X, i.e., Y 1 ( u ) ≥ Y 0 ( u ) , then PN is identifiable whenever the causal effect P ( y | do ( x )) is identifiable and, moreover,

The first term on the r.h.s. of (61) is the familiar excess risk ratio (ERR) that epidemiologists have been using as a surrogate for PN in court cases ( Cole, 1997 , Robins and Greenland, 1989b ). The second term represents the correction needed to account for confounding bias, that is, P ( y | do ( x ′ )) ≠ P ( y | x ′ ).

This suggests that monotonicity and unconfoundedness were tacitly assumed by the many authors who proposed or derived ERR as a measure for the “fraction of exposed cases that are attributable to the exposure” ( Greenland, 1999 ).

Equation (61) thus provides a more refined measure of causation, which can be used in situations where the causal effect P ( y | do ( x )) can be estimated from either randomized trials or graph-assisted observational studies (e.g., through Theorem 3 or Eq. (25) ). It can also be shown ( Tian and Pearl, 2000 ) that the expression in (61) provides a lower bound for PN in the general, nonmonotonic case. (See also ( Robins and Greenland, 1989a ).) In particular, the tight upper and lower bounds on PN are given by:

It is worth noting that, in drug related litigation, it is not uncommon to obtain data from both experimental and observational studies. The former is usually available at the manufacturer or the agency that approved the drug for distribution (e.g., FDA), while the latter is easy to obtain by random surveys of the population. In such cases, the standard lower bound used by epidemiologists to establish legal responsibility, the Excess Risk Ratio, can be improved substantially using the corrective term of Eq. (61) . Likewise, the upper bound of Eq. (62) can be used to exonerate drug-makers from legal responsibility. Cai and Kuroki (2006) analyzed the statistical properties of PN.

Pearl (2000a , p. 302) shows that combining data from experimental and observational studies which, taken separately, may indicate no causal relations between X and Y , can nevertheless bring the lower bound of Eq. (62) to unity, thus implying causation with probability one .

Such extreme results dispel all fears and trepidations concerning the empirical content of counterfactuals ( Dawid, 2000 , Pearl, 2000b ). They demonstrate that a quantity PN which at first glance appears to be hypothetical, ill-defined, untestable and, hence, unworthy of scientific analysis is nevertheless definable, testable and, in certain cases, even identifiable. Moreover, the fact that, under certain combination of data, and making no assumptions whatsoever, an important legal claim such as “the plaintiff would be alive had he not taken the drug” can be ascertained with probability approaching one, is a remarkable tribute to formal analysis.

Another counterfactual quantity that has been fully characterized recently is the Effect of Treatment on the Treated (ETT):

ETT has been used in econometrics to evaluate the effectiveness of social programs on their participants ( Heckman, 1992 ) and has long been the target of research in epidemiology, where it came to be known as “the effect of exposure on the exposed,” or “standardized morbidity” ( Miettinen, 1974 ; Greenland and Robins, 1986 ).

Shpitser and Pearl (2009) have derived a complete characterization of those models in which ETT can be identified from either experimental or observational studies. They have shown that, despite its blatant counterfactual character, (e.g., “I just took an aspirin, perhaps I shouldn’t have?”) ETT can be evaluated from experimental studies in many, though not all cases. It can also be evaluated from observational studies whenever a sufficient set of covariates can be measured that satisfies the back-door criterion and, more generally, in a wide class of graphs that permit the identification of conditional interventions.

These results further illuminate the empirical content of counterfactuals and their essential role in causal analysis. They prove once again the triumph of logic and analysis over traditions that a-priori exclude from the analysis quantities that are not testable in isolation. Most of all, they demonstrate the effectiveness and viability of the scientific approach to causation whereby the dominant paradigm is to model the activities of Nature, rather than those of the experimenter. In contrast to the ruling paradigm of conservative statistics, we begin with relationships that we know in advance will never be estimated, tested or falsified. Only after assembling a host of such relationships and judging them to faithfully represent our theory about how Nature operates, we ask whether the parameter of interest, crisply defined in terms of those theoretical relationships, can be estimated consistently from empirical data and how. It often does, to the credit of progressive statistics.

7. Conclusions

Traditional statistics is strong in devising ways of describing data and inferring distributional parameters from sample. Causal inference requires two additional ingredients: a science-friendly language for articulating causal knowledge, and a mathematical machinery for processing that knowledge, combining it with data and drawing new causal conclusions about a phenomenon. This paper surveys recent advances in causal analysis from the unifying perspective of the structural theory of causation and shows how statistical methods can be supplemented with the needed ingredients. The theory invokes non-parametric structural equations models as a formal and meaningful language for defining causal quantities, formulating causal assumptions, testing identifiability, and explicating many concepts used in causal discourse. These include: randomization, intervention, direct and indirect effects, confounding, counterfactuals, and attribution. The algebraic component of the structural language coincides with the potential-outcome framework, and its graphical component embraces Wright’s method of path diagrams. When unified and synthesized, the two components offer statistical investigators a powerful and comprehensive methodology for empirical research.

1 By “untested” I mean untested using frequency data in nonexperimental studies.

2 Clearly, P ( Y = y | do ( X = x )) is equivalent to P ( Yx = y ). This is what we normally assess in a controlled experiment, with X randomized, in which the distribution of Y is estimated for each level x of X .

3 Linear relations are used here for illustration purposes only; they do not represent typical disease-symptom relations but illustrate the historical development of path analysis. Additionally, we will use standardized variables, that is, zero mean and unit variance.

4 Additional implications called “dormant independence” ( Shpitser and Pearl, 2008 ) may be deduced from some graphs with correlated errors ( Verma and Pearl, 1990 ).

5 A simple proof of the Causal Markov Theorem is given in Pearl (2000a , p. 30). This theorem was first presented in Pearl and Verma (1991) , but it is implicit in the works of Kiiveri, Speed, and Carlin (1984) and others. Corollary 1 was named “Manipulation Theorem” in Spirtes et al. (1993) , and is also implicit in Robins’ (1987) G -computation formula. See Lauritzen (2001) .

6 For clarity, we drop the (superfluous) subscript 0 from x 0 and z 2 0 .

7 Before applying this criterion, one may delete from the causal graph all nodes that are not ancestors of Y .

8 The reason for this fundamental limitation is that no death case can be tested twice, with and without treatment. For example, if we measure equal proportions of deaths in the treatment and control groups, we cannot tell how many death cases are actually attributable to the treatment itself; it is quite possible that many of those who died under treatment would be alive if untreated and, simultaneously, many of those who survived with treatment would have died if not treated.

9 Connections between structural equations and a restricted class of counterfactuals were first recognized by Simon and Rescher (1966) . These were later generalized by Balke and Pearl (1995) , using surgeries ( Eq. (29) ), thus permitting endogenous variables to serve as counterfactual antecedents. The term “surgery definition” was used in Pearl (2000a , Epilogue) and criticized by Cartwright (2007) and Heckman (2005) , (see Pearl (2009b , pp. 362–3, 374–9 for rebuttals)).

10 The distinction between general, or population-level causes (e.g., “Drinking hemlock causes death”) and singular or unit-level causes (e.g., “Socrates’ drinking hemlock caused his death”), which many philosophers have regarded as irreconcilable ( Eells, 1991 ), introduces no tension at all in the structural theory. The two types of sentences differ merely in the level of situation-specific information that is brought to bear on a problem, that is, in the specificity of the evidence e that enters the quantity P ( Y x = y | e ). When e includes all factors u , we have a deterministic, unit-level causation on our hand; when e contains only a few known attributes (e.g., age, income, occupation etc.) while others are assigned probabilities, a population-level analysis ensues.

11 For example, “The probability is 80% that Joe belongs to the class of patients who will be cured if they take the drug and die otherwise.”

12 Note that β in Eq. (1) , the incremental causal effect of X on Y , is defined procedurally by β ≜ E ( Y | d o ( x 0 + 1 ) ) − E ( Y | d o ( x 0 ) ) = ∂ ∂ x E ( Y | d o ( x ) ) = ∂ ∂ x E ( Y x ) . Naturally, all attempts to give β statistical interpretation have ended in frustrations ( Holland, 1988 , Whittaker, 1990 , Wermuth, 1992 , Wermuth and Cox, 1993 ), some persisting well into the 21st century ( Sobel, 2008 ).

13 These equivalence classes were later called “principal stratification” by Frangakis and Rubin (2002) . Looser bounds were derived earlier by Robins (1989) and Manski (1990) .

14 The notation Y ⊥⊥ X | Z stands for the conditional independence relationship P ( Y = y , X = x | Z = z ) = P ( Y = y | Z = z ) P ( X = x | Z = z ) ( Dawid, 1979 ).

15 Inquisitive readers are invited to guess whether X z ⊥⊥ Z | Y holds in Fig. 2(a) , then reflect on why causality is so slow in penetrating statistical education.

16 The opaqueness of counterfactual independencies explains why many researchers within the potential-outcome camp are unaware of the fact that adding a covariate to the analysis (e.g., Z 3 in Fig. 4 , Z in Fig. 5 may actually increase confounding bias in propensity-score matching. Paul Rosenbaum, for example, writes: “there is little or no reason to avoid adjustment for a true covariate, a variable describing subjects before treatment” ( Rosenbaum, 2002 , p. 76). Rubin (2009) goes as far as stating that refraining from conditioning on an available measurement is “nonscientific ad hockery” for it goes against the tenets of Bayesian philosophy (see ( Pearl, 2009c , d , Heckman and Navarro-Lozano, 2004 ) for a discussion of this fallacy).

17 Such symbiosis is now standard in epidemiology research ( Robins, 2001 , Petersen, Sinisi, and van der Laan, 2006 , VanderWeele and Robins, 2007 , Hafeman and Schwartz, 2009 , VanderWeele, 2009 ) yet still lacking in econometrics ( Heckman, 2008 , Imbens and Wooldridge, 2009 ).

18 All articles cited above define the direct and indirect effects through their regressional interpretations; I am not aware of any article in this tradition that formally adapts a causal interpretation, free of estimation-specific parameterization.

19 Such symbiosis is now standard in epidemiology research ( Robins, 2001 , Petersen et al., 2006 , VanderWeele and Robins, 2007 , Hafeman and Schwartz, 2009 , VanderWeele, 2009 ) and is making its way slowly toward the social and behavioral sciences.

* Portions of this paper are adapted from Pearl (2000a , 2009a , b) ; I am indebted to Elja Arjas, Sander Greenland, David MacKinnon, Patrick Shrout, and many readers of the UCLA Causality Blog ( http://www.mii.ucla.edu/causality/ ) for reading and commenting on various segments of this manuscript, and especially to Erica Moodie and David Stephens for their thorough editorial input. This research was supported in parts by NIH grant #1R01 LM009961-01, NSF grant #IIS-0914211, and ONR grant #N000-14-09-1-0665.

• Angrist J, Imbens G, Rubin D. “Identification of causal effects using instrumental variables (with comments),” Journal of the American Statistical Association. 1996; 91 :444–472. doi: 10.2307/2291629. [ CrossRef ] [ Google Scholar ]
• Arah O.2008 “The role of causal reasoning in understanding Simpson’s paradox, Lord’s paradox, and the suppression effect: Covariate selection in the analysis of observational studies,” Emerging Themes in Epidemiology 4doi: 10.1186/1742–7622–5–5, online at < http://www.ete-online.com/content/5/1/5 >. [ PMC free article ] [ PubMed ]
• Arjas E, Parner J. “Causal reasoning from longitudinal data,” Scandinavian Journal of Statistics. 2004; 31 :171–187. doi: 10.1111/j.1467-9469.2004.02-134.x. [ CrossRef ] [ Google Scholar ]
• Avin C, Shpitser I, Pearl J. Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence IJCAI-05. Edinburgh, UK: Morgan-Kaufmann Publishers; 2005. “Identifiability of path-specific effects,” pp. 357–363. [ Google Scholar ]
• Balke A, Pearl J. “Counterfactuals and policy analysis in structural models,” In: Besnard P, Hanks S, editors. Uncertainty in Artificial Intelligence 11. San Francisco: Morgan Kaufmann; 1995. pp. 11–18. [ Google Scholar ]
• Balke A, Pearl J. “Bounds on treatment effects from studies with imperfect compliance,” Journal of the American Statistical Association. 1997; 92 :1172–1176. doi: 10.2307/2965583. [ CrossRef ] [ Google Scholar ]
• Baron R, Kenny D. “The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations,” Journal of Personality and Social Psychology. 1986; 51 :1173–1182. doi: 10.1037/0022-3514.51.6.1173. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Berkson J. “Limitations of the application of fourfold table analysis to hospital data,” Biometrics Bulletin. 1946; 2 :47–53. doi: 10.2307/3002000. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Bollen K. Structural Equations with Latent Variables. New York: John Wiley; 1989. [ Google Scholar ]
• Brent R, Lok L. “A fishing buddy for hypothesis generators,” Science. 2005; 308 :523–529. doi: 10.1126/science.1110535. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Cai Z, Kuroki M. “Variance estimators for three ‘probabilities of causation’,” Risk Analysis. 2006; 25 :1611–1620. doi: 10.1111/j.1539-6924.2005.00696.x. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Cai Z, Kuroki M. “On identifying total effects in the presence of latent variables and selection bias,” In: McAllester DA, Myllymäki P, editors. Uncertainty in Artificial Intelligence, Proceedings of the Twenty-Fourth Conference. Arlington, VA: AUAI; 2008. pp. 62–69. [ Google Scholar ]
• Cartwright N. Hunting Causes and Using Them: Approaches in Philosophy and Economics. New York, NY: Cambridge University Press; 2007. [ Google Scholar ]
• Chalak K, White H.2006 “An extended class of instrumental variables for the estimation of causal effects,” Technical Report Discussion Paper, UCSD, Department of Economics.
• Chickering D, Pearl J. “A clinician’s tool for analyzing noncompliance,” Computing Science and Statistics. 1997; 29 :424–431. [ Google Scholar ]
• Cole P. “Causality in epidemiology, health policy, and law,” Journal of Marketing Research. 1997; 27 :10279–10285. [ Google Scholar ]
• Cole S, Hernán M. “Fallibility in estimating direct effects,” International Journal of Epidemiology. 2002; 31 :163–165. doi: 10.1093/ije/31.1.163. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Cox D. The Planning of Experiments. NY: John Wiley and Sons; 1958. [ Google Scholar ]
• Cox D, Wermuth N. “Causality: A statistical view,” International Statistical Review. 2004; 72 :285–305. [ Google Scholar ]
• Dawid A. “Conditional independence in statistical theory,” Journal of the Royal Statistical Society, Series B. 1979; 41 :1–31. [ Google Scholar ]
• Dawid A. “Causal inference without counterfactuals (with comments and rejoinder),” Journal of the American Statistical Association. 2000; 95 :407–448. doi: 10.2307/2669377. [ CrossRef ] [ Google Scholar ]
• Dawid A. “Influence diagrams for causal modelling and inference,” International Statistical Review. 2002; 70 :161–189. doi: 10.1111/j.1751-5823.2002.tb00354.x. [ CrossRef ] [ Google Scholar ]
• Duncan O. Introduction to Structural Equation Models. New York: Academic Press; 1975. [ Google Scholar ]
• Eells E. Probabilistic Causality. Cambridge, MA: Cambridge University Press; 1991. [ Google Scholar ]
• Frangakis C, Rubin D. “Principal stratification in causal inference,” Biometrics. 2002; 1 :21–29. doi: 10.1111/j.0006-341X.2002.00021.x. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Glymour M, Greenland S. “Causal diagrams,” In: Rothman K, Greenland S, Lash T, editors. Modern Epidemiology. 3rd edition. Philadelphia, PA: Lippincott Williams & Wilkins; 2008. pp. 183–209. [ Google Scholar ]
• Goldberger A. “Structural equation models in the social sciences,” Econometrica: Journal of the Econometric Society. 1972; 40 :979–1001. [ Google Scholar ]
• Goldberger A. “Structural equation models: An overview,” In: Goldberger A, Duncan O, editors. Structural Equation Models in the Social Sciences. New York, NY: Seminar Press; 1973. pp. 1–18. [ Google Scholar ]
• Greenland S. “Relation of probability of causation, relative risk, and doubling dose: A methodologic error that has become a social problem,” American Journal of Public Health. 1999; 89 :1166–1169. doi: 10.2105/AJPH.89.8.1166. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Greenland S, Pearl J, Robins J. “Causal diagrams for epidemiologic research,” Epidemiology. 1999; 10 :37–48. doi: 10.1097/00001648-199901000-00008. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Greenland S, Robins J. “Identifiability, exchangeability, and epidemiological confounding,” International Journal of Epidemiology. 1986; 15 :413–419. doi: 10.1093/ije/15.3.413. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Haavelmo T.1943 “The statistical implications of a system of simultaneous equations,” Econometrica 11 1–12.reprinted in Hendry DF, Morgan MS. The Foundations of Econometric Analysis Cambridge University Press; 477–490.1995 10.2307/1905714 [ CrossRef ] [ Google Scholar ]
• Hafeman D, Schwartz S. “Opening the black box: A motivation for the assessment of mediation,” International Journal of Epidemiology. 2009; 3 :838–845. doi: 10.1093/ije/dyn372. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Halpern J.1998 “Axiomatizing causal reasoning,” Cooper G, Moral S. Uncertainty in Artificial Intelligence San Francisco, CA: Morgan Kaufmann; 202–210.also Journal of Artificial Intelligence Research 12 3 17–37.2000 [ Google Scholar ]
• Heckman J. “Randomization and social policy evaluation,” In: Manski C, Garfinkle I, editors. Evaluations: Welfare and Training Programs. Cambridge, MA: Harvard University Press; 1992. pp. 201–230. [ Google Scholar ]
• Heckman J. “The scientific model of causality,” Sociological Methodology. 2005; 35 :1–97. doi: 10.1111/j.0081-1750.2006.00163.x. [ CrossRef ] [ Google Scholar ]
• Heckman J. “Econometric causality,” International Statistical Review. 2008; 76 :1–27. doi: 10.1111/j.1751-5823.2007.00024.x. [ CrossRef ] [ Google Scholar ]
• Heckman J, Navarro-Lozano S. “Using matching, instrumental variables, and control functions to estimate economic choice models,” The Review of Economics and Statistics. 2004; 86 :30–57. doi: 10.1162/003465304323023660. [ CrossRef ] [ Google Scholar ]
• Heckman J, Vytlacil E. “Structural equations, treatment effects and econometric policy evaluation,” Econometrica. 2005; 73 :669–738. doi: 10.1111/j.1468-0262.2005.00594.x. [ CrossRef ] [ Google Scholar ]
• Hernán M, Cole S. “Invited commentary: Causal diagrams and measurement bias,” American Journal of Epidemiology. 2009; 170 :959–962. doi: 10.1093/aje/kwp293. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Holland P. “Causal inference, path analysis, and recursive structural equations models,” In: Clogg C, editor. Sociological Methodology. Washington, DC: American Sociological Association; 1988. pp. 449–484. [ CrossRef ] [ Google Scholar ]
• Hurwicz L.1950 “Generalization of the concept of identification,” Koopmans T. Statistical Inference in Dynamic Economic Models Cowles Commission, Monograph 10New York: Wiley; 245–257. [ Google Scholar ]
• Imai K, Keele L, Yamamoto T.2008 “Identification, inference, and sensitivity analysis for causal mediation effects,” Technical reportDepartment of Politics, Princton University [ Google Scholar ]
• Imbens G, Wooldridge J. “Recent developments in the econometrics of program evaluation,” Journal of Economic Literature. 2009; 47 :5–86. doi: 10.1257/jel.47.1.5. [ CrossRef ] [ Google Scholar ]
• Judd C, Kenny D. “Process analysis: Estimating mediation in treatment evaluations,” Evaluation Review. 1981; 5 :602–619. doi: 10.1177/0193841X8100500502. [ CrossRef ] [ Google Scholar ]
• Kaufman S, Kaufman J, MacLenose R. “Analytic bounds on causal risk differences in directed acyclic graphs involving three observed binary variables,” Journal of Statistical Planning and Inference. 2009; 139 :3473–3487. doi: 10.1016/j.jspi.2009.03.024. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Kiiveri H, Speed T, Carlin J. “Recursive causal models,” Journal of Australian Math Society. 1984; 36 :30–52. doi: 10.1017/S1446788700027312. [ CrossRef ] [ Google Scholar ]
• Koopmans T. “Identification problems in econometric model construction,” In: Hood W, Koopmans T, editors. Studies in Econometric Method. New York: Wiley; 1953. pp. 27–48. [ Google Scholar ]
• Kuroki M, Miyakawa M. “Identifiability criteria for causal effects of joint interventions,” Journal of the Royal Statistical Society. 1999; 29 :105–117. [ Google Scholar ]
• Lauritzen S. Graphical Models. Oxford: Clarendon Press; 1996. [ Google Scholar ]
• Lauritzen S. “Causal inference from graphical models,” In: Cox D, Kluppelberg C, editors. Complex Stochastic Systems. Boca Raton, FL: Chapman and Hall/CRC Press; 2001. pp. 63–107. [ Google Scholar ]
• Lindley D. “Seeing and doing: The concept of causation,” International Statistical Review. 2002; 70 :191–214. doi: 10.1111/j.1751-5823.2002.tb00355.x. [ CrossRef ] [ Google Scholar ]
• MacKinnon D, Fairchild A, Fritz M. “Mediation analysis,” Annual Review of Psychology. 2007a; 58 :593–614. doi: 10.1146/annurev.psych.58.110405.085542. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• MacKinnon D, Lockwood C, Brown C, Wang W, Hoffman J. “The intermediate endpoint effect in logistic and probit regression,” Clinical Trials. 2007b; 4 :499–513. doi: 10.1177/1740774507083434. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Manski C. “Nonparametric bounds on treatment effects,” American Economic Review, Papers and Proceedings. 1990; 80 :319–323. [ Google Scholar ]
• Marschak J.1950 “Statistical inference in economics,” Koopmans T. Statistical Inference in Dynamic Economic Models New York: Wiley; 1–50.cowles Commission for Research in Economics, Monograph 10. [ Google Scholar ]
• Meek C, Glymour C. “Conditioning and intervening,” British Journal of Philosophy Science. 1994; 45 :1001–1021. doi: 10.1093/bjps/45.4.1001. [ CrossRef ] [ Google Scholar ]
• Miettinen O. “Proportion of disease caused or prevented by a given exposure, trait, or intervention,” Journal of Epidemiology. 1974; 99 :325–332. [ PubMed ] [ Google Scholar ]
• Morgan S, Winship C. Counterfactuals and Causal Inference: Methods and Principles for Social Research (Analytical Methods for Social Research) New York, NY: Cambridge University Press; 2007. [ Google Scholar ]
• Muller D, Judd C, Yzerbyt V. “When moderation is mediated and mediation is moderated,” Journal of Personality and Social Psychology. 2005; 89 :852–863. doi: 10.1037/0022-3514.89.6.852. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Neyman J. “On the application of probability theory to agricultural experiments. Essay on principles. Section 9,” Statistical Science. 1923; 5 :465–480. [ Google Scholar ]
• Pearl J. Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufmann; 1988. [ Google Scholar ]
• Pearl J. “Comment: Graphical models, causality, and intervention,” Statistical Science. 1993a; 8 :266–269. doi: 10.1214/ss/1177010894. [ CrossRef ] [ Google Scholar ]
• Pearl J.1993b “Mediating instrumental variables,” Technical Report R-210, < http://ftp.cs.ucla.edu/pub/stat_ser/R210.pdf >, Department of Computer Science, University of California, Los Angeles.
• Pearl J. “Causal diagrams for empirical research,” Biometrika. 1995; 82 :669–710. doi: 10.1093/biomet/82.4.669. [ CrossRef ] [ Google Scholar ]
• Pearl J. “Graphs, causality, and structural equation models,” Sociological Methods and Research. 1998; 27 :226–284. doi: 10.1177/0049124198027002004. [ CrossRef ] [ Google Scholar ]
• Pearl J. Causality: Models, Reasoning, and Inference. second ed. New York: Cambridge University Press; 2000a. 2009. [ Google Scholar ]
• Pearl J. “Comment on A.P. Dawid’s, Causal inference without counterfactuals,” Journal of the American Statistical Association. 2000b; 95 :428–431. doi: 10.2307/2669380. [ CrossRef ] [ Google Scholar ]
• Pearl J. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence. San Francisco, CA: Morgan Kaufmann; 2001. “Direct and indirect effects,” pp. 411–420. [ Google Scholar ]
• Pearl J. “Robustness of causal claims,” In: Chickering M, Halpern J, editors. Proceedings of the Twentieth Conference Uncertainty in Artificial Intelligence. Arlington, VA: AUAI Press; 2004. pp. 446–453. [ Google Scholar ]
• Pearl J.2009a “Causal inference in statistics: An overview,” Statistics Surveys 3 96–146.< http://www.i-journals.org/ss/viewarticle.php?id=57 >. 10.1214/09-SS057 [ CrossRef ] [ Google Scholar ]
• Pearl J. Causality: Models, Reasoning, and Inference. second edition New York: Cambridge University Press; 2009b. [ Google Scholar ]
• Pearl J.2009c “Letter to the editor: Remarks on the method of propensity scores,” Statistics in Medicine 28 1415–1416.< http://ftp.cs.ucla.edu/pub/stat_ser/r345-sim.pdf >. 10.1002/sim.3521 [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Pearl J.2009d “Myth, confusion, and science in causal analysis,” Technical Report R-348Department of Computer Science, University of California; Los Angeles, CA: < http://ftp.cs.ucla.edu/pub/stat_ser/r348.pdf >. [ Google Scholar ]
• Pearl J.2009e “On a class of bias-amplifying covariates that endanger effect estimates,” Technical Report R-346Department of Computer Science, University of California; Los Angeles, CA: < http://ftp.cs.ucla.edu/pub/stat_ser/r346.pdf >. [ Google Scholar ]
• Pearl J.2009f “On measurement bias in causal inference,” Technical Report R-357< http://ftp.cs.ucla.edu/pub/stat_ser/r357.pdf >, Department of Computer Science, University of California; Los Angeles [ Google Scholar ]
• Pearl J, Paz A.2009 “Confounding equivalence in observational studies,” Technical Report R-343Department of Computer Science, University of California; Los Angeles, CA: < http://ftp.cs.ucla.edu/pub/stat_ser/r343.pdf >. [ Google Scholar ]
• Pearl J, Robins J. “Probabilistic evaluation of sequential plans from causal models with hidden variables,” In: Besnard P, Hanks S, editors. Uncertainty in Artificial Intelligence 11. San Francisco: Morgan Kaufmann; 1995. pp. 444–453. [ Google Scholar ]
• Pearl J, Verma T. “A theory of inferred causation,” In: Allen J, Fikes R, Sandewall E, editors. Principles of Knowledge Representation and Reasoning: Proceedings of the Second International Conference. San Mateo, CA: Morgan Kaufmann; 1991. pp. 441–452. [ Google Scholar ]
• Petersen M, Sinisi S, van der Laan M. “Estimation of direct causal effects,” Epidemiology. 2006; 17 :276–284. doi: 10.1097/01.ede.0000208475.99429.2d. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Richard J. “Models with several regimes and changes in exogeneity,” Review of Economic Studies. 1980; 47 :1–20. doi: 10.2307/2297101. [ CrossRef ] [ Google Scholar ]
• Robertson D. “The common sense of cause in fact,” Texas Law Review. 1997; 75 :1765–1800. [ Google Scholar ]
• Robins J. “A new approach to causal inference in mortality studies with a sustained exposure period – applications to control of the healthy workers survivor effect,” Mathematical Modeling. 1986; 7 :1393–1512. doi: 10.1016/0270-0255(86)90088-6. [ CrossRef ] [ Google Scholar ]
• Robins J. “A graphical approach to the identification and estimation of causal parameters in mortality studies with sustained exposure periods,” Journal of Chronic Diseases. 1987; 40 :139S–161S. doi: 10.1016/S0021-9681(87)80018-8. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Robins J. “The analysis of randomized and non-randomized aids treatment trials using a new approach to causal inference in longitudinal studies,” In: Sechrest L, Freeman H, Mulley A, editors. Health Service Research Methodology: A Focus on AIDS. Washington, DC: NCHSR, U.S. Public Health Service; 1989. pp. 113–159. [ Google Scholar ]
• Robins J. “Testing and estimation of directed effects by reparameterizing directed acyclic with structural nested models,” In: Glymour C, Cooper G, editors. Computation, Causation, and Discovery. Cambridge, MA: AAAI/MIT Press; 1999. pp. 349–405. [ Google Scholar ]
• Robins J. “Data, design, and background knowledge in etiologic inference,” Epidemiology. 2001; 12 :313–320. doi: 10.1097/00001648-200105000-00011. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Robins J, Greenland S. “Estimability and estimation of excess and etiologic fractions,” Statistics in Medicine. 1989a; 8 :845–859. doi: 10.1002/sim.4780080709. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Robins J, Greenland S. “The probability of causation under a stochastic model for individual risk,” Biometrics. 1989b; 45 :1125–1138. doi: 10.2307/2531765. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Robins J, Greenland S. “Identifiability and exchangeability for direct and indirect effects,” Epidemiology. 1992; 3 :143–155. doi: 10.1097/00001648-199203000-00013. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Rosenbaum P. Observational Studies. second edition New York: Springer-Verlag; 2002. [ Google Scholar ]
• Rosenbaum P, Rubin D. “The central role of propensity score in observational studies for causal effects,” Biometrika. 1983; 70 :41–55. doi: 10.1093/biomet/70.1.41. [ CrossRef ] [ Google Scholar ]
• Rothman K. “Causes,” American Journal of Epidemiology. 1976; 104 :587–592. [ PubMed ] [ Google Scholar ]
• Rubin D. “Estimating causal effects of treatments in randomized and non-randomized studies,” Journal of Educational Psychology. 1974; 66 :688–701. doi: 10.1037/h0037350. [ CrossRef ] [ Google Scholar ]
• Rubin D. “Direct and indirect causal effects via potential outcomes,” Scandinavian Journal of Statistics. 2004; 31 :161–170. doi: 10.1111/j.1467-9469.2004.02-123.x. [ CrossRef ] [ Google Scholar ]
• Rubin D. “Causal inference using potential outcomes: Design, modeling, decisions,” Journal of the American Statistical Association. 2005; 100 :322–331. doi: 10.1198/016214504000001880. [ CrossRef ] [ Google Scholar ]
• Rubin D. “The design versus the analysis of observational studies for causal effects: Parallels with the design of randomized trials,” Statistics in Medicine. 2007; 26 :20–36. doi: 10.1002/sim.2739. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Rubin D. “Author’s reply: Should observational studies be designed to allow lack of balance in covariate distributions across treatment group?” Statistics in Medicine. 2009; 28 :1420–1423. doi: 10.1002/sim.3565. [ CrossRef ] [ Google Scholar ]
• Shpitser I, Pearl J. “Identification of conditional interventional distributions,” In: Dechter R, Richardson T, editors. Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence. Corvallis, OR: AUAI Press; 2006. pp. 437–444. [ Google Scholar ]
• Shpitser I, Pearl J. Proceedings of the Twenty-Third Conference on Artificial Intelligence. Menlo Park, CA: AAAI Press; 2008. “Dormant independence,” pp. 1081–1087. [ Google Scholar ]
• Shpitser I, Pearl J. Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence. Montreal, Quebec: AUAI Press; 2009. “Effects of treatment on the treated: Identification and generalization,” [ Google Scholar ]
• Shrier I.2009 “Letter to the editor: Propensity scores,” Statistics in Medicine 28 1317–1318.see also Pearl 2009< http://ftp.cs.ucla.edu/pub/stat_ser/r348.pdf >. 10.1002/sim.3554 [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Shrout P, Bolger N. “Mediation in experimental and nonexperimental studies: New procedures and recommendations,” Psychological Methods. 2002; 7 :422–445. doi: 10.1037/1082-989X.7.4.422. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Simon H. “Causal ordering and identifiability,” In: Hood WC, Koopmans T, editors. Studies in Econometric Method. New York, NY: Wiley and Sons, Inc; 1953. pp. 49–74. [ Google Scholar ]
• Simon H, Rescher N. “Cause and counterfactual,” Philosophy and Science. 1966; 33 :323–340. doi: 10.1086/288105. [ CrossRef ] [ Google Scholar ]
• Sobel M. “Causal inference in statistical models of the process of socioeconomic achievement,” Sociological Methods & Research. 1998; 27 :318–348. doi: 10.1177/0049124198027002006. [ CrossRef ] [ Google Scholar ]
• Sobel M. “Identification of causal parameters in randomized studies with mediating variables,” Journal of Educational and Behavioral Statistics. 2008; 33 :230–231. doi: 10.3102/1076998607307239. [ CrossRef ] [ Google Scholar ]
• Spirtes P, Glymour C, Scheines R. Causation, Prediction, and Search. New York: Springer-Verlag; 1993. [ Google Scholar ]
• Spirtes P, Glymour C, Scheines R. Causation, Prediction, and Search. 2nd edition Cambridge, MA: MIT Press; 2000. [ Google Scholar ]
• Stock J, Watson M. Introduction to Econometrics. New York: Addison Wesley; 2003. [ Google Scholar ]
• Strotz R, Wold H. “Recursive versus nonrecursive systems: An attempt at synthesis,” Econometrica. 1960; 28 :417–427. doi: 10.2307/1907731. [ CrossRef ] [ Google Scholar ]
• Suppes P. A Probabilistic Theory of Causality. Amsterdam: North-Holland Publishing Co; 1970. [ Google Scholar ]
• Tian J, Paz A, Pearl J.1998 “Finding minimal separating sets,” Technical Report R-254, University of California; Los Angeles, CA [ Google Scholar ]
• Tian J, Pearl J. “Probabilities of causation: Bounds and identification,” Annals of Mathematics and Artificial Intelligence. 2000; 28 :287–313. doi: 10.1023/A:1018912507879. [ CrossRef ] [ Google Scholar ]
• Tian J, Pearl J. Proceedings of the Eighteenth National Conference on Artificial Intelligence. Menlo Park, CA: AAAI Press/The MIT Press; 2002. “A general identification condition for causal effects,” pp. 567–573. [ Google Scholar ]
• VanderWeele T. “Marginal structural models for the estimation of direct and indirect effects,” Epidemiology. 2009; 20 :18–26. doi: 10.1097/EDE.0b013e31818f69ce. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• VanderWeele T, Robins J. “Four types of effect modification: A classification based on directed acyclic graphs,” Epidemiology. 2007; 18 :561–568. doi: 10.1097/EDE.0b013e318127181b. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Verma T, Pearl J.1990 “Equivalence and synthesis of causal models,” Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence Cambridge, MA: 220–227.also in Bonissone P, Henrion M, Kanal LN, Lemmer JF. Uncertainty in Artificial Intelligence 6 Elsevier Science Publishers, B.V. 255–268.1991 [ Google Scholar ]
• Wermuth N.1992 “On block-recursive regression equations,” Brazilian Journal of Probability and Statistics (with discussion) 6 1–56. [ Google Scholar ]
• Wermuth N, Cox D. “Linear dependencies represented by chain graphs,” Statistical Science. 1993; 8 :204–218. doi: 10.1214/ss/1177010887. [ CrossRef ] [ Google Scholar ]
• Whittaker J. Graphical Models in Applied Multivariate Statistics. Chichester, England: John Wiley; 1990. [ Google Scholar ]
• Wilkinson L, the Task Force on Statistical Inference and APA Board of Scientific Affairs “Statistical methods in psychology journals: Guidelines and explanations,” American Psychologist. 1999; 54 :594–604. doi: 10.1037/0003-066X.54.8.594. [ CrossRef ] [ Google Scholar ]
• Woodward J. Making Things Happen. New York, NY: Oxford University Press; 2003. [ Google Scholar ]
• Wooldridge J. Econometric Analysis of Cross Section and Panel Data. Cambridge and London: MIT Press; 2002. [ Google Scholar ]
• Wooldridge J.2009 “Should instrumental variables be used as matching variables?” Technical Report < https://www.msu.edu/~ec/faculty/wooldridge/current%20research/treat1r6.pdf >Michigan State University, MI [ Google Scholar ]
• Wright S. “Correlation and causation,” Journal of Agricultural Research. 1921; 20 :557–585. [ Google Scholar ]

Causal Research (Explanatory research)

Causal research, also known as explanatory research is conducted in order to identify the extent and nature of cause-and-effect relationships. Causal research can be conducted in order to assess impacts of specific changes on existing norms, various processes etc.

Causal studies focus on an analysis of a situation or a specific problem to explain the patterns of relationships between variables. Experiments  are the most popular primary data collection methods in studies with causal research design.

The presence of cause cause-and-effect relationships can be confirmed only if specific causal evidence exists. Causal evidence has three important components:

1. Temporal sequence . The cause must occur before the effect. For example, it would not be appropriate to credit the increase in sales to rebranding efforts if the increase had started before the rebranding.

2. Concomitant variation . The variation must be systematic between the two variables. For example, if a company doesn’t change its employee training and development practices, then changes in customer satisfaction cannot be caused by employee training and development.

3. Nonspurious association . Any covarioaton between a cause and an effect must be true and not simply due to other variable. In other words, there should be no a ‘third’ factor that relates to both, cause, as well as, effect.

The table below compares the main characteristics of causal research to exploratory and descriptive research designs: [1]

Main characteristics of research designs

 Examples of Causal Research (Explanatory Research)

The following are examples of research objectives for causal research design:

• To assess the impacts of foreign direct investment on the levels of economic growth in Taiwan
• To analyse the effects of re-branding initiatives on the levels of customer loyalty
• To identify the nature of impact of work process re-engineering on the levels of employee motivation

Advantages of Causal Research (Explanatory Research)

• Causal studies may play an instrumental role in terms of identifying reasons behind a wide range of processes, as well as, assessing the impacts of changes on existing norms, processes etc.
• Causal studies usually offer the advantages of replication if necessity arises
• This type of studies are associated with greater levels of internal validity due to systematic selection of subjects

Disadvantages of Causal Research (Explanatory Research)

• Coincidences in events may be perceived as cause-and-effect relationships. For example, Punxatawney Phil was able to forecast the duration of winter for five consecutive years, nevertheless, it is just a rodent without intellect and forecasting powers, i.e. it was a coincidence.
• It can be difficult to reach appropriate conclusions on the basis of causal research findings. This is due to the impact of a wide range of factors and variables in social environment. In other words, while casualty can be inferred, it cannot be proved with a high level of certainty.
• It certain cases, while correlation between two variables can be effectively established; identifying which variable is a cause and which one is the impact can be a difficult task to accomplish.

My e-book,  The Ultimate Guide to Writing a Dissertation in Business Studies: a step by step assistance  contains discussions of theory and application of research designs. The e-book also explains all stages of the  research process  starting from the  selection of the research area  to writing personal reflection. Important elements of dissertations such as  research philosophy ,  research approach ,  methods of data collection ,  data analysis  and  sampling  are explained in this e-book in simple words.

John Dudovskiy

[1] Source: Zikmund, W.G., Babin, J., Carr, J. & Griffin, M. (2012) “Business Research Methods: with Qualtrics Printed Access Card” Cengage Learning

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Statistics By Jim

Making statistics intuitive

Causation in Statistics: Hill’s Criteria

By Jim Frost 11 Comments

Causation indicates that an event affects an outcome. Do fatty diets cause heart problems? If you study for a test, does it cause you to get a higher score?

In statistics , causation is a bit tricky. As you’ve no doubt heard, correlation doesn’t necessarily imply causation. An association or correlation between variables simply indicates that the values vary together. It does not necessarily suggest that changes in one variable cause changes in the other variable. Proving causality can be difficult.

If correlation does not prove causation, what statistical test do you use to assess causality? That’s a trick question because no statistical analysis can make that determination. In this post, learn about why you want to determine causation and how to do that.

Relationships and Correlation vs. Causation

The expression is, “correlation does not imply causation.” Consequently, you might think that it applies to things like Pearson’s correlation coefficient . And, it does apply to that statistic. However, we’re really talking about relationships between variables in a broader context. Pearson’s is for two continuous variables . However, a relationship can involve different types of variables such as categorical variables , counts, binary data, and so on.

For example, in a medical experiment, you might have a categorical variable that defines which treatment group subjects belong to—control group, placebo group, and several different treatment groups. If the health outcome is a continuous variable, you can assess the differences between group means. If the means differ by group, then you can say that mean health outcomes depend on the treatment group. There’s a correlation, or relationship, between the type of treatment and health outcome. Or, maybe we have the treatment groups and the outcome is binary, say infected and not infected. In that case, we’d compare group proportions of the infected/not infected between groups to determine whether treatment correlates with infection rates.

Through this post, I’ll refer to correlation and relationships in this broader sense—not just literal correlation coefficients . But relationships between variables, such as differences between group means and proportions, regression coefficients , associations between pairs of categorical variables , and so on.

Why Determining Causality Is Important

If you’re only predicting events, not trying to understand why they happen, and do not want to alter the outcomes, correlation can be perfectly fine. For example, ice cream sales correlate with shark attacks. If you just need to predict the number of shark attacks, ice creams sales might be a good thing to measure even though it’s not causing the shark attacks.

However, if you want to reduce the number of attacks, you’ll need to find something that genuinely causes a change in the attacks. As far as I know, sharks don’t like ice cream!

There are many occasions where you want to affect the outcome. For example, you might want to do the following:

• Improve health by using medicine, exercising, or flu vaccinations .
• Reducing the risk of adverse outcomes, such as procedures for reducing manufacturing defects.
• Improving outcomes, such as studying for a test.

For intentional changes in one variable to affect the outcome variable, there must be a causal relationship between the variables. After all, if studying does not cause an increase in test scores, there’s no point for studying. If the medicine doesn’t cause an improvement in your health or ward off disease, there’s no reason to take it.

Before you can state that some course of action will improve your outcomes, you must be sure that a causal relationship exists between your variables.

Confounding Variables and Their Role in Causation

How does it come to be that variables are correlated but do not have a causal relationship? A common reason is a confounding variable that creates a spurious correlation. A confounding variable correlates with both of your variables of interest. It’s possible that the confounding variable might be the real causal factor ! Let’s go through the ice cream and shark attack example.

In this example, the number of people at the beach is a confounding variable. A confounding variable correlates with both variables of interest—ice cream and shark attacks in our example.

In the diagram below, imagine that as the number of people increases, ice cream sales also tend to increase. In turn, more people at the beach cause shark attacks to increase. The correlation structure creates an apparent, or spurious, correlation between ice cream sales and shark attacks, but it isn’t causation.

Confounders are common reasons for associations between variables that are not causally connected.

Related post : Confounding Variables Can Bias Your Results

Causation and Hypothesis Tests

Before moving on to determining whether a relationship is causal, let’s take a moment to reflect on why statistically significant hypothesis test results do not signify causation.

Hypothesis tests are inferential procedures . They allow you to use relatively small samples to draw conclusions about entire populations. For the topic of causation, we need to understand what statistical significance means.

When you see a relationship in sample data, whether it is a correlation coefficient, a difference between group means, or a regression coefficient, hypothesis tests help you determine whether your sample provides sufficient evidence to conclude that the relationship exists in the population . You can see it in your sample, but you need to know whether it exists in the population. It’s possible that random sampling error (i.e., luck of the draw) produced the “relationship” in your sample.

Statistical significance indicates that you have sufficient evidence to conclude that the relationship you observe in the sample also exists in the population.

That’s it. It doesn’t address causality at all.

Related post : Understanding P-values and Statistical Significance

Hill’s Criteria of Causation

Determining whether a causal relationship exists requires far more in-depth subject area knowledge and contextual information than you can include in a hypothesis test. In 1965, Austin Hill, a medical statistician, tackled this question in a paper* that’s become the standard. While he introduced it in the context of epidemiological research, you can apply the ideas to other fields.

Hill describes nine criteria to help establish causal connections. The goal is to satisfy as many criteria possible. No single criterion is sufficient. However, it’s often impossible to meet all the criteria. These criteria are an exercise in critical thought. They show you how to think about determining causation and highlight essential qualities to consider.

Studies can take steps to increase the strength of their case for a causal relationship, which statisticians call internal validity . To learn more about this, read my post about internal and external validity .

A strong, statistically significant relationship is more likely to be causal. The idea is that causal relationships are likely to produce statistical significance. If you have significant results, at the very least you have reason to believe that the relationship in your sample also exists in the population—which is a good thing. After all, if the relationship only appears in your sample, you don’t have anything meaningful! Correlation still does not imply causation, but a statistically significant relationship is a good starting point.

However, there are many more criteria to satisfy! There’s a critical caveat for this criterion as well. Confounding variables can mask a correlation that actually exists. They can also create the appearance of correlation where causation doesn’t exist, as shown with the ice cream and shark attack example. A strong relationship is simply a hint.

Consistency and causation

When there is a real, causal connection, the result should be repeatable. Other experimenters in other locations should be able to produce the same results. It’s not one and done. Replication builds up confidence that the relationship is causal. Preferably, the replication efforts use other methods, researchers, and locations.

In my post with five tips for using p-values without being misled , I emphasize the need for replication.

Specificity

It’s easier to determine that a relationship is causal if you can rule out other explanations. I write about ruling out other explanations in my posts about randomized experiments and observational studies. In a more general sense, it’s essential to study the literature, consider other plausible hypotheses, and, hopefully, be able to rule them out or otherwise control for them. You need to be sure that what you’re studying is causing the observed change rather than something else of which you’re unaware.

It’s important to note that you don’t need to prove that your variable of interest is the only factor that affects the outcome. For example, smoking causes lung cancer, but it’s not the only thing that causes it. However, you do need to perform experiments that account for other relevant factors and be able to attribute some causation to your variable of interest specifically.

For example, in regression analysis , you control for other factors by including them in the model .

Temporality and causation

Causes should precede effects. Ensure that what you consider to be the cause occurs before the effect . Sometimes it can be challenging to determine which way causality runs. Hill uses the following example. It’s possible that a particular diet leads to an abdominal disease. However, it’s also possible that the disease leads to specific dietary habits.

The Granger Causality Test assesses potential causality by determining whether earlier values in one time series predicts later values in another time series. Analysts say that time series A Granger-causes time series B when significant statistical tests indicate that values in series A predict future values of series B.

Despite being called a “causality test,” it really is only a test of prediction. After all, the increase of Christmas card sales Granger-causes Christmas!

Temporality is just one aspect of causality!

Biological Gradient

Hill was a biologist, hence the focus on biological questions. He suggests that for a genuinely causal relationship, there should be a dose-response type of relationship. If a little bit of exposure causes a little bit of change, a larger exposure should cause more change. Hill uses cigarette smoking and lung cancer as an example—greater amounts of smoking are linked to a greater risk of lung cancer. You can apply the same type of thinking in other fields. Does more studying lead to even higher scores?

However, be aware that the relationship might not remain linear. As the dose increases beyond a threshold, the response can taper off. You can check for this by modeling curvature in regression analysis .

Plausibility

If you can find a plausible mechanism that explains the causal nature of the relationship, it supports the notion of a causal relationship. For example, biologists understand how antibiotics inhibit microbes on a biological level. However, Hill points out that you have to be careful because there are limits to scientific knowledge at any given moment. A causal mechanism might not be known at the time of the study even if one exists. Consequently, Hill says, “we should not demand” that a study meets this requirement.

Coherence and causation

The probability that a relationship is causal is higher when it is consistent with related causal relationships that are generally known and accepted as facts. If your results outright disagree with accepted facts, it’s more likely to be correlation. Assess causality in the broader context of related theory and knowledge.

Experiments and causation

Randomized experiments are the best way to identify causal relationships. Experimenters control the treatment (or factors involved), randomly assign the subjects, and help manage other sources of variation. Hill calls satisfying this criterion the strongest support for causation. However, randomized experiments are not always possible as I write about in my post about observational studies. Learn more about Experimental Design: Definition, Types and Examples .

Related posts : Randomized Experiments and Observational Studies

If there is an accepted, causal relationship that is similar to a relationship in your research, it supports causation for the current study. Hill writes, “With the effects of thalidomide and rubella before us we would surely be ready to accept slighter but similar evidence with another drug or another viral disease in pregnancy.”

Determining whether a correlation also represents causation requires much deliberation. Properly designing experiments and using statistical procedures can help you make that determination. But there are many other factors to consider.

Use your critical thinking and subject-area expertise to think about the big picture. If there is a causal relationship, you’d expect to see consistent results that have been replicated, other causes have been ruled out, the results fit with established theory and other findings, there is a plausible mechanism, and the cause precedes the effect.

Austin Bradford Hill, “The Environment and Disease: Association or Causation?,” Proceedings of the Royal Society of Medicine , 58 (1965), 295-300.

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Reader Interactions

December 2, 2020 at 9:06 pm

I believe there is a logical flaw in the movie “Good Will Hunting”. Specifically, in the scene where psychologist Dr. Sean Maguire (Robin Williams) tells Will (Matt Damon) about the first time he met his wife, there seems to be an implied assumption that if Sean had gone to “the game” (Game 6 of the World Series in 1975), instead of staying at the bar where he had just met his future wife, then the very famous home run hit by Carlton Fisk would still have occurred. I contend that if Sean had gone to the game, the game would have played out completely differently, and the famous home run which actually occurred would not have occurred – that’s not to say that some other famous home run could not have occurred. It seems to be clear that neither characters Sean nor Will understand this – and I contend these two supposedly brilliant people would have known better! It is certainly clear that neither Matt Damon nor Ben Affleck (the writers) understand this. What do you think?

August 24, 2019 at 8:00 pm

Hi Jim Thanks for the great site and content. Being new to statistics I am finding it daunting to understand all of these concepts. I have read most of the articles in the basics section and whilst I am gaining some insights I feel like I need to take a step back in order to move forward. Could you recommend some resources for a rank beginner such as my self? Maybe some books that you read when you where starting out that where useful. I am really keen to jump in and start doing some statistics but I am wondering if it is even possible for someone like me to do so. To clearly define my question where is the best place to start?? I realize this doesn’t really relate to the above article but hopefully this question might be useful to others as well. Thanks.

August 25, 2019 at 2:45 pm

I’m glad that my website has been helpful! I do understand your desire to get the pick picture specifically for starting out. In just about a week, September 3rd to be exact, I’m launching a new ebook that does just that. The book is titled Introduction to Statistics: An Intuitive Guide for Analyzing Data and Unlocking Discoveries . My goal is to provide the big picture about the field of statistics. It covers the basics of data analysis up to larger issues such as using experiments and data to make discoveries.

To be sure that you receive the latest about this book, please subscribe to my email list using the form in the right column of every page in my website.

August 16, 2019 at 12:55 am

Jim , I am new to stats and find ur blog very useful. Yet , I am facing an issue of very low R square values , as low as 1 percent, 3 percent… do we still hold these values valid? Any references on research while accepting such low values . request ur valuable inputs please.

August 17, 2019 at 4:11 pm

Low R-squared can be a problem. It depends on several other factors. Are any independent variables significant? Is the F-test of overall significance significant?

I have posts about this topic and answers those questions. Please read: Low R-squared values and F-test of overall significance .

If you have further questions, please post them in the comments section of the relevant post. It helps keep the questions and answers organized for other readers. Thanks!

June 27, 2019 at 11:23 am

Thank you so much for your website. It has helped me tremendously with my stats, particularly regression. I have a question concerning correlation testing. I have a continuous dependent variable, quality of life, and 3 independent variables, which are categorical (education = 4 levels, marital status = 3 levels, stress = 3 levels). How can I test for a relationship among the dependent and independent variables? Thank you Jim.

June 27, 2019 at 1:30 pm

You can use either ANOVA or OLS regression to assess the relationship between categorical IVs to a continuous DV.

I write about this in my ebook, Regression Analysis: An Intuitive Guide . I recommend you get that ebook to learn about how it works with categorical IVs. I discuss that in detail in the ebook. Unfortunately, I don’t have a blog post to point you towards.

Best of luck with your analysis!

June 25, 2019 at 3:24 pm

great post, Jim. Thanks!

June 25, 2019 at 11:32 am

Useful post

June 24, 2019 at 4:51 am

Very nice and interesting post. And very educational. Many thanks for your efforts!

June 24, 2019 at 10:13 am

Thank you very much! I appreciate the kind words!

Comments and Questions Cancel reply

Bivariate Analysis: Associations, Hypotheses, and Causal Stories

• Open Access
• First Online: 04 October 2022

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• Mark Tessler 2  

Part of the book series: SpringerBriefs in Sociology ((BRIEFSSOCY))

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Every day, we encounter various phenomena that make us question how, why, and with what implications they vary. In responding to these questions, we often begin by considering bivariate relationships, meaning the way that two variables relate to one another. Such relationships are the focus of this chapter.

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3.1 Description, Explanation, and Causal Stories

There are many reasons why we might be interested in the relationship between two variables. Suppose we observe that some of the respondents interviewed in Arab Barometer surveys and other surveys report that they have thought about emigrating, and we are interested in this variable. We may want to know how individuals’ consideration of emigration varies in relation to certain attributes or attitudes. In this case, our goal would be descriptive , sometimes described as the mapping of variance. Our goal may also or instead be explanation , such as when we want to know why individuals have thought about emigrating.

Description

Description means that we seek to increase our knowledge and refine our understanding of a single variable by looking at whether and how it varies in relation to one or more other variables. Descriptive information makes a valuable contribution when the structure and variance of an important phenomenon are not well known, or not well known in relation to other important variables.

Returning to the example about emigration, suppose you notice that among Jordanians interviewed in 2018, 39.5 percent of the 2400 men and women interviewed reported that they have considered the possibility of emigrating.

Our objective may be to discover what these might-be migrants look like and what they are thinking. We do this by mapping the variance of emigration across attributes and orientations that provide some of this descriptive information, with the descriptions themselves each expressed as bivariate relationships. These relationships are also sometimes labeled “associations” or “correlations” since they are not considered causal relationships and are not concerned with explanation.

Of the 39.5 percent of Jordanians who told interviewers that they have considered emigrating, 57.3 percent are men and 42.7 percent are women. With respect to age, 34 percent are age 29 or younger and 19.2 percent are age 50 or older. It might have been expected that a higher percentage of respondents age 29 or younger would have considered emigrating. In fact, however, 56 percent of the 575 men and women in this age category have considered emigrating. And with respect to destination, the Arab country most frequently mentioned by those who have considered emigration is the UAE, named by 17 percent, followed by Qatar at 10 percent and Saudi Arabia at 9.8 percent. Non-Arab destinations were mentioned more frequently, with Turkey named by 18.1 percent, Canada by 21.1 percent, and the U.S. by 24.2 percent.

With the variables sex, age, and prospective destination added to the original variable, which is consideration of emigration, there are clearly more than two variables under consideration. But the variables are described two at a time and so each relationship is bivariate.

These bivariate relationships, between having considered emigration on the one hand and sex, age, and prospective destination on the other, provide descriptive information that is likely to be useful to analysts, policymakers, and others concerned with emigration. They tell, or begin to tell, as noted above, what might-be migrants look like and what they are thinking. Still additional insight may be gained by adding descriptive bivariate relationships for Jordanians interviewed in a different year to those interviewed in 2018. In addition, of course, still more information and possibly a more refined understanding, may be gained by examining the attributes and orientations of prospective emigrants who are citizens of other Arab (and perhaps also non-Arab) countries.

With a focus on description, these bivariate relationships are not constructed to shed light on explanation, that is to contribute to causal stories that seek to account for variance and tell why some individuals but not others have considered the possibility of emigrating. In fact, however, as useful as bivariate relationships that provide descriptive information may be, researchers usually are interested as much if not more in bivariate relationships that express causal stories and purport to provide explanations.

Explanation and Causal Stories

There is a difference in the origins of bivariate relationships that seek to provide descriptive information and those that seek to provide explanatory information. The former can be thought to be responding to what questions: What characterizes potential emigrants? What do they look like? What are their thoughts about this or that subject? If the objective is description, a researcher collects and uses her data to investigate the relationship between two variables without a specific and firm prediction about the relationship between them. Rather, she simply wonders about the “what” questions listed above and believes that finding out the answers will be instructive. In this case, therefore, she selects the bivariate relationships to be considered based on what she thinks it will be useful to know, and not based on assessing the accuracy of a previously articulated causal story that specifies the strength and structure of the effect that one variable has on the other.

A researcher is often interested in causal stories and explanation, however, and this does usually begin with thinking about the relationship between two variables, one of which is the presumed cause and the other the presumed effect. The presumed cause is the independent variable, and the presumed effect is the dependent variable . Offering evidence that there is a strong relationship between two variables is not sufficient to demonstrate that the variables are likely to be causally related, but it is a necessary first step. In this respect it is a point of departure for the fuller, probably multivariate analysis, required to persuasively argue that a relationship is likely to be causal. In addition, as discussed in Chap. 4 , multivariate analysis often not only strengthens the case for inferring that a relationship is causal, but also provides a more elaborate and more instructive causal story. The foundation, however, on which a multivariate analysis aimed at causal inference is built, is a bivariate relationship composed of a presumed independent variable and a presumed dependent variable.

A hypothesis that posits a causal relationship between two variables is not the same as a causal story, although the two are of course closely connected. The former specifies a presumed cause, a presumed determinant of variance on the dependent variable. It probably also specifies the structure of the relationship, such as linear as opposed to non-linear, or positive (also called direct) as opposed to negative (also called inverse).

On the other hand, a causal story describes in more detail what the researcher believes is actually taking place in the relationship between the variables in her hypothesis; and accordingly, why she thinks this involves causality. A causal story provides a fuller account of operative processes, processes that the hypothesis references but does not spell out. These processes may, for example, involve a pathway or a mechanism that tells how it is that the independent variable causes and thus accounts for some of the variance on the dependent variable. Expressed yet another way, the causal story describes the researcher’s understandings, or best guesses, about the real world, understandings that have led her to believe, and then propose for testing, that there is a causal connection between her variables that deserves investigation. The hypothesis itself does not tell this story; it is rather a short formulation that references and calls attention to the existence, or hypothesized existence, of a causal story. Research reports present the causal story as well as the hypothesis, as the hypothesis is often of limited interpretability without the causal story.

A causal story is necessary for causal inference. It enables the researcher to formulate propositions that purport to explain rather than merely describe or predict. There may be a strong relationship between two variables, and if this is the case, it will be possible to predict with relative accuracy the value, or score, of one variable from knowledge of the value, or score, of the other variable. Prediction is not explanation, however. To explain, or attribute causality, there must be a causal story to which a hypothesized causal relationship is calling attention.

An instructive illustration is provided by a recent study of Palestinian participation in protest activities that express opposition to Israeli occupation. Footnote 1 There is plenty of variance on the dependent variable: There are many young Palestinians who take part in these activities, and there are many others who do not take part. Education is one of the independent variables that the researcher thought would be an important determinant of participation, and so she hypothesized that individuals with more education would be more likely to participate in protest activities than individuals with less education.

But why would the researcher think this? The answer is provided by the causal story. To the extent that this as yet untested story is plausible, or preferably, persuasive, at least in the eyes of the investigator, it gives the researcher a reason to believe that education is indeed a determinant of participation in protest activities in Palestine. By spelling out in some detail how and why the hypothesized independent variable, education in this case, very likely impacts a person’s decision about whether or not to protest, the causal story provides a rationale for the researcher’s hypothesis.

In the case of Palestinian participation in protest activities, another investigator offered an insightful causal story about the ways that education pushes toward greater participation, with emphasis on its role in communication and coordination. Footnote 2 Schooling, as the researcher theorizes and subsequently tests, integrates young Palestinians into a broader institutional environment that facilitates mass mobilizations and lowers informational and organizational barriers to collective action. More specifically, she proposes that those individuals who have had at least a middle school education, compared to those who have not finished middle school, have access to better and more reliable sources of information, which, among other things, enables would-be protesters to assess risks. More schooling also makes would-be protesters better able to forge inter-personal relationships and establish networks that share information about needs, opportunities, and risks, and that in this way facilitate engaging in protest activities in groups, rather than on an individual basis. This study offers some additional insights to be discussed later.

The variance motivating the investigation of a causal story may be thought of as the “variable of interest,” and it may be either an independent variable or a dependent variable. It is a variable of interest because the way that it varies poses a question, or puzzle, that a researcher seeks to investigate. It is the dependent variable in a bivariate relationship if the researcher seeks to know why this variable behaves, or varies, as it does, and in pursuit of this objective, she will seek to identify the determinants and drivers that account for this variance. The variable of interest is an independent variable in a particular research project if the researcher seeks to know what difference it makes—on what does its variance have an impact, of what other variable or variables is it a driver or determinant.

The variable in which a researcher is initially interested, that is to say the variable of interest, can also be both a dependent variable and an independent variable. Returning to the variable pertaining to consideration of emigration, but this time with country as the unit of analysis, the variance depicted in Table 3.1 provides an instructive example. The data are based on Arab Barometer surveys conducted in 2018–2019, and the table shows that there is substantial variation across twelve countries. Taking the countries together, the mean percentage of citizens that have thought about relocating to another country is 30.25 percent. But in fact, there is very substantial variation around this mean. Kuwait is an outlier, with only 8 percent having considered emigration. There are also countries in which only 21 percent or 22 percent of the adult population have thought about this, figures that may be high in absolute terms but are low relative to other Arab countries. At the other end of the spectrum are countries in which 45 percent or even 50 percent of the citizens report having considered leaving their country and relocating elsewhere.

The very substantial variance shown in Table 3.1 invites reflection on both the causes and the consequences of this country-level variable, aggregate thinking about emigration. As a dependent variable, the cross-country variance brings the question of why the proportion of citizens that have thought about emigrating is higher in some countries than in others; and the search for an answer begins with the specification of one or more bivariate relationships, each of which links this dependent variable to a possible cause or determinant. As an independent variable, the cross-country variance brings the question of what difference does it make—of what is it a determinant or driver and what are the consequences for a country if more of its citizens, rather than fewer, have thought about moving to another country.

3.2 Hypotheses and Formulating Hypotheses

Hypotheses emerge from the research questions to which a study is devoted. Accordingly, a researcher interested in explanation will have something specific in mind when she decides to hypothesize and then evaluate a bivariate relationship in order to determine whether, and if so how, her variable of interest is related to another variable. For example, if the researcher’s variable of interest is attitude toward gender equality and one of her research questions asks why some people support gender equality and others do not, she might formulate the hypothesis below to see if education provides part of the answer.

Hypothesis 1. Individuals who are better educated are more likely to support gender equality than are individuals who are less well-educated.

The usual case, and the preferred case, is for an investigator to be specific about the research questions she seeks to answer, and then to formulate hypotheses that propose for testing part of the answer to one or more of these questions. Sometimes, however, a researcher will proceed without formulating specific hypotheses based on her research questions. Sometimes she will simply look at whatever relationships between her variable of interest and a second variable her data permit her to identify and examine, and she will then follow up and incorporate into her study any findings that turn out to be significant and potentially instructive. This is sometimes described as allowing the data to “speak.” When this hit or miss strategy of trial and error is used in bivariate and multivariate analysis, findings that are significant and potentially instructive are sometimes described as “grounded theory.” Some researchers also describe the latter process as “inductive” and the former as “deductive.”

Although the inductive, atheoretical approach to data analysis might yield some worthwhile findings that would otherwise have been missed, it can sometimes prove misleading, as you may discover relationships between variables that happened by pure chance and are not instructive about the variable of interest or research question. Data analysis in research aimed at explanation should be, in most cases, preceded by the formulation of one or more hypotheses. In this context, when the focus is on bivariate relationships and the objective is explanation rather than description, each hypothesis will include a dependent variable and an independent variable and make explicit the way the researcher thinks the two are, or probably are, related. As discussed, the dependent variable is the presumed effect; its variance is what a hypothesis seeks to explain. The independent variable is the presumed cause; its impact on the variance of another variable is what the hypothesis seeks to determine.

Hypotheses are usually in the form of if-then, or cause-and-effect, propositions. They posit that if there is variance on the independent variable, the presumed cause, there will then be variance on the dependent variable, the presumed effect. This is because the former impacts the latter and causes it to vary.

An illustration of formulating hypotheses is provided by a study of voting behavior in seven Arab countries: Algeria, Bahrain, Jordan, Lebanon, Morocco, Palestine, and Yemen. Footnote 3 The variable of interest in this individual-level study is electoral turnout, and prominent among the research questions is why some citizens vote and others do not. The dependent variable in the hypotheses proposed in response to this question is whether a person did or did not vote in the country’s most recent parliamentary election. The study initially proposed a number of hypotheses, which include the two listed here and which would later be tested with data from Arab Barometer surveys in the seven countries in 2006–2007. We will return to this illustration later in this chapter.

Hypothesis 1: Individuals who have used clientelist networks in the past are more likely to turn out to vote than are individuals who have not used clientelist networks in the past.

Hypothesis 2: Individuals with a positive evaluation of the economy are more likely to vote than are individuals with a negative evaluation of the economy.

Another example pertaining to voting, which this time is hypothetical but might be instructively tested with Arab Barometer data, considers the relationship between perceived corruption and turning out to vote at the individual level of analysis.

The normal expectation in this case would be that perceptions of corruption influence the likelihood of voting. Even here, however, competing causal relationships are plausible. More perceived corruption might increase the likelihood of voting, presumably to register discontent with those in power. But greater perceived corruption might also actually reduce the likelihood of voting, presumably in this case because the would-be voter sees no chance that her vote will make a difference. But in this hypothetical case, even the direction of the causal connection might be ambiguous. If voting is complicated, cumbersome, and overly bureaucratic, it might be that the experience of voting plays a role in shaping perceptions of corruption. In cases like this, certain variables might be both independent and dependent variables, with causal influence pushing in both directions (often called “endogeneity”), and the researcher will need to carefully think through and be particularly clear about the causal story to which her hypothesis is designed to call attention.

The need to assess the accuracy of these hypotheses, or any others proposed to account for variance on a dependent variable, will guide and shape the researcher’s subsequent decisions about data collection and data analysis. Moreover, in most cases, the finding produced by data analysis is not a statement that the hypothesis is true or that the hypothesis is false. It is rather a statement that the hypothesis is probably true or it is probably false. And more specifically still, when testing a hypothesis with quantitative data, it is often a statement about the odds, or probability, that the researcher will be wrong if she concludes that the hypothesis is correct—if she concludes that the independent variable in the hypothesis is indeed a significant determinant of the variance on the dependent variable. The lower the probability of being wrong, of course, the more confident a researcher can be in concluding, and reporting, that her data and analysis confirm her hypothesis.

Exercise 3.1

Hypotheses emerge from the research questions to which a study is devoted. Thinking about one or more countries with which you are familiar: (a) Identify the independent and dependent variables in each of the example research questions below. (b) Formulate at least one hypothesis for each question. Make sure to include your expectations about the directionality of the relationship between the two variables; is it positive/direct or negative/inverse? (c) In two or three sentences, describe a plausible causal story to which each of your hypotheses might call attention.

Does religiosity affect people’s preference for democracy?

Does preference for democracy affect the likelihood that a person will vote? Footnote 4

Exercise 3.2

Since its establishment in 2006, the Arab Barometer has, as of spring 2022, conducted 68 social and political attitude surveys in the Middle East and North Africa. It has conducted one or more surveys in 16 different Arab countries, and it has recorded the attitudes, values, and preferences of more than 100,000 ordinary citizens.

The Arab Barometer website ( arabbarometer.org ) provides detailed information about the Barometer itself and about the scope, methodology, and conduct of its surveys. Data from the Barometer’s surveys can be downloaded in either SPSS, Stata, or csv format. The website also contains numerous reports, articles, and summaries of findings.

In addition, the Arab Barometer website contains an Online Data Analysis Tool that makes it possible, without downloading any data, to find the distribution of responses to any question asked in any country in any wave. The tool is found in the “Survey Data” menu. After selecting the country and wave of interest, click the “See Results” tab to select the question(s) for which you want to see the response distributions. Click the “Cross by” tab to see the distributions of respondents who differ on one of the available demographic attributes.

The charts below present, in percentages, the response distributions of Jordanians interviewed in 2018 to two questions about gender equality. Below the charts are questions that you are asked to answer. These questions pertain to formulating hypotheses and to the relationship between hypotheses and causal stories.

For each of the two distributions, do you think (hypothesize) that the attitudes of Jordanian women are:

About the same as those of Jordanian men

More favorable toward gender equality than those of Jordanian men

Less favorable toward gender equality than those of Jordanian men

For each of the two distributions, do you think (hypothesize) that the attitudes of younger Jordanians are:

About the same as those of older Jordanians

More favorable toward gender equality than those of older Jordanians

Less favorable toward gender equality than those of older Jordanians

Restate your answers to Questions 1 and 2 as hypotheses.

Give the reasons for your answers to Questions 1 and 2. In two or three sentences, make explicit the presumed causal story on which your hypotheses are based.

Using the Arab Barometer’s Online Analysis Tool, check to see whether your answers to Questions 1 and 2 are correct. For those instances in which an answer is incorrect, suggest in a sentence or two a causal story on which the correct relationship might be based.

In which other country surveyed by the Arab Barometer in 2018 do you think the distributions of responses to the questions about gender equality are very similar to the distributions in Jordan? What attributes of Jordan and the other country informed your selection of the other country?

In which other country surveyed by the Arab Barometer in 2018 do you think the distributions of responses to the questions about gender equality are very different from the distributions in Jordan? What attributes of Jordan and the other country informed your selection of the other country?

Using the Arab Barometer’s Online Analysis Tool, check to see whether your answers to Questions 6 and 7 are correct. For those instances in which an answer is incorrect, suggest in a sentence or two a causal story on which the correct relationship might be based.

We will shortly return to and expand the discussion of probabilities and of hypothesis testing more generally. First, however, some additional discussion of hypothesis formulation is in order. Three important topics will be briefly considered. The first concerns the origins of hypotheses; the second concerns the criteria by which the value of a particular hypothesis or set of hypotheses should be evaluated; and the third, requiring a bit more discussion, concerns the structure of the hypothesized relationship between an independent variable and a dependent variable, or between any two variables that are hypothesized to be related.

Origins of Hypotheses

Where do hypotheses come from? How should an investigator identify independent variables that may account for much, or at least some, of the variance on a dependent variable that she has observed and in which she is interested? Or, how should an investigator identify dependent variables whose variance has been determined, presumably only in part, by an independent variable whose impact she deems it important to assess.

Previous research is one place the investigator may look for ideas that will shape her hypotheses and the associated causal stories. This may include previous hypothesis-testing research, and this can be particularly instructive, but it may also include less systematic and structured observations, reports, and testimonies. The point, very simply, is that the investigator almost certainly is not the first person to think about, and offer information and insight about, the topic and questions in which the researcher herself is interested. Accordingly, attention to what is already known will very likely give the researcher some guidance and ideas as she strives for originality and significance in delineating the relationship between the variables in which she is interested.

Consulting previous research will also enable the researcher to determine what her study will add to what is already known—what it will contribute to the collective and cumulative work of researchers and others who seek to reduce uncertainty about a topic in which they share an interest. Perhaps the researcher’s study will fill an important gap in the scientific literature. Perhaps it will challenge and refine, or perhaps even place in doubt, distributions and explanations of variance that have thus far been accepted. Or perhaps her study will produce findings that shed light on the generalizability or scope conditions of previously accepted variable relationships. It need not do any of these things, but that will be for the researcher to decide, and her decision will be informed by knowledge of what is already known and reflection on whether and in what ways her study should seek to add to that body of knowledge.

Personal experience will also inform the researcher’s search for meaningful and informative hypotheses. It is almost certainly the case that a researcher’s interest in a topic in general, and in questions pertaining to this topic in particular, have been shaped by her own experience. The experience itself may involve many different kinds of connections or interactions, some more professional and work-related and some flowing simply and perhaps unintentionally from lived experience. The hypotheses about voting mentioned earlier, for example, might be informed by elections the researcher has witnessed and/or discussions with friends and colleagues about elections, their turnout, and their fairness. Or perhaps the researcher’s experience in her home country has planted questions about the generalizability of what she has witnessed at home.

All of this is to some extent obvious. But the take-away is that an investigator should not endeavor to set aside what she has learned about a topic in the name of objectivity, but rather, she should embrace whatever personal experience has taught her as she selects and refines the puzzles and propositions she will investigate. Should it happen that her experience leads her to incorrect or perhaps distorted understandings, this will be brought to light when her hypotheses are tested. It is in the testing that objectivity is paramount. In hypothesis formation, by contrast, subjectivity is permissible, and, in fact, it may often be unavoidable.

A final arena in which an investigator may look for ideas that will shape her hypotheses overlaps with personal experience and is also to some extent obvious. This is referenced by terms like creativity and originality and is perhaps best captured by the term “sociological imagination.” The take-away here is that hypotheses that deserve attention and, if confirmed, will provide important insights, may not all be somewhere out in the environment waiting to be found, either in the relevant scholarly literature or in recollections about relevant personal experience. They can and sometimes will be the product of imagination and wondering, of discernments that a researcher may come upon during moments of reflection and deliberation.

As in the case of personal experience, the point to be retained is that hypothesis formation may not only be a process of discovery, of finding the previous research that contains the right information. Hypothesis formation may also be a creative process, a process whereby new insights and proposed original understandings are the product of an investigator’s intellect and sociological imagination.

Crafting Valuable Hypotheses

What are the criteria by which the value of a hypothesis or set of hypotheses should be evaluated? What elements define a good hypothesis? Some of the answers to these questions that come immediately to mind pertain to hypothesis testing rather than hypothesis formation. A good hypothesis, it might be argued, is one that is subsequently confirmed. But whether or not a confirmed hypothesis makes a positive contribution depends on the nature of the hypothesis and goals of the research. It is possible that a researcher will learn as much, and possibly even more, from findings that lead to rejection of a hypothesis. In any event, findings, whatever they may be, are valuable only to the extent that the hypothesis being tested is itself worthy of study.

Two important considerations, albeit somewhat obvious ones, are that a hypothesis should be non-trivial and non-obvious. If a proposition is trivial, suggesting a variable relationship with little or no significance, discovering whether and how the variables it brings together are related will not make a meaningful contribution to knowledge about the determinants and/or impact of the variance at the heart of the researcher’s concern. Few will be interested in findings, however rigorously derived, about a trivial proposition. The same is true of an obvious hypothesis, obvious being an attribute that makes a proposition trivial. As stated, these considerations are themselves somewhat obvious, barely deserving mention. Nevertheless, an investigator should self-consciously reflect on these criteria when formulating hypotheses. She should be sure that she is proposing variable relationships that are neither trivial nor obvious.

A third criterion, also somewhat obvious but nonetheless essential, has to do with the significance and salience of the variables being considered. Will findings from research about these variables be important and valuable, and perhaps also useful? If the primary variable of interest is a dependent variable, meaning that the primary goal of the research is to account for variance, then the significance and salience of the dependent variable will determine the value of the research. Similarly, if the primary variable of interest is an independent variable, meaning that the primary goal of the research is to determine and assess impact, then the significance and salience of the independent variable will determine the value of the research.

These three criteria—non-trivial, non-obvious, and variable importance and salience—are not very different from one another. They collectively mean that the researcher must be able to specify why and how the testing of her hypothesis, or hypotheses, will make a contribution of value. Perhaps her propositions are original or innovative; perhaps knowing whether they are true or false makes a difference or will be of practical benefit; perhaps her findings add something specific and identifiable to the body of existing scholarly literature on the subject. While calling attention to these three connected and overlapping criteria might seem unnecessary since they are indeed somewhat obvious, it remains the case that the value of a hypothesis, regardless of whether or not it is eventually confirmed, is itself important to consider, and an investigator should, therefore, know and be able to articulate the reasons and ways that consideration of her hypothesis, or hypotheses, will indeed be of value.

Hypothesizing the Structure of a Relationship

Relevant in the process of hypothesis formation are, as discussed, questions about the origins of hypotheses and the criteria by which the value of any particular hypothesis or set of hypotheses will be evaluated. Relevant, too, is consideration of the structure of a hypothesized variable relationship and the causal story to which that relationship is believed to call attention.

The point of departure in considering the structure of a hypothesized variable relationship is an understanding that such a relationship may or may not be linear. In a direct, or positive, linear relationship, each increase in the independent variable brings a constant increase in the dependent variable. In an inverse, or negative, linear relationship, each increase in the independent variable brings a constant decrease in the dependent variable. But these are only two of the many ways that an independent variable and a dependent variable may be related, or hypothesized to be related. This is easily illustrated by hypotheses in which level of education or age is the independent variable, and this is relevant in hypothesis formation because the investigator must be alert to and consider the possibility that the variables in which she is interested are in fact related in a non-linear way.

Consider, for example, the relationship between age and support for gender equality, the latter measured by an index based on several questions about the rights and behavior of women that are asked in Arab Barometer surveys. A researcher might expect, and might therefore want to hypothesize, that an increase in age brings increased support for, or alternatively increased opposition to, gender equality. But these are not the only possibilities. Likely, perhaps, is the possibility of a curvilinear relationship, in which case increases in age bring increases in support for gender equality until a person reaches a certain age, maybe 40, 45, or 50, after which additional increases in age bring decreases in support for gender equality. Or the researcher might hypothesize that the curve is in the opposite direction, that support for gender equality initially decreases as a function of age until a particular age is reached, after which additional increases in age bring an increase in support.

Of course, there are also other possibilities. In the case of education and gender equality, for example, increased education may initially have no impact on attitudes toward gender equality. Individuals who have not finished primary school, those who have finished primary school, and those who have gone somewhat beyond primary school and completed a middle school program may all have roughly the same attitudes toward gender equality. Thus, increases in education, within a certain range of educational levels, are not expected to bring an increase or a decrease in support for gender equality. But the level of support for gender equality among high school graduates may be higher and among university graduates may be higher still. Accordingly, in this hypothetical illustration, an increase in education does bring increased support for gender equality but only beginning after middle school.

A middle school level of education is a “floor” in this example. Education does not begin to make a difference until this floor is reached, and thereafter it does make a difference, with increases in education beyond middle school bringing increases in support for gender equality. Another possibility might be for middle school to be a “ceiling.” This would mean that increases in education through middle school would bring increases in support for gender equality, but the trend would not continue beyond middle school. In other words, level of education makes a difference and appears to have explanatory power only until, and so not after, this ceiling is reached. This latter pattern was found in the study of education and Palestinian protest activity discussed earlier. Increases in education through middle school brought increases in the likelihood that an individual would participate in demonstrations and protests of Israeli occupation. However, additional education beyond middle school was not associated with greater likelihood of taking part in protest activities.

This discussion of variation in the structure of a hypothesized relationship between two variables is certainly not exhaustive, and the examples themselves are straightforward and not very complicated. The purpose of the discussion is, therefore, to emphasize that an investigator must be open to and think through the possibility and plausibility of different kinds of relationships between her two variables, that is to say, relationships with different structures. Bivariate relationships with several different kinds of structures are depicted visually by the scatter plots in Fig. 3.4 .

These possibilities with respect to structure do not determine the value of a proposed hypothesis. As discussed earlier, the value of a proposed relationship depends first and foremost on the importance and salience of the variable of interest. Accordingly, a researcher should not assume that the value of a hypothesis varies as a function of the degree to which it posits a complicated variable relationship. More complicated hypotheses are not necessarily better or more correct. But while she should not strive for or give preference to variable relationships that are more complicated simply because they are more complicated, she should, again, be alert to the possibility that a more complicated pattern does a better job of describing the causal connection between the two variables in the place and time in which she is interested.

This brings the discussion of formulating hypotheses back to our earlier account of causal stories. In research concerned with explanation and causality, a hypothesis for the most part is a simplified stand-in for a causal story. It represents the causal story, as it were. Expressing this differently, the hypothesis states the causal story’s “bottom line;” it posits that the independent variable is a determinant of variance on the dependent variable, and it identifies the structure of the presumed relationship between the independent variable and the dependent variable. But it does not describe the interaction between the two variables in a way that tells consumers of the study why the researcher believes that the relationship involves causality rather than an association with no causal implications. This is left to the causal story, which will offer a fuller account of the way the presumed cause impacts the presumed effect.

3.3 Describing and Visually Representing Bivariate Relationships

Once a researcher has collected or otherwise obtained data on the variables in a bivariate relationship she wishes to examine, her first step will be to describe the variance on each of the variables using the univariate statistics described in Chap. 2 . She will need to understand the distribution on each variable before she can understand how these variables vary in relation to one another. This is important whether she is interested in description or wishes to explore a bivariate causal story.

Once she has described each one of the variables, she can turn to the relationship between them. She can prepare and present a visual representation of this relationship, which is the subject of the present section. She can also use bivariate statistical tests to assess the strength and significance of the relationship, which is the subject of the next section of this chapter.

Contingency Tables

Contingency tables are used to display the relationship between two categorical variables. They are similar to the univariate frequency distributions described in Chap. 2 , the difference being that they juxtapose the two univariate distributions and display the interaction between them. Also called cross-tabulation tables, the cells of the table may present frequencies, row percentages, column percentages, and/or total percentages. Total frequencies and/or percentages are displayed in a total row and a total column, each one of which is the same as the univariate distribution of one of the variables taken alone.

Table 3.2 , based on Palestinian data from Wave V of the Arab Barometer, crosses gender and the average number of hours watching television each day. Frequencies are presented in the cells of the table. In the cell showing the number of Palestinian men who do not watch television at all, row percentage, column percentage, and total percentage are also presented. Note that total percentage is based on the 10 cells showing the two variables taken together, which are summed in the lower right-hand cell. Thus, total percent for this cell is 342/2488 = 13.7. Only frequencies are given in the other cells of the table; but in a full table, these four figures – frequency, row percent, column percent and total percent – would be given in every cell.

Exercise 3.3

Compute the row percentage, the column percentage, and the total percentage in the cell showing the number of Palestinian women who do not watch television at all.

Describe the relationship between gender and watching television among Palestinians that is shown in the table. Do the television watching habits of Palestinian men and women appear to be generally similar or fairly different? You might find it helpful to convert the frequencies in other cells to row or column percentages.

Stacked Column Charts and Grouped Bar Charts

Stacked column charts and grouped bar charts are used to visually describe how two categorical variables, or one categorical and one continuous variable, relate to one another. Much like contingency tables, they show the percentage or count of each category of one variable within each category of the second variable. This information is presented in columns stacked on each other or next to each other. The charts below show the number of male Palestinians and the number of female Palestinians who watch television for a given number of hours each day. Each chart presents the same information as the other chart and as the contingency table shown above (Fig. 3.1 ).

Stacked column charts and grouped bar charts comparing Palestinian men and Palestinian women on hours watching television

Box Plots and Box and Whisker Plots

Box plots, box and whisker plots, and other types of plots can also be used to show the relationship between one categorical variable and one continuous variable. They are particularly useful for showing how spread out the data are. Box plots show five important numbers in a variable’s distribution: the minimum value; the median; the maximum value; and the first and third quartiles (Q1 and Q2), which represent, respectively, the number below which are 25 percent of the distribution’s values and the number below which are 75 percent of the distribution’s values. The minimum value is sometimes called the lower extreme, the lower bound, or the lower hinge. The maximum value is sometimes called the upper extreme, the upper bound, or the upper hinge. The middle 50 percent of the distribution, the range between Q1 and Q3 that represents the “box,” constitutes the interquartile range (IQR). In box and whisker plots, the “whiskers” are the short perpendicular lines extending outside the upper and lower quartiles. They are included to indicate variability below Q1 and above Q3. Values are usually categorized as outliers if they are less than Q1 − IQR*1.5 or greater than Q3 + IQR*1.5. A visual explanation of a box and whisker plot is shown in Fig. 3.2a and an example of a box plot that uses actual data is shown in Fig. 3.2b .

The box plot in Fig. 3.2b uses Wave V Arab Barometer data from Tunisia and shows the relationship between age, a continuous variable, and interpersonal trust, a dichotomous categorical variable. The line representing the median value is shown in bold. Interpersonal trust, sometimes known as generalized trust, is an important personal value. Previous research has shown that social harmony and prospects for democracy are greater in societies in which most citizens believe that their fellow citizens for the most part are trustworthy. Although the interpersonal trust variable is dichotomous in Fig. 3.2b , the variance in interpersonal trust can also be measured by a set of ordered categories or a scale that yields a continuous measure, the latter not being suitable for presentation by a box plot. Figure 3.2b shows that the median age of Tunisians who are trusting is slightly higher than the median age of Tunisians who are mistrustful of other people. Notice also that the box plot for the mistrustful group has an outlier.

( a ) A box and whisker plot. ( b ) Box plot comparing the ages of trusting and mistrustful Tunisians in 2018

Line plots may be used to visualize the relationship between two continuous variables or a continuous variable and a categorical variable. They are often used when time, or a variable related to time, is one of the two variables. If a researcher wants to show whether and how a variable changes over time for more than one subgroup of the units about which she has data (looking at men and women separately, for example), she can include multiple lines on the same plot, with each line showing the pattern over time for a different subgroup. These lines will generally be distinguished from each other by color or pattern, with a legend provided for readers.

Line plots are a particularly good way to visualize a relationship if an investigator thinks that important events over time may have had a significant impact. The line plot in Fig. 3.3 shows the average support for gender equality among men and among women in Tunisia from 2013 to 2018. Support for gender equality is a scale based on four questions related to gender equality in the three waves of the Arab Barometer. An answer supportive of gender equality on a question adds +.5 to the scale and an answer unfavorable to gender equality adds −.5 to the scale. Accordingly, a scale score of 2 indicates maximum support for gender equality and a scale score of −2 indicates maximum opposition to gender equality.

Line plot showing level of support for gender equality among Tunisian women and men in 2013, 2016, and 2018

Scatter Plots

Scatter plots are used to visualize a bivariate relationship when both variables are numerical. The independent variable is put on the x-axis, the horizontal axis, and the dependent variable is put on the y-axis, the vertical axis. Each data point becomes a dot in the scatter plot’s two-dimensional field, with its precise location being the point at which its value on the x-axis intersects with its value on the y-axis. The scatter plot shows how the variables are related to one another, including with respect to linearity, direction, and other aspects of structure. The scatter plots in Fig. 3.4 illustrate a strong positive linear relationship, a moderately strong negative linear relationship, a strong non-linear relationship, and a pattern showing no relationship. Footnote 5 If the scatter plot displays no visible and clear pattern, as in the lower left hand plot shown in Fig. 3.4 , the scatter plot would indicate that the independent variable, by itself, has no meaningful impact on the dependent variable.

Scatter plots showing bivariate relationships with different structures

Scatter plots are also a good way to identify outliers—data points that do not follow a pattern that characterizes most of the data. These are also called non-scalar types. Figure 3.5 shows a scatter plot with outliers.

Outliers can be informative, making it possible, for example, to identify the attributes of cases for which the measures of one or both variables are unreliable and/or invalid. Nevertheless, the inclusion of outliers may not only distort the assessment of measures, raising unwarranted doubts about measures that are actually reliable and valid for the vast majority of cases, they may also bias bivariate statistics and make relationships seem weaker than they really are for most cases. For this reason, researchers sometimes remove outliers prior to testing a hypothesis. If one does this, it is important to have a clear definition of what is an outlier and to justify the removal of the outlier, both using the definition and perhaps through substantive analysis. There are several mathematical formulas for identifying outliers, and researchers should be aware of these formulas and their pros and cons if they plan to remove outliers.

If there are relatively few outliers, perhaps no more than 5–10 percent of the cases, it may be justifiable to remove them in order to better discern the relationship between the independent variable and the dependent variable. If outliers are much more numerous, however, it is probably because there is not a significant relationship between the two variables being considered. The researcher might in this case find it instructive to introduce a third variable and disaggregate the data. Disaggregation will be discussed in Chap. 4 .

A scatter plot with outliers marked in red

Exercise 3.4 Exploring Hypotheses through Visualizing Data: Exercise with the Arab Barometer Online Analysis Tool

Go to the Arab Barometer Online Analysis Tool ( https://www.arabbarometer.org/survey-data/data-analysis-tool/ )

Select Wave V and a country that interests you

Select “See Results”

Select “Social, Cultural and Religious topics”

Select “Religion: frequency: pray”

Questions: What does the distribution of this variable look like? How would you describe the variance?

Click on “Cross by,” then

Select “Show all variables”

Select “Kind of government preferable” and click

Select “Options,” then “Show % over Row total,” then “Apply”

Questions: Does there seem to be a relationship between religiosity and preference for democracy? If so, what might explain the relationship you observe—what is a plausible causal story? Is it consistent with the hypothesis you wrote for Exercise 3.1?

What other variables could be used to measure religiosity and preference for democracy? Explore your hypothesis using different items from the list of Arab Barometer variables

Do these distributions support the previous results you found? Do you learn anything additional about the relationship between religiosity and preference for democracy?

Now it is your turn to explore variables and variable relationships that interest you!

Pick two variables that interest you from the list of Arab Barometer variables. Are they continuous or categorical? Ordinal or nominal? (Hint: Most Arab Barometer variables are categorical, even if you might be tempted to think of them as continuous. For example, age is divided into the ordinal categories 18–29, 30–49, and 50 and more.)

Do you expect there to be a relationship between the two variables? If so, what do you think will be the structure of that relationship, and why?

Select the wave (year) and the country that interest you

Select one of your two variables of interest

Click on “Cross by,” and then select your second variable of interest.

On the left side of the page, you’ll see a contingency table. On the right side at the top, you’ll see several options to graphically display the relationship between your two variables. Which type of graph best represents the relationship between your two variables of interest?

Do the two variables seem to be independent of each other, or do you think there might be a relationship between them? Is the relationship you see similar to what you had expected

3.4 Probabilities and Type I and Type II Errors

As in visual presentations of bivariate relationships, selecting the appropriate measure of association or bivariate statistical test depends on the types of the two variables. The data on both variables may be categorical; the data on both may be continuous; or the data may be categorical on one variable and continuous on the other variable. These characteristics of the data will guide the way in which our presentation of these measures and tests is organized. Before briefly describing some specific measures of association and bivariate statistical tests, however, it is necessary to lay a foundation by introducing a number of terms and concepts. Relevant here are the distinction between population and sample and the notions of the null hypothesis, of Type I and Type II errors, and of probabilities and confidence intervals. As concepts, or abstractions, these notions may influence the way a researcher thinks about drawing conclusions about a hypothesis from qualitative data, as was discussed in Chap. 2 . In their precise meaning and application, however, these terms and concepts come into play when hypothesis testing involves the statistical analysis of quantitative data.

To begin, it is important to distinguish between, on the one hand, the population of units—individuals, countries, ethnic groups, political movements, or any other unit of analysis—in which the researcher is interested and about which she aspires to advance conclusions and, on the other hand, the units on which she has actually acquired the data to be analyzed. The latter, the units on which she actually has data, is her sample. In cases where the researcher has collected or obtained data on all of the units in which she is interested, there is no difference between the sample and the population, and drawing conclusions about the population based on the sample is straightforward. Most often, however, a researcher does not possess data on all of the units that make up the population in which she is interested, and so the possibility of error when making inferences about the population based on the analysis of data in the sample requires careful and deliberate consideration.

This concern for error is present regardless of the size of the sample and the way it was constructed. The likelihood of error declines as the size of the sample increases and thus comes closer to representing the full population. It also declines if the sample was constructed in accordance with random or other sampling procedures designed to maximize representation. It is useful to keep these criteria in mind when looking at, and perhaps downloading and using, Arab Barometer data. The Barometer’s website gives information about the construction of each sample. But while it is possible to reduce the likelihood of error when characterizing the population from findings based on the sample, it is not possible to eliminate entirely the possibility of erroneous inference. Accordingly, a researcher must endeavor to make the likelihood of this kind of error as small as possible and then decide if it is small enough to advance conclusions that apply to the population as well as the sample.

The null hypothesis, frequently designated as H0, is a statement to the effect that there is no meaningful and significant relationship between the independent variable and the dependent variable in a hypothesis, or indeed between two variables even if the relationship between them has not been formally specified in a hypothesis and does not purport to be causal or explanatory. The null hypothesis may or may not be stated explicitly by an investigator, but it is nonetheless present in her thinking; it stands in opposition to the hypothesized variable relationship. In a point and counterpoint fashion, the hypothesis, H1, posits that the variables are significantly related, and the null hypothesis, H0, replies and says no, they are not significantly related. It further says that they are not related in any meaningful way, neither in the way proposed in H1 nor in any other way that could be proposed.

Based on her analysis, the researcher needs to determine whether her findings permit rejecting the null hypothesis and concluding that there is indeed a significant relationship between the variables in her hypothesis, concluding in effect that the research hypothesis, H1, has been confirmed. This is most relevant and important when the investigator is basing her analysis on some but not all of the units to which her hypothesis purports to apply—when she is analyzing the data in her sample but seeks to advance conclusions that apply to the population in which she is interested. The logic here is that the findings produced by an analysis of some of the data, the data she actually possesses, may be different than the findings her analysis would hypothetically produce were she able to use data from very many more, or ideally even all, of the units that make up her population of interest.

This means, of course, that there will be uncertainty as the researcher adjudicates between H0 and H1 on the basis of her data. An analysis of these data may suggest that there is a strong and significant relationship between the variables in H1. And the stronger the relationship, the more unlikely it is that the researcher’s sample is a subset of a population characterized by H0 and that, therefore, the researcher may consider H1 to have been confirmed. Yet, it remains at least possible that the researcher’s sample, although it provides strong support for H1, is actually a subset of a population characterized by the null hypothesis. This may be unlikely, but it is not impossible, and so, therefore, to consider H1 to have been confirmed is to run the risk, at least a small risk, of what is known as a Type I error. A Type I error is made when a researcher accepts a research hypothesis that is actually false, when she judges to be true a hypothesis that does not characterize the population of which her sample is a subset. Because of the possibility of a Type I error, even if quite unlikely, researchers will often write something like “We can reject the null hypothesis,” rather than “We can confirm our hypothesis.”

Another analysis related to voter turnout provides a ready illustration. In the Arab Barometer Wave V surveys in 12 Arab countries, Footnote 6 13,899 respondents answered a question about voting in the most recent parliamentary election. Of these, 46.6 percent said they had voted, and the remainder, 53.4 percent, said they had not voted in the last parliamentary election. Footnote 7 Seeking to identify some of the determinants of voting—the attitudes and experiences of an individual that increase the likelihood that she will vote, the researcher might hypothesize that a judgment that the country is going in the right direction will push toward voting. More formally:

H1. An individual who believes that her country is going in the right direction is more likely to vote in a national election than is an individual who believes her country is going in the wrong direction.

Arab Barometer surveys provide data with which to test this proposition, and in fact there is a difference associated with views about the direction in which the country is going. Among those who judged that their country is going in the right direction, 52.4 percent voted in the last parliamentary election. By contrast, among those who judged that their country is going in the wrong direction, only 43.8 percent voted in the last parliamentary election.

This illustrates the choice a researcher faces when deciding what to conclude from a study. Does the analysis of her data from a subset of her population of interest confirm or not confirm her hypothesis? In this example, based on Arab Barometer data, the findings are in the direction of her hypothesis, and differences in voting associated with views about the direction the country is going do not appear to be trivial. But are these differences big enough to justify the conclusion that judgements about the country’s path going forward are a determinant of voting, one among others of course, in the population from which her sample was drawn? In other words, although this relationship clearly characterizes the sample, it is unclear whether it characterizes the researcher’s population of interest, the population from which the sample was drawn.

Unless the researcher can gather data on the entire population of eligible voters, or at least almost all of this population, it is not possible to entirely eliminate uncertainty when the researcher makes inferences about the population of voters based on findings from the subset, or sample, of voters on which she has data. She can either conclude that her findings are sufficiently strong and clear to propose that the pattern she has observed characterizes the population as well, and that H1 is therefore confirmed; or she can conclude that her findings are not strong enough to make such an inference about the population, and that H1, therefore, is not confirmed. Either conclusion could be wrong, and so there is a chance of error no matter which conclusion the researcher advances.

The terms Type I error and Type II error are often used to designate the possible error associated with each of these inferences about the population based on the sample. Type I error refers to the rejection of a true null hypothesis. This means, in other words, that the investigator could be wrong if she concludes that her finding of a strong, or at least fairly strong, relationship between her variables characterizes Arab voters in the 12 countries in general, and if she thus judges H1 to have been confirmed when the population from which her sample was drawn is in fact characterized by H0. Type II error refers to acceptance of a false null hypothesis. This means, in other words, that the investigator could be wrong if she concludes that her finding of a somewhat weak relationship, or no relationship at all, between her variables characterizes Arab voters in the 12 countries in general, and that she thus judges H0 to be true when the population from which her sample was drawn is in fact characterized by H1.

In statistical analyses of quantitative data, decisions about whether to risk a Type I error or a Type II error are usually based on probabilities. More specifically, they are based on the probability of a researcher being wrong if she concludes that the variable relationship—or hypothesis in most cases—that characterizes her data, meaning her sample, also characterizes the population on which the researcher hopes her sample and data will shed light. To say this in yet another way, she computes the odds that her sample does not represent the population of which it is a subset; or more specifically still, she computes the odds that from a population that is characterized by the null hypothesis she could have obtained, by chance alone, a subset of the population, her sample, that is not characterized by the null hypothesis. The lower the odds, or probability, the more willing the researcher will be to risk a Type I error.

There are numerous statistical tests that are used to compute such probabilities. The nature of the data and the goals of the analysis will determine the specific test to be used in a particular situation. Most of these tests, frequently called tests of significance or tests of statistical significance, provide output in the form of probabilities, which always range from 0 to 1. The lower the value, meaning the closer to 0, the less likely it is that a researcher has collected and is working with data that produce findings that differ from what she would find were she to somehow have data on the entire population. Another way to think about this is the following:

If the researcher provisionally assumes that the population is characterized by the null hypothesis with respect to the variable relationship under study, what is the probability of obtaining from that population, by chance alone, a subset or sample that is not characterized by the null hypothesis but instead shows a strong relationship between the two variables;

The lower the probability value, meaning the closer to 0, the less likely it is that the researcher’s data, which support H1, have come from a population that is characterized by H0;

The lower the probability that her sample could have come from a population characterized by H0, the lower the possibility that the researcher will be wrong, that she will make a Type I error, if she rejects the null hypothesis and accepts that the population, as well as her sample, is characterized by H1;

When the probability value is low, the chance of actually making a Type I error is small. But while small, the possibility of an error cannot be entirely eliminated.

If it helps you to think about probability and Type I and Type II error, imagine that you will be flipping a coin 100 times and your goal is to determine whether the coin is unbiased, H0, or biased in favor of either heads or tails, H1. How many times more than 50 would heads have to come up before you would be comfortable concluding that the coin is in fact biased in favor of heads? Would 60 be enough? What about 65? To begin to answer these questions, you would want to know the odds of getting 60 or 65 heads from a coin that is actually unbiased, a coin that would come up heads and come up tails roughly the same number of times if it were flipped many more than 100 times, maybe 1000 times, maybe 10,000. With this many flips, would the ratio of heads to tails even out. The lower the odds, the less likely it is that the coin is unbiased. In this analogy, you can think of the mathematical calculations about an unbiased coin’s odds of getting heads as the population, and your actual flips of the coin as the sample.

But exactly how low does the probability of a Type I error have to be for a researcher to run the risk of rejecting H0 and accepting that her variables are indeed related? This depends, of course, on the implications of being wrong. If there are serious and harmful consequences of being wrong, of accepting a research hypothesis that is actually false, the researcher will reject H0 and accept H1 only if the odds of being wrong, of making a Type I error, are very low.

There are some widely used probability values, which define what are known as “confidence intervals,” that help researchers and those who read their reports to think about the likelihood that a Type I error is being made. In the social sciences, rejecting H0 and running the risk of a Type I error is usually thought to require a probability value of less than .05, written as p < .05. The less stringent value of p < .10 is sometimes accepted as sufficient for rejecting H0, although such a conclusion would be advanced with caution and when the consequences of a Type I error are not very harmful. Frequently considered safer, meaning that the likelihood of accepting a false hypothesis is lower, are p < .01 and p < .001. The next section introduces and briefly describes some of the bivariate statistics that may be used to calculate these probabilities.

3.5 Measures of Association and Bivariate Statistical Tests

The following section introduces some of the bivariate statistical tests that can be used to compute probabilities and test hypotheses. The accounts are not very detailed. They will provide only a general overview and refresher for readers who are already fairly familiar with bivariate statistics. Readers without this familiarity are encouraged to consult a statistics textbook, for which the accounts presented here will provide a useful guide. While the account below will emphasize calculating these test statistics by hand, it is also important to remember that they can be calculated with the assistance of statistical software as well. A discussion of statistical software is available in Appendix 4.

Parametric and Nonparametric Statistics

Parametric and nonparametric are two broad classifications of statistical procedures. A parameter in statistics refers to an attribute of a population. For example, the mean of a population is a parameter. Parametric statistical tests make certain assumptions about the shape of the distribution of values in a population from which a sample is drawn, generally that it is normally distributed, and about its parameters, that is to say the means and standard deviations of the assumed distributions. Nonparametric statistical procedures rely on no or very few assumptions about the shape or parameters of the distribution of the population from which the sample was drawn. Chi-squared is the only nonparametric statistical test among the tests described below.

Degrees of Freedom

Degrees of freedom (df) is the number of values in the calculation of a statistic that are free to vary. Statistical software programs usually give degrees of freedom in the output, so it is generally unnecessary to know the number of the degrees of freedom in advance. It is nonetheless useful to understand what degrees of freedom represent. Consistent with the definition above, it is the number of values that are not predetermined, and thus are free to vary, within the variables used in a statistical test.

This is illustrated by the contingency tables below, which are constructed to examine the relationship between two categorical variables. The marginal row and column totals are known since these are just the univariate distributions of each variable. df = 1 for Table 3.3a , which is a 4-cell table. You can enter any one value in any one cell, but thereafter the values of all the other three cells are determined. Only one number is not free to vary and thus not predetermined. df = 2 for Table 3.3b , which is a 6-cell table. You can enter any two values in any two cells, but thereafter the values of all the other cells are determined. Only two numbers are free to vary and thus not predetermined. For contingency tables, the formula for calculating df is:

Chi-Squared

Chi-squared, frequently written X 2 , is a statistical test used to determine whether two categorical variables are significantly related. As noted, it is a nonparametric test. The most common version of the chi-squared test is the Pearson chi-squared test, which gives a value for the chi-squared statistic and permits determining as well a probability value, or p-value. The magnitude of the statistic and of the probability value are inversely correlated; the higher the value of the chi-squared statistic, the lower the probability value, and thus the lower the risk of making a Type I error—of rejecting a true null hypothesis—when asserting that the two variables are strongly and significantly related.

The simplicity of the chi-squared statistic permits giving a little more detail in order to illustrate several points that apply to bivariate statistical tests in general. The formula for computing chi-squared is given below, with O being the observed (actual) frequency in each cell of a contingency table for two categorical variables and E being the frequency that would be expected in each cell if the two variables are not related. Put differently, the distribution of E values across the cells of the two-variable table constitutes the null hypothesis, and chi-squared provides a number that expresses the magnitude of the difference between an investigator’s actual observed values and the values of E.

The computation of chi-squared involves the following procedures, which are illustrated using the data in Table 3.4 .

The values of O in the cells of the table are based on the data collected by the investigator. For example, Table 3.4 shows that of the 200 women on whom she collected information, 85 are majoring in social science.

The value of E for each cell is computed by multiplying the marginal total of the column in which the cell is located by the marginal total of the row in which the cell is located divided by N, N being the total number of cases. For the female students majoring in social science in Table 3.4 , this is: 200 * 150/400 = 30,000/400 = 75. For the female students majoring in math and natural science in Table 3.4 , this is: 200 * 100/400 = 20,000/400 = 50.

The difference between the value of O and the value of E is computed for each cell using the formula for chi-squared. For the female students majoring in social science in Table 3.4 , this is: (85–75) 2 /75 = 10 2 /75 = 100/75 = 1.33. For the female students majoring in math and natural science, the value resulting from the application of the chi-squared is: (45–50) 2 /50 = 5 2 /75 = 25/75 = .33.

The values in each cell of the table resulting from the application of the chi-squared formula are summed (Σ). This chi-squared value expresses the magnitude of the difference between a distribution of values indicative of the null hypothesis and what the investigator actually found about the relationship between gender and field of study. In Table 3.4 , the cell for female students majoring in social science adds 1.33 to the sum of the values in the eight cells, the cell for female students majoring in math and natural science adds .33 to the sum, and so forth for the remaining six cells.

A final point to be noted, which applies to many other statistical tests as well, is that the application of chi-squared and other bivariate (and multivariate) statistical tests yields a value with which can be computed the probability that an observed pattern does not differ from the null hypothesis and that a Type I error will be made if the null hypothesis is rejected and the research hypothesis is judged to be true. The lower the probability, of course, the lower the likelihood of an error if the null hypothesis is rejected.

Prior to the advent of computer assisted statistical analysis, the value of the statistic and the number of degrees of freedom were used to find the probability value in a table of probability values in an appendix in most statistics books. At present, however, the probability value, or p-value, and also the degrees of freedom, are routinely given as part of the output when analysis is done by one of the available statistical software packages.

Table 3.5 shows the relationship between economic circumstance and trust in the government among 400 ordinary citizens in a hypothetical country. The observed data were collected to test the hypothesis that greater wealth pushes people toward greater trust and less wealth pushes people toward lesser trust. In the case of all three patterns, the probability that the null hypothesis is true is very low. All three patterns have the same high chi-squared value and low probability value. Thus, the chi-squared and p-values show only that the patterns all differ significantly from what would be expected were the null hypothesis true. They do not show whether the data support the hypothesized variable relationship or any other particular relationship.

As the three patterns in Table 3.5 show, variable relationships with very different structures can yield similar or even identical statistical test and probability values, and thus these tests provide only some of the information a researcher needs to draw conclusions about her hypothesis. To draw the right conclusion, it may also be necessary for the investigator to “look at” her data. For example, as Table 3.5 suggests, looking at a tabular or visual presentation of the data may also be needed to draw the proper conclusion about how two variables are related.

How would you describe the three patterns shown in the table, each of which differs significantly from the null hypothesis? Which pattern is consistent with the research hypothesis? How would you describe the other two patterns? Try to visualize a plot of each pattern.

Pearson Correlation Coefficient

The Pearson correlation coefficient, more formally known as the Pearson product-moment correlation, is a parametric measure of linear association. It gives a numerical representation of the strength and direction of the relationship between two continuous numerical variables. The coefficient, which is commonly represented as r , will have a value between −1 and 1. A value of 1 means that there is a perfect positive, or direct, linear relationship between the two variables; as one variable increases, the other variable consistently increases by some amount. A value of −1 means that there is a perfect negative, or inverse, linear relationship; as one variable increases, the other variable consistently decreases by some amount. A value of 0 means that there is no linear relationship; as one variable increases, the other variable neither consistently increases nor consistently decreases.

It is easy to think of relationships that might be assessed by a Pearson correlation coefficient. Consider, for example, the relationship between age and income and the proposition that as age increases, income consistently increases or consistently decreases as well. The closer a coefficient is to 1 or −1, the greater the likelihood that the data on which it is based are not the subset of a population in which age and income are unrelated, meaning that the population of interest is not characterized by the null hypothesis. Coefficients very close to 1 or −1 are rare; although it depends on the number of units on which the researcher has data and also on the nature of the variables. Coefficients higher than .3 or lower than −.03 are frequently high enough, in absolute terms, to yield a low probability value and justify rejecting the null hypothesis. The relationship in this case would be described as “statistically significant.”

Exercise 3.5

Estimating Correlation Coefficients from scatter plots

Look at the scatter plots in Fig. 3.4 and estimate the correlation coefficient that the bivariate relationship shown in each scatter plot would yield.

Explain the basis for each of your estimates of the correlation coefficient.

Spearman’s Rank-Order Correlation Coefficient

The Spearman’s rank-order correlation coefficient is a nonparametric version of the Pearson product-moment correlation . Spearman’s correlation coefficient, (ρ, also signified by r s ) measures the strength and direction of the association between two ranked variables.

Bivariate Regression

Bivariate regression is a parametric measure of association that, like correlation analysis, assesses the strength and direction of the relationship between two variables. Also, like correlation analysis, regression assumes linearity. It may give misleading results if used with variable relationships that are not linear.

Regression is a powerful statistic that is widely used in multivariate analyses. This includes ordinary least squares (OLS) regression, which requires that the dependent variable be continuous and assumes linearity; binary logistic regression, which may be used when the dependent variable is dichotomous; and ordinal logistic regression, which is used with ordinal dependent variables. The use of regression in multivariate analysis will be discussed in the next chapter. In bivariate analysis, regression analysis yields coefficients that indicate the strength and direction of the relationship between two variables. Researchers may opt to “standardize” these coefficients. Standardized coefficients from a bivariate regression are the same as the coefficients produced by Pearson product-moment correlation analysis.

The t-test, also sometimes called a “difference of means” test, is a parametric statistical test that compares the means of two variables and determines whether they are different enough from each other to reject the null hypothesis and risk a Type I error. The dependent variable in a t-test must be continuous or ordinal—otherwise the investigator cannot calculate a mean. The independent variable must be categorical since t-tests are used to compare two groups.

An example, drawing again on Arab Barometer data, tests the relationship between voting and support for democracy. The hypothesis might be that men and women who voted in the last parliamentary election are more likely than men and women who did not vote to believe that democracy is suitable for their country. Whether a person did or did not vote would be the categorical independent variable, and the dependent variable would be the response to a question like, “To what extent do you think democracy is suitable for your country?” The question about democracy asked respondents to situate their views on a 11-point scale, with 0 indicating completely unsuitable and 10 indicating completely suitable.

Focusing on Tunisia in 2018, Arab Barometer Wave V data show that the mean response on the 11-point suitability question is 5.11 for those who voted and 4.77 for those who did not vote. Is this difference of .34 large enough to be statistically significant? A t-test will determine the probability of getting a difference of this magnitude from a population of interest, most likely all Tunisians of voting age, in which there is no difference between voters and non-voters in views about the suitability of democracy for Tunisia. In this example, the t-test showed p < .086. With this p-value, which is higher than the generally accepted standard of .05, a researcher cannot with confidence reject the null hypotheses, and she is unable, therefore, to assert that the proposed relationship has been confirmed.

This question can also be explored at the country level of analysis with, for example, regime type as the independent variable. In this illustration, the hypothesis is that citizens of monarchies are more likely than citizens of republics to believe that democracy is suitable for their country. Of course, a researcher proposing this hypothesis would also advance an associated causal story that provides the rationale for the hypothesis and specifies what is really being tested. To test this proposition, an investigator might merge data from surveys in, say, three monarchies, perhaps Morocco, Jordan, and Kuwait, and then also merge data from surveys in three republics, perhaps Algeria, Egypt, and Iraq. A t-test would then be used to compare the means of people in republics and people in monarchies and give the p-value.

A similar test, the Wilcoxon-Mann-Whitney test, is a nonparametric test that does not require that the dependent variable be normally distributed.

Analysis of variance, or ANOVA, is closely related to the t-test. It may be used when the dependent variable is continuous and the independent variable is categorical. A one-way ANOVA compares the mean and variance values of a continuous dependent variable in two or more categories of a categorical independent variable in order to determine if the latter affects the former.

ANOVA calculates the F-ratio based on the variance between the groups and the variance within each group. The F-ratio can then be used to calculate a p-value. However, if there are more than two categories of the independent variable, the ANOVA test will not indicate which pairs of categories differ enough to be statistically significant, making it necessary, again, to look at the data in order to draw correct conclusions about the structure of the bivariate relationships. Two-way ANOVA is used when an investigator has more than two variables.

Table 3.6 presents a summary list of the visual representations and bivariate statistical tests that have been discussed. It reminds readers of the procedures that can be used when both variables are categorical, when both variables are numerical/continuous, and when one variable is categorical and one variable is numerical/continuous.

Bivariate Statistics and Causal Inference

It is important to remember that bivariate statistical tests only assess the association or correlation between two variables. The tests described above can help a researcher estimate how much confidence her hypothesis deserves and, more specifically, the probability that any significant variable relationships she has found characterize the larger population from which her data were drawn and about which she seeks to offer information and insight.

The finding that two variables in a hypothesized relationship are related to a statistically significant degree is not evidence that the relationship is causal, only that the independent variable is related to the dependent variable. The finding is consistent with the causal story that the hypothesis represents, and to that extent, it offers support for this story. Nevertheless, there are many reasons why an observed statistically significant relationship might be spurious. The correlation might, for example, reflect the influence of one or more other and uncontrolled variables. This will be discussed more fully in the next chapter. The point here is simply that bivariate statistics do not, by themselves, address the question of whether a statistically significant relationship between two variables is or is not a causal relationship.

Only an Introductory Overview

As has been emphasized throughout, this chapter seeks only to offer an introductory overview of the bivariate statistical tests that may be employed when an investigator seeks to assess the relationship between two variables. Additional information will be presented in Chap. 4 . The focus in Chap. 4 will be on multivariate analysis, on analyses involving three or more variables. In this case again, however, the chapter will provide only an introductory overview. The overviews in the present chapter and the next provide a foundation for understanding social statistics, for understanding what statistical analyses involve and what they seek to accomplish. This is important and valuable in and of itself. Nevertheless, researchers and would-be researchers who intend to incorporate statistical analyses into their investigations, perhaps to test hypotheses and decide whether to risk a Type I error or a Type II error, will need to build on this foundation and become familiar with the contents of texts on social statistics. If this guide offers a bird’s eye view, researchers who implement these techniques will also need to expose themselves to the view of the worm at least once.

Chapter 2 makes clear that the concept of variance is central and foundational for much and probably most data-based and quantitative social science research. Bivariate relationships, which are the focus of the present chapter, are building blocks that rest on this foundation. The goal of this kind of research is very often the discovery of causal relationships, relationships that explain rather than merely describe or predict. Such relationships are also frequently described as accounting for variance. This is the focus of Chap. 4 , and it means that there will be, first, a dependent variable, a variable that expresses and captures the variance to be explained, and then, second, an independent variable, and possibly more than one independent variable, that impacts the dependent variable and causes it to vary.

Bivariate relationships are at the center of this enterprise, establishing the empirical pathway leading from the variance discussed in Chap. 2 to the causality discussed in Chap. 4 . Finding that there is a significant relationship between two variables, a statistically significant relationship, is not sufficient to establish causality, to conclude with confidence that one of the variables impacts the other and causes it to vary. But such a finding is necessary.

The goal of social science inquiry that investigates the relationship between two variables is not always explanation. It might be simply to describe and map the way two variables interact with one another. And there is no reason to question the value of such research. But the goal of data-based social science research is very often explanation; and while the inter-relationships between more than two variables will almost always be needed to establish that a relationship is very likely to be causal, these inter-relationships can only be examined by empirics that begin with consideration of a bivariate relationship, a relationship with one variable that is a presumed cause and one variable that is a presumed effect.

Against this background, with the importance of two-variable relationships in mind, the present chapter offers a comprehensive overview of bivariate relationships, including but not only those that are hypothesized to be causally related. The chapter considers the origin and nature of hypotheses that posit a particular relationship between two variables, a causal relationship if the larger goal of the research is explanation and the delineation of a causal story to which the hypothesis calls attention. This chapter then considers how a bivariate relationship might be described and visually represented, and thereafter it discusses how to think about and determine whether the two variables actually are related.

Presenting tables and graphs to show how two variables are related and using bivariate statistics to assess the likelihood that an observed relationship differs significantly from the null hypothesis, the hypothesis of no relationship, will be sufficient if the goal of the research is to learn as much as possible about whether and how two variables are related. And there is plenty of excellent research that has this kind of description as its primary objective, that makes use for purposes of description of the concepts and procedures introduced in this chapter. But there is also plenty of research that seeks to explain, to account for variance, and for this research, use of these concepts and procedures is necessary but not sufficient. For this research, consideration of a two-variable relationship, the focus of the present chapter, is a necessary intermediate step on a pathway that leads from the observation of variance to explaining how and why that variance looks and behaves as it does.

Dana El Kurd. 2019. “Who Protests in Palestine? Mobilization Across Class Under the Palestinian Authority.” In Alaa Tartir and Timothy Seidel, eds. Palestine and Rule of Power: Local Dissent vs. International Governance . New York: Palgrave Macmillan.

Yael Zeira. 2019. The Revolution Within: State Institutions and Unarmed Resistance in Palestine . New York: Cambridge University Press.

Carolina de Miguel, Amaney A. Jamal, and Mark Tessler. 2015. “Elections in the Arab World: Why do citizens turn out?” Comparative Political Studies 48, (11): 1355–1388.

Question 1: Independent variable is religiosity; dependent variable is preference for democracy. Example of hypothesis for Question 1: H1. More religious individuals are more likely than less religious individuals to prefer democracy to other political systems. Question 2: Independent variable is preference for democracy; dependent variable is turning out to vote. Example of hypothesis for Question 2: H2. Individuals who prefer democracy to other political systems are more likely than individuals who do not prefer democracy to other political systems to turn out to vote.

Mike Yi. “A complete Guide to Scatter Plots,” posted October 16, 2019 and seen at https://chartio.com/learn/charts/what-is-a-scatter-plot/

The countries are Algeria, Egypt, Iraq, Jordan, Kuwait, Lebanon, Libya, Morocco, Palestine, Sudan, Tunisia, and Yemen. The Wave V surveys were conducted in 2018–2019.

Not considered in this illustration are the substantial cross-country differences in voter turnout. For example, 63.6 of the Lebanese respondents reported voting, whereas in Algeria the proportion who reported voting was only 20.3 percent. In addition to testing hypotheses about voting in which the individual is the unit of analysis, country could also be the unit of analysis, and hypotheses seeking to account for country-level variance in voting could be formulated and tested.

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Tessler, M. (2023). Bivariate Analysis: Associations, Hypotheses, and Causal Stories. In: Social Science Research in the Arab World and Beyond. SpringerBriefs in Sociology. Springer, Cham. https://doi.org/10.1007/978-3-031-13838-6_3

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Causal Hypothesis

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In scientific research, understanding causality is key to unraveling the intricacies of various phenomena. A causal hypothesis is a statement that predicts a cause-and-effect relationship between variables in a study. It serves as a guide to study design, data collection, and interpretation of results. This thesis statement segment aims to provide you with clear examples of causal hypotheses across diverse fields, along with a step-by-step guide and useful tips for formulating your own. Let’s delve into the essential components of constructing a compelling causal hypothesis.

What is Causal Hypothesis?

A causal hypothesis is a predictive statement that suggests a potential cause-and-effect relationship between two or more variables. It posits that a change in one variable (the independent or cause variable) will result in a change in another variable (the dependent or effect variable). The primary goal of a causal hypothesis is to determine whether one event or factor directly influences another. This type of Simple hypothesis is commonly tested through experiments where one variable can be manipulated to observe the effect on another variable.

What is an example of a Causal Hypothesis Statement?

Example 1: If a person increases their physical activity (cause), then their overall health will improve (effect).

Explanation: Here, the independent variable is the “increase in physical activity,” while the dependent variable is the “improvement in overall health.” The hypothesis suggests that by manipulating the level of physical activity (e.g., by exercising more), there will be a direct effect on the individual’s health.

Other examples can range from the impact of a change in diet on weight loss, the influence of class size on student performance, or the effect of a new training method on employee productivity. The key element in all causal hypotheses is the proposed direct relationship between cause and effect.

100 Causal Hypothesis Statement Examples

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Causal hypotheses predict cause-and-effect relationships, aiming to understand the influence one variable has on another. Rooted in experimental setups, they’re essential for deriving actionable insights in many fields. Delve into these 100 illustrative examples to understand the essence of causal relationships.

• Dietary Sugar & Weight Gain: Increased sugar intake leads to weight gain.
• Exercise & Mental Health: Regular exercise improves mental well-being.
• Sleep & Productivity: Lack of adequate sleep reduces work productivity.
• Class Size & Learning: Smaller class sizes enhance student understanding.
• Smoking & Lung Disease: Regular smoking causes lung diseases.
• Pesticides & Bee Decline: Use of certain pesticides leads to bee population decline.
• Stress & Hair Loss: Chronic stress accelerates hair loss.
• Music & Plant Growth: Plants grow better when exposed to classical music.
• UV Rays & Skin Aging: Excessive exposure to UV rays speeds up skin aging.
• Reading & Vocabulary: Regular reading improves vocabulary breadth.
• Video Games & Reflexes: Playing video games frequently enhances reflex actions.
• Air Pollution & Respiratory Issues: High levels of air pollution increase respiratory diseases.
• Green Spaces & Happiness: Living near green spaces improves overall happiness.
• Yoga & Blood Pressure: Regular yoga practices lower blood pressure.
• Meditation & Stress Reduction: Daily meditation reduces stress levels.
• Social Media & Anxiety: Excessive social media use increases anxiety in teenagers.
• Alcohol & Liver Damage: Regular heavy drinking leads to liver damage.
• Training & Job Efficiency: Intensive training improves job performance.
• Seat Belts & Accident Survival: Using seat belts increases chances of surviving car accidents.
• Soft Drinks & Bone Density: High consumption of soft drinks decreases bone density.
• Homework & Academic Performance: Regular homework completion improves academic scores.
• Organic Food & Health Benefits: Consuming organic food improves overall health.
• Fiber Intake & Digestion: Increased dietary fiber enhances digestion.
• Therapy & Depression Recovery: Regular therapy sessions improve depression recovery rates.
• Financial Education & Savings: Financial literacy education increases personal saving rates.
• Brushing & Dental Health: Brushing teeth twice a day reduces dental issues.
• Carbon Emission & Global Warming: Higher carbon emissions accelerate global warming.
• Afforestation & Climate Stability: Planting trees stabilizes local climates.
• Ad Exposure & Sales: Increased product advertisement boosts sales.
• Parental Involvement & Academic Success: Higher parental involvement enhances student academic performance.
• Hydration & Skin Health: Regular water intake improves skin elasticity and health.
• Caffeine & Alertness: Consuming caffeine increases alertness levels.
• Antibiotics & Bacterial Resistance: Overuse of antibiotics leads to increased antibiotic-resistant bacteria.
• Pet Ownership & Loneliness: Having pets reduces feelings of loneliness.
• Fish Oil & Cognitive Function: Regular consumption of fish oil improves cognitive functions.
• Noise Pollution & Sleep Quality: High levels of noise pollution degrade sleep quality.
• Exercise & Bone Density: Weight-bearing exercises increase bone density.
• Vaccination & Disease Prevention: Proper vaccination reduces the incidence of related diseases.
• Laughter & Immune System: Regular laughter boosts the immune system.
• Gardening & Stress Reduction: Engaging in gardening activities reduces stress levels.
• Travel & Cultural Awareness: Frequent travel increases cultural awareness and tolerance.
• High Heels & Back Pain: Prolonged wearing of high heels leads to increased back pain.
• Junk Food & Heart Disease: Excessive junk food consumption increases the risk of heart diseases.
• Mindfulness & Anxiety Reduction: Practicing mindfulness lowers anxiety levels.
• Online Learning & Flexibility: Online education offers greater flexibility to learners.
• Urbanization & Wildlife Displacement: Rapid urbanization leads to displacement of local wildlife.
• Vitamin C & Cold Recovery: High doses of vitamin C speed up cold recovery.
• Team Building Activities & Work Cohesion: Regular team-building activities improve workplace cohesion.
• Multitasking & Productivity: Multitasking reduces individual task efficiency.
• Protein Intake & Muscle Growth: Increased protein consumption boosts muscle growth in individuals engaged in strength training.
• Mentoring & Career Progression: Having a mentor accelerates career progression.
• Fast Food & Obesity Rates: High consumption of fast food leads to increased obesity rates.
• Deforestation & Biodiversity Loss: Accelerated deforestation results in significant biodiversity loss.
• Language Learning & Cognitive Flexibility: Learning a second language enhances cognitive flexibility.
• Red Wine & Heart Health: Moderate red wine consumption may benefit heart health.
• Public Speaking Practice & Confidence: Regular public speaking practice boosts confidence.
• Fasting & Metabolism: Intermittent fasting can rev up metabolism.
• Plastic Usage & Ocean Pollution: Excessive use of plastics leads to increased ocean pollution.
• Peer Tutoring & Academic Retention: Peer tutoring improves academic retention rates.
• Mobile Usage & Sleep Patterns: Excessive mobile phone use before bed disrupts sleep patterns.
• Green Spaces & Mental Well-being: Living near green spaces enhances mental well-being.
• Organic Foods & Health Outcomes: Consuming organic foods leads to better health outcomes.
• Art Exposure & Creativity: Regular exposure to art boosts creativity.
• Gaming & Hand-Eye Coordination: Engaging in video games improves hand-eye coordination.
• Prenatal Music & Baby’s Development: Exposing babies to music in the womb enhances their auditory development.
• Dark Chocolate & Mood Enhancement: Consuming dark chocolate can elevate mood.
• Urban Farms & Community Engagement: Establishing urban farms promotes community engagement.
• Reading Fiction & Empathy Levels: Reading fiction regularly increases empathy.
• Aerobic Exercise & Memory: Engaging in aerobic exercises sharpens memory.
• Meditation & Blood Pressure: Regular meditation can reduce blood pressure.
• Classical Music & Plant Growth: Plants exposed to classical music show improved growth.
• Pollution & Respiratory Diseases: Higher pollution levels increase respiratory diseases’ incidence.
• Parental Involvement & Child’s Academic Success: Direct parental involvement in schooling enhances children’s academic success.
• Sugar Intake & Tooth Decay: High sugar intake is directly proportional to tooth decay.
• Physical Books & Reading Comprehension: Reading physical books improves comprehension better than digital mediums.
• Daily Journaling & Self-awareness: Maintaining a daily journal enhances self-awareness.
• Robotics Learning & Problem-solving Skills: Engaging in robotics learning fosters problem-solving skills in students.
• Forest Bathing & Stress Relief: Immersion in forest environments (forest bathing) reduces stress levels.
• Reusable Bags & Environmental Impact: Using reusable bags reduces environmental pollution.
• Affirmations & Self-esteem: Regularly reciting positive affirmations enhances self-esteem.
• Local Produce Consumption & Community Economy: Buying and consuming local produce boosts the local economy.
• Sunlight Exposure & Vitamin D Levels: Regular sunlight exposure enhances Vitamin D levels in the body.
• Group Study & Learning Enhancement: Group studies can enhance learning compared to individual studies.
• Active Commuting & Fitness Levels: Commuting by walking or cycling improves overall fitness.
• Foreign Film Watching & Cultural Understanding: Watching foreign films increases understanding and appreciation of different cultures.
• Craft Activities & Fine Motor Skills: Engaging in craft activities enhances fine motor skills.
• Listening to Podcasts & Knowledge Expansion: Regularly listening to educational podcasts broadens one’s knowledge base.
• Outdoor Play & Child’s Physical Development: Encouraging outdoor play accelerates physical development in children.
• Thrift Shopping & Sustainable Living: Choosing thrift shopping promotes sustainable consumption habits.
• Nature Retreats & Burnout Recovery: Taking nature retreats aids in burnout recovery.
• Virtual Reality Training & Skill Acquisition: Using virtual reality for training accelerates skill acquisition in medical students.
• Pet Ownership & Loneliness Reduction: Owning a pet significantly reduces feelings of loneliness among elderly individuals.
• Intermittent Fasting & Metabolism Boost: Practicing intermittent fasting can lead to an increase in metabolic rate.
• Bilingual Education & Cognitive Flexibility: Being educated in a bilingual environment improves cognitive flexibility in children.
• Urbanization & Loss of Biodiversity: Rapid urbanization contributes to a loss of biodiversity in the surrounding environment.
• Recycled Materials & Carbon Footprint Reduction: Utilizing recycled materials in production processes reduces a company’s overall carbon footprint.
• Artificial Sweeteners & Appetite Increase: Consuming artificial sweeteners might lead to an increase in appetite.
• Green Roofs & Urban Temperature Regulation: Implementing green roofs in urban buildings contributes to moderating city temperatures.
• Remote Work & Employee Productivity: Adopting a remote work model can boost employee productivity and job satisfaction.
• Sensory Play & Child Development: Incorporating sensory play in early childhood education supports holistic child development.

Causal Hypothesis Statement Examples in Research

Research hypothesis often delves into understanding the cause-and-effect relationships between different variables. These causal hypotheses attempt to predict a specific effect if a particular cause is present, making them vital for experimental designs.

• Artificial Intelligence & Job Market: Implementation of artificial intelligence in industries causes a decline in manual jobs.
• Online Learning Platforms & Traditional Classroom Efficiency: The introduction of online learning platforms reduces the efficacy of traditional classroom teaching methods.
• Nano-technology & Medical Treatment Efficacy: Using nano-technology in drug delivery enhances the effectiveness of medical treatments.
• Genetic Editing & Lifespan: Advancements in genetic editing techniques directly influence the lifespan of organisms.
• Quantum Computing & Data Security: The rise of quantum computing threatens the security of traditional encryption methods.
• Space Tourism & Aerospace Advancements: The demand for space tourism propels advancements in aerospace engineering.
• E-commerce & Retail Business Model: The surge in e-commerce platforms leads to a decline in the traditional retail business model.
• VR in Real Estate & Buyer Decisions: Using virtual reality in real estate presentations influences buyer decisions more than traditional methods.
• Biofuels & Greenhouse Gas Emissions: Increasing biofuel production directly reduces greenhouse gas emissions.
• Crowdfunding & Entrepreneurial Success: The availability of crowdfunding platforms boosts the success rate of start-up enterprises.

Causal Hypothesis Statement Examples in Epidemiology

Epidemiology is a study of how and why certain diseases occur in particular populations. Causal hypotheses in this field aim to uncover relationships between health interventions, behaviors, and health outcomes.

• Vaccine Introduction & Disease Eradication: The introduction of new vaccines directly leads to the reduction or eradication of specific diseases.
• Urbanization & Rise in Respiratory Diseases: Increased urbanization causes a surge in respiratory diseases due to pollution.
• Processed Foods & Obesity Epidemic: The consumption of processed foods is directly linked to the rising obesity epidemic.
• Sanitation Measures & Cholera Outbreaks: Implementing proper sanitation measures reduces the incidence of cholera outbreaks.
• Tobacco Consumption & Lung Cancer: Prolonged tobacco consumption is the primary cause of lung cancer among adults.
• Antibiotic Misuse & Antibiotic-Resistant Strains: Misuse of antibiotics leads to the evolution of antibiotic-resistant bacterial strains.
• Alcohol Consumption & Liver Diseases: Excessive and regular alcohol consumption is a leading cause of liver diseases.
• Vitamin D & Rickets in Children: A deficiency in vitamin D is the primary cause of rickets in children.
• Airborne Pollutants & Asthma Attacks: Exposure to airborne pollutants directly triggers asthma attacks in susceptible individuals.
• Sedentary Lifestyle & Cardiovascular Diseases: Leading a sedentary lifestyle is a significant risk factor for cardiovascular diseases.

Causal Hypothesis Statement Examples in Psychology

In psychology, causal hypotheses explore how certain behaviors, conditions, or interventions might influence mental and emotional outcomes. These hypotheses help in deciphering the intricate web of human behavior and cognition.

• Childhood Trauma & Personality Disorders: Experiencing trauma during childhood increases the risk of developing personality disorders in adulthood.
• Positive Reinforcement & Skill Acquisition: The use of positive reinforcement accelerates skill acquisition in children.
• Sleep Deprivation & Cognitive Performance: Lack of adequate sleep impairs cognitive performance in adults.
• Social Isolation & Depression: Prolonged social isolation is a significant cause of depression among teenagers.
• Mindfulness Meditation & Stress Reduction: Regular practice of mindfulness meditation reduces symptoms of stress and anxiety.
• Peer Pressure & Adolescent Risk Taking: Peer pressure significantly increases risk-taking behaviors among adolescents.
• Parenting Styles & Child’s Self-esteem: Authoritarian parenting styles negatively impact a child’s self-esteem.
• Multitasking & Attention Span: Engaging in multitasking frequently leads to a reduced attention span.
• Childhood Bullying & Adult PTSD: Individuals bullied during childhood have a higher likelihood of developing PTSD as adults.
• Digital Screen Time & Child Development: Excessive digital screen time impairs cognitive and social development in children.

Causal Inference Hypothesis Statement Examples

Causal inference is about deducing the cause-effect relationship between two variables after considering potential confounders. These hypotheses aim to find direct relationships even when other influencing factors are present.

• Dietary Habits & Chronic Illnesses: Even when considering genetic factors, unhealthy dietary habits increase the chances of chronic illnesses.
• Exercise & Mental Well-being: When accounting for daily stressors, regular exercise improves mental well-being.
• Job Satisfaction & Employee Turnover: Even when considering market conditions, job satisfaction inversely relates to employee turnover.
• Financial Literacy & Savings Behavior: When considering income levels, financial literacy is directly linked to better savings behavior.
• Online Reviews & Product Sales: Even accounting for advertising spends, positive online reviews boost product sales.
• Prenatal Care & Child Health Outcomes: When considering genetic factors, adequate prenatal care ensures better health outcomes for children.
• Teacher Qualifications & Student Performance: Accounting for socio-economic factors, teacher qualifications directly influence student performance.
• Community Engagement & Crime Rates: When considering economic conditions, higher community engagement leads to lower crime rates.
• Eco-friendly Practices & Brand Loyalty: Accounting for product quality, eco-friendly business practices boost brand loyalty.
• Mental Health Support & Workplace Productivity: Even when considering workload, providing mental health support enhances workplace productivity.

What are the Characteristics of Causal Hypothesis

Causal hypotheses are foundational in many research disciplines, as they predict a cause-and-effect relationship between variables. Their unique characteristics include:

• Cause-and-Effect Relationship: The core of a causal hypothesis is to establish a direct relationship, indicating that one variable (the cause) will bring about a change in another variable (the effect).
• Testability: They are formulated in a manner that allows them to be empirically tested using appropriate experimental or observational methods.
• Specificity: Causal hypotheses should be specific, delineating clear cause and effect variables.
• Directionality: They typically demonstrate a clear direction in which the cause leads to the effect.
• Operational Definitions: They often use operational definitions, which specify the procedures used to measure or manipulate variables.
• Temporal Precedence: The cause (independent variable) always precedes the effect (dependent variable) in time.

What is a causal hypothesis in research?

In research, a causal hypothesis is a statement about the expected relationship between variables, or explanation of an occurrence, that is clear, specific, testable, and falsifiable. It suggests a relationship in which a change in one variable is the direct cause of a change in another variable. For instance, “A higher intake of Vitamin C reduces the risk of common cold.” Here, Vitamin C intake is the independent variable, and the risk of common cold is the dependent variable.

What is the difference between causal and descriptive hypothesis?

• Causal Hypothesis: Predicts a cause-and-effect relationship between two or more variables.
• Descriptive Hypothesis: Describes an occurrence, detailing the characteristics or form of a particular phenomenon.
• Causal: Consuming too much sugar can lead to diabetes.
• Descriptive: 60% of adults in the city exercise at least thrice a week.
• Causal: To establish a causal connection between variables.
• Descriptive: To give an accurate portrayal of the situation or fact.
• Causal: Often involves experiments.
• Descriptive: Often involves surveys or observational studies.

How do you write a Causal Hypothesis? – A Step by Step Guide

• Identify Your Variables: Pinpoint the cause (independent variable) and the effect (dependent variable). For instance, in studying the relationship between smoking and lung health, smoking is the independent variable while lung health is the dependent variable.
• State the Relationship: Clearly define how one variable affects another. Does an increase in the independent variable lead to an increase or decrease in the dependent variable?
• Be Specific: Avoid vague terms. Instead of saying “improved health,” specify the type of improvement like “reduced risk of cardiovascular diseases.”
• Use Operational Definitions: Clearly define any terms or variables in your hypothesis. For instance, define what you mean by “regular exercise” or “high sugar intake.”
• Ensure It’s Testable: Your hypothesis should be structured so that it can be disproven or supported by data.
• Review Existing Literature: Check previous research to ensure that your hypothesis hasn’t already been tested, and to ensure it’s plausible based on existing knowledge.
• Draft Your Hypothesis: Combine all the above steps to write a clear, concise hypothesis. For instance: “Regular exercise (defined as 150 minutes of moderate exercise per week) decreases the risk of cardiovascular diseases.”

Tips for Writing Causal Hypothesis

• Simplicity is Key: The clearer and more concise your hypothesis, the easier it will be to test.
• Avoid Absolutes: Using words like “all” or “always” can be problematic. Few things are universally true.
• Seek Feedback: Before finalizing your hypothesis, get feedback from peers or mentors.
• Stay Objective: Base your hypothesis on existing literature and knowledge, not on personal beliefs or biases.
• Revise as Needed: As you delve deeper into your research, you may find the need to refine your hypothesis for clarity or specificity.
• Falsifiability: Always ensure your hypothesis can be proven wrong. If it can’t be disproven, it can’t be validated either.
• Avoid Circular Reasoning: Ensure that your hypothesis doesn’t assume what it’s trying to prove. For example, “People who are happy have a positive outlook on life” is a circular statement.
• Specify Direction: In causal hypotheses, indicating the direction of the relationship can be beneficial, such as “increases,” “decreases,” or “leads to.”

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Analysing near-miss incidents in construction: a systematic literature review.

1. Introduction

• Q 1 —Are near-miss events in construction industry the subject of scientific research?
• Q 2 —What methods have been employed thus far to obtain information on near misses and systems for recording incidents in construction companies?
• Q 3 —What methods have been used to analyse the information and figures obtained?
• Q 4 —What are the key aspects of near misses in the construction industry that have been of interest to the researchers?

2. Definition of Near-Miss Events

3. research methodology, 4.1. a statistical analysis of publications, 4.2. methods used to obtain information about near misses, 4.2.1. traditional methods.

• Traditional registration forms
• Computerized systems for the recording of events
• Surveys and interviews

4.2.2. Real-Time Monitoring Systems

• Employee-tracking systems
• Video surveillance systems
• Wearable technology
• Motion sensors

4.3. Methods Used to Analyse the Information and Figures That Have Been Obtained

4.3.1. quantitative and qualitative statistical methods, 4.3.2. analysis using artificial intelligence (ai), 4.3.3. building information modelling, 4.4. key aspects of near-miss investigations in the construction industry, 4.4.1. occupational risk assessment, 4.4.2. causes of hazards in construction, 4.4.3. time series of near misses, 4.4.4. material factors of construction processes, 4.5. a comprehensive overview of the research questions and references on near misses in the construction industry, 5. discussion, 5.1. interest of researchers in near misses in construction (question 1), 5.2. methods used to obtain near-miss information (question 2), 5.3. methods used to analyse the information and data sets (question 3), 5.4. key aspects of near-miss investigations in the construction industry (question 4), 6. conclusions.

• A quantitative analysis of the Q 1 question has revealed a positive trend, namely that there is a growing interest among researchers in studying near misses in construction. The greatest interest in NM topics is observed in the United States of America, China, the United Kingdom, Australia, Hong Kong, and Germany. Additionally, there has been a recent emergence of interest in Poland. The majority of articles are mainly published in journals such as Safety Science (10), Journal of Construction Engineering and Management (8), and Automation in Construction (5);
• The analysis of question Q 2 illustrates that traditional paper-based event registration systems are currently being superseded by advanced IT systems. However, both traditional and advanced systems are subject to the disadvantage of relying on employee-reported data, which introduces a significant degree of uncertainty regarding in the quality of the information provided. A substantial proportion of the data and findings presented in the studies was obtained through surveys and interviews. The implementation of real-time monitoring systems is becoming increasingly prevalent in construction sites. The objective of such systems is to provide immediate alerts in the event of potential hazards, thereby preventing a significant number of near misses. Real-time monitoring systems employ a range of technologies, including ultrasonic technology, radio frequency identification (RFID), inertial measurement units (IMUs), real-time location systems (RTLSs), industrial cameras, wearable technology, motion sensors, and advanced IT technologies, among others;
• The analysis of acquired near-miss data is primarily conducted through the utilisation of quantitative and qualitative statistical methods, as evidenced by the examination of the Q 3 question. In recent years, research utilising artificial intelligence (AI) has made significant advances. The most commonly employed artificial intelligence techniques include text mining, machine learning, and artificial neural networks. The growing deployment of Building Information Modelling (BIM) technology has precipitated a profound transformation in the safety management of construction sites, with the advent of sophisticated tools for the identification and management of hazardous occurrences;
• In response to question Q 4 , the study of near misses in the construction industry has identified several key aspects that have attracted the attention of researchers. These include the utilisation of both quantitative and qualitative methodologies for risk assessment, the analysis of the causes of hazards, the identification of accident precursors through the creation of time series, and the examination of material factors pertaining to construction processes. Researchers are focusing on the utilisation of both databases and advanced technologies, such as real-time location tracking, for the assessment and analysis of occupational risks. Techniques such as Analytic Hierarchy Process (AHP) and clustering facilitate a comprehensive assessment and categorisation of incidents, thereby enabling the identification of patterns and susceptibility to specific types of accidents. Moreover, the impact of a company’s safety climate and organisational culture on the frequency and characteristics of near misses represents a pivotal area of investigation. The findings of this research indicate that effective safety management requires a holistic approach that integrates technology, risk management and safety culture, with the objective of reducing accidents and enhancing overall working conditions on construction sites.

7. Gaps and Future Research Directions, Limitations

• Given the diversity and variability of construction sites and the changing conditions and circumstances of work, it is essential to create homogeneous clusters of near misses and to analyse the phenomena within these clusters. The formation of such clusters may be contingent upon the direct causes of the events in question;
• Given the inherently dynamic nature of construction, it is essential to analyse time series of events that indicate trends in development and safety levels. The numerical characteristics of these trends may be used to construct predictive models for future accidents and near misses;
• The authors have identified potential avenues for future research, which could involve the development of mathematical models using techniques such as linear regression, artificial intelligence, and machine learning. The objective of these models is to predict the probable timing of occupational accidents within defined incident categories, utilising data from near misses. Moreover, efforts are being made to gain access to the hazardous incident recording systems of different construction companies, with a view to facilitating comparison of the resulting data;
• One significant limitation of near-miss research is the lack of an integrated database that encompasses a diverse range of construction sites and construction work. A data resource of this nature would be of immense value for the purpose of conducting comprehensive analyses and formulating effective risk management strategies. This issue can be attributed to two factors: firstly, the reluctance of company managers to share their databases with researchers specialising in risk assessment, and secondly, the reluctance of employees to report near-miss incidents. Such actions may result in adverse consequences for employees, including disciplinary action or negative perceptions from managers. This consequently results in the recording of only a subset of incidents, thereby distorting the true picture of safety on the site.

Author Contributions

Institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

• Occupational Risk|Safety and Health at Work EU-OSHA. Available online: https://osha.europa.eu/en/tools-and-resources/eu-osha-thesaurus/term/70194i (accessed on 28 June 2023).
• Guo, S.; Zhou, X.; Tang, B.; Gong, P. Exploring the Behavioral Risk Chains of Accidents Using Complex Network Theory in the Construction Industry. Phys. A Stat. Mech. Its Appl. 2020 , 560 , 125012. [ Google Scholar ] [ CrossRef ]
• Woźniak, Z.; Hoła, B. The Structure of near Misses and Occupational Accidents in the Polish Construction Industry. Heliyon 2024 , 10 , e26410. [ Google Scholar ] [ CrossRef ]
• Li, X.; Sun, W.; Fu, H.; Bu, Q.; Zhang, Z.; Huang, J.; Zang, D.; Sun, Y.; Ma, Y.; Wang, R.; et al. Schedule Risk Model of Water Intake Tunnel Construction Considering Mood Factors and Its Application. Sci. Rep. 2024 , 14 , 3857. [ Google Scholar ] [ CrossRef ]
• Li, X.; Huang, J.; Li, C.; Luo, N.; Lei, W.; Fan, H.; Sun, Y.; Chen, W. Study on Construction Resource Optimization and Uncertain Risk of Urban Sewage Pipe Network. Period. Polytech. Civ. Eng. 2022 , 66 , 335–343. [ Google Scholar ] [ CrossRef ]
• Central Statistical Office Central Statistical Office/Thematic Areas/Labor Market/Working Conditions/Accidents at Work/Accidents at Work in the 1st Quarter of 2024. Available online: https://stat.gov.pl/obszary-tematyczne/rynek-pracy/warunki-pracy-wypadki-przy-pracy/wypadki-przy-pracy-w-1-kwartale-2024-roku,3,55.html (accessed on 17 July 2024).
• Manzo, J. The $ 5 Billion Cost of Construction Fatalities in the United States: A 50 State Comparison ; The Midwest Economic Policy Institute (MEPI): Saint Paul, MN, USA, 2017. [ Google Scholar ]
• Sousa, V.; Almeida, N.M.; Dias, L.A. Risk-Based Management of Occupational Safety and Health in the Construction Industry—Part 1: Background Knowledge. Saf. Sci. 2014 , 66 , 75–86. [ Google Scholar ] [ CrossRef ]
• Amirah, N.A.; Him, N.F.N.; Rashid, A.; Rasheed, R.; Zaliha, T.N.; Afthanorhan, A. Fostering a Safety Culture in Manufacturing through Safety Behavior: A Structural Equation Modelling Approach. J. Saf. Sustain. 2024; in press . [ Google Scholar ] [ CrossRef ]
• Heinrich, H.W. Industrial Accident Prevention ; A Scientific Approach; McGraw-Hill: New York, NY, USA, 1931. [ Google Scholar ]
• Near Miss Definition Per OSHA—What Is a Near Miss? Available online: https://safetystage.com/osha-compliance/near-miss-definition-osha/ (accessed on 17 August 2024).
• Cambraia, F.B.; Saurin, T.A.; Formoso, C.T. Identification, Analysis and Dissemination of Information on near Misses: A Case Study in the Construction Industry. Saf. Sci. 2010 , 48 , 91–99. [ Google Scholar ] [ CrossRef ]
• Tan, J.; Li, M. How to Achieve Accurate Accountability under Current Administrative Accountability System for Work Safety Accidents in Chemical Industry in China: A Case Study on Major Work Safety Accidents during 2010–2020. J. Chin. Hum. Resour. Manag. 2022 , 13 , 26–40. [ Google Scholar ] [ CrossRef ]
• Wu, W.; Gibb, A.G.F.; Li, Q. Accident Precursors and near Misses on Construction Sites: An Investigative Tool to Derive Information from Accident Databases. Saf. Sci. 2010 , 48 , 845–858. [ Google Scholar ] [ CrossRef ]
• Janicak, C.A. Fall-Related Deaths in the Construction Industry. J. Saf. Res. 1998 , 29 , 35–42. [ Google Scholar ] [ CrossRef ]
• Li, H.; Yang, X.; Wang, F.; Rose, T.; Chan, G.; Dong, S. Stochastic State Sequence Model to Predict Construction Site Safety States through Real-Time Location Systems. Saf. Sci. 2016 , 84 , 78–87. [ Google Scholar ] [ CrossRef ]
• Yang, K.; Aria, S.; Ahn, C.R.; Stentz, T.L. Automated Detection of Near-Miss Fall Incidents in Iron Workers Using Inertial Measurement Units. In Proceedings of the Construction Research Congress 2014: Construction in a Global Network, Atlanta, GA, USA, 19–21 May 2014; pp. 935–944. [ Google Scholar ] [ CrossRef ]
• Raviv, G.; Fishbain, B.; Shapira, A. Analyzing Risk Factors in Crane-Related near-Miss and Accident Reports. Saf. Sci. 2017 , 91 , 192–205. [ Google Scholar ] [ CrossRef ]
• Zhao, X.; Zhang, M.; Cao, T. A Study of Using Smartphone to Detect and Identify Construction Workers’ near-Miss Falls Based on ANN. In Proceedings of the Nondestructive Characterization and Monitoring of Advanced Materials, Aerospace, Civil Infrastructure, and Transportation XII, Denver, CO, USA, 4–8 March 2018; p. 80. [ Google Scholar ] [ CrossRef ]
• Santiago, K.; Yang, X.; Ruano-Herreria, E.C.; Chalmers, J.; Cavicchia, P.; Caban-Martinez, A.J. Characterising near Misses and Injuries in the Temporary Agency Construction Workforce: Qualitative Study Approach. Occup. Environ. Med. 2020 , 77 , 94–99. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• What Is OSHA’s Definition of a Near Miss. Available online: https://www.osha.com/blog/near-miss-definition (accessed on 4 August 2023).
• Martins, I. Investigation of Occupational Accidents and Diseases a Practical Guide for Labour Inspectors ; International Labour Office: Geneva, Switzerland, 2015. [ Google Scholar ]
• National Safety Council. Near Miss Reporting Systems ; National Safety Council: Singapore, 2013. [ Google Scholar ]
• PKN PN-ISO 45001:2018-06 ; Occupational Health and Safety Management Systems—Requirements with Guidance for Use. CRC Press: Boca Raton, FL, USA, 2019.
• PKN PN-N-18001:2004 ; Occupational Health and Safety Management Systems—Requirements. CRC Press: Boca Raton, FL, USA, 2004.
• World Health Organisation. WHO Draft GuiDelines for Adverse Event Reporting and Learning Systems ; World Health Organisation: Geneva, Switzerland, 2005. [ Google Scholar ]
• International Atomic Energy Agency IAEA Satety Glossary. Terminology Used in Nuclear Safety and Radiation Protection: 2007 Edition ; International Atomic Energy Agency: Vienna, Austria, 2007. [ Google Scholar ]
• Marks, E.; Teizer, J.; Hinze, J. Near Miss Reporting Program to Enhance Construction Worker Safety Performance. In Proceedings of the Construction Research Congress 2014: Construction in a Global Network, Atlanta, GA, USA, 19 May 2014; pp. 2315–2324. [ Google Scholar ] [ CrossRef ]
• Gnoni, M.G.; Saleh, J.H. Near-Miss Management Systems and Observability-in-Depth: Handling Safety Incidents and Accident Precursors in Light of Safety Principles. Saf. Sci. 2017 , 91 , 154–167. [ Google Scholar ] [ CrossRef ]
• Thoroman, B.; Goode, N.; Salmon, P. System Thinking Applied to near Misses: A Review of Industry-Wide near Miss Reporting Systems. Theor. Issues Ergon. Sci. 2018 , 19 , 712–737. [ Google Scholar ] [ CrossRef ]
• Gnoni, M.G.; Tornese, F.; Guglielmi, A.; Pellicci, M.; Campo, G.; De Merich, D. Near Miss Management Systems in the Industrial Sector: A Literature Review. Saf. Sci. 2022 , 150 , 105704. [ Google Scholar ] [ CrossRef ]
• Bird, F. Management Guide to Loss Control ; Loss Control Publications: Houston, TX, USA, 1975. [ Google Scholar ]
• Zimmermann. Bauer International Norms and Identity ; Zimmermann: Sydney, NSW, Australia, 2006; pp. 5–21. [ Google Scholar ]
• Arslan, M.; Cruz, C.; Ginhac, D. Semantic Trajectory Insights for Worker Safety in Dynamic Environments. Autom. Constr. 2019 , 106 , 102854. [ Google Scholar ] [ CrossRef ]
• Arslan, M.; Cruz, C.; Ginhac, D. Visualizing Intrusions in Dynamic Building Environments for Worker Safety. Saf. Sci. 2019 , 120 , 428–446. [ Google Scholar ] [ CrossRef ]
• Zhou, C.; Chen, R.; Jiang, S.; Zhou, Y.; Ding, L.; Skibniewski, M.J.; Lin, X. Human Dynamics in Near-Miss Accidents Resulting from Unsafe Behavior of Construction Workers. Phys. A Stat. Mech. Its Appl. 2019 , 530 , 121495. [ Google Scholar ] [ CrossRef ]
• Chen, F.; Wang, C.; Wang, J.; Zhi, Y.; Wang, Z. Risk Assessment of Chemical Process Considering Dynamic Probability of near Misses Based on Bayesian Theory and Event Tree Analysis. J. Loss Prev. Process Ind. 2020 , 68 , 104280. [ Google Scholar ] [ CrossRef ]
• Wright, L.; Van Der Schaaf, T. Accident versus near Miss Causation: A Critical Review of the Literature, an Empirical Test in the UK Railway Domain, and Their Implications for Other Sectors. J. Hazard. Mater. 2004 , 111 , 105–110. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Saleh, J.H.; Saltmarsh, E.A.; Favar, F.M.; Loı¨c Brevault, L. Accident Precursors, near Misses, and Warning Signs: Critical Review and Formal Definitions within the Framework of Discrete Event Systems. Reliab. Eng. Syst. Saf. 2013 , 114 , 148–154. [ Google Scholar ] [ CrossRef ]
• Fred, A. Manuele Reviewing Heinrich. Am. Soc. Saf. Prof. 2011 , 56 , 52–61. [ Google Scholar ]
• Love, P.E.D.; Tenekedjiev, K. Understanding Near-Miss Count Data on Construction Sites Using Greedy D-Vine Copula Marginal Regression: A Comment. Reliab. Eng. Syst. Saf. 2022 , 217 , 108021. [ Google Scholar ] [ CrossRef ]
• Jan van Eck, N.; Waltman, L. VOSviewer Manual ; Universiteit Leiden: Leiden, The Netherlands, 2015. [ Google Scholar ]
• Scopus. Content Coverage Guide ; Elsevier: Amsterdam, The Netherlands, 2023; pp. 1–24. [ Google Scholar ]
• Lukic, D.; Littlejohn, A.; Margaryan, A. A Framework for Learning from Incidents in the Workplace. Saf. Sci. 2012 , 50 , 950–957. [ Google Scholar ] [ CrossRef ]
• Teizer, J.; Cheng, T. Proximity Hazard Indicator for Workers-on-Foot near Miss Interactions with Construction Equipment and Geo-Referenced Hazard Area. Autom. Constr. 2015 , 60 , 58–73. [ Google Scholar ] [ CrossRef ]
• Zong, L.; Fu, G. A Study on Designing No-Penalty Reporting System about Enterprise Staff’s near Miss. Adv. Mater. Res. 2011 , 255–260 , 3846–3851. [ Google Scholar ] [ CrossRef ]
• Golovina, O.; Teizer, J.; Pradhananga, N. Heat Map Generation for Predictive Safety Planning: Preventing Struck-by and near Miss Interactions between Workers-on-Foot and Construction Equipment. Autom. Constr. 2016 , 71 , 99–115. [ Google Scholar ] [ CrossRef ]
• Zou, P.X.W.; Lun, P.; Cipolla, D.; Mohamed, S. Cloud-Based Safety Information and Communication System in Infrastructure Construction. Saf. Sci. 2017 , 98 , 50–69. [ Google Scholar ] [ CrossRef ]
• Hinze, J.; Godfrey, R. An Evaluation of Safety Performance Measures for Construction Projects. J. Constr. Res. 2011 , 4 , 5–15. [ Google Scholar ] [ CrossRef ]
• Construction Inspection Software|IAuditor by SafetyCulture. Available online: https://safetyculture.com/construction/ (accessed on 25 August 2023).
• Incident Reporting Made Easy|Safety Compliance|Mobile EHS Solutions. Available online: https://www.safety-reports.com/lp/safety/incident/ (accessed on 25 August 2023).
• Wu, F.; Wu, T.; Yuce, M.R. An Internet-of-Things (IoT) Network System for Connected Safety and Health Monitoring Applications. Sensors 2019 , 19 , 21. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Fang, W.; Luo, H.; Xu, S.; Love, P.E.D.; Lu, Z.; Ye, C. Automated Text Classification of Near-Misses from Safety Reports: An Improved Deep Learning Approach. Adv. Eng. Inform. 2020 , 44 , 101060. [ Google Scholar ] [ CrossRef ]
• Gatti, U.C.; Lin, K.-Y.; Caldera, C.; Chiang, R. Exploring the Relationship between Chronic Sleep Deprivation and Safety on Construction Sites. In Proceedings of the Construction Research Congress 2014: Construction in a Global Network, Atlanta, GA, USA, 19–24 May 2014; pp. 1772–1781. [ Google Scholar ] [ CrossRef ]
• Hon, C.K.H.; Chan, A.P.C.; Yam, M.C.H. Relationships between Safety Climate and Safety Performance of Building Repair, Maintenance, Minor Alteration, and Addition (RMAA) Works. Saf. Sci. 2014 , 65 , 10–19. [ Google Scholar ] [ CrossRef ]
• Oni, O.; Olanrewaju, A.; Cheen, K.S. Accidents at construction sites and near-misses: A constant problem. Int. Struct. Eng. Constr. 2022 , 9 , 2022. [ Google Scholar ] [ CrossRef ]
• Wu, W.; Yang, H.; Chew, D.A.S.; Yang, S.-H.; Gibb, A.G.F.; Li, Q. Towards an Autonomous Real-Time Tracking System of near-Miss Accidents on Construction Sites. Autom. Constr. 2010 , 19 , 134–141. [ Google Scholar ] [ CrossRef ]
• Aria, S.S.; Yang, K.; Ahn, C.R.; Vuran, M.C. Near-Miss Accident Detection for Ironworkers Using Inertial Measurement Unit Sensors. In Proceedings of the International Symposium on Automation and Robotics in Construction, ISARC 2014, Sydney, Australia, 9–11 July 2014; Volume 31, pp. 854–859. [ Google Scholar ] [ CrossRef ]
• Hasanzadeh, S.; Garza, J.M. de la Productivity-Safety Model: Debunking the Myth of the Productivity-Safety Divide through a Mixed-Reality Residential Roofing Task. J. Constr. Eng. Manag. 2020 , 146 , 04020124. [ Google Scholar ] [ CrossRef ]
• Teizer, J. Magnetic Field Proximity Detection and Alert Technology for Safe Heavy Construction Equipment Operation. In Proceedings of the 32nd International Symposium on Automation and Robotics in Construction, Oulu, Finland, 15–18 June 2015. [ Google Scholar ] [ CrossRef ]
• Mohajeri, M.; Ardeshir, A.; Banki, M.T.; Malekitabar, H. Discovering Causality Patterns of Unsafe Behavior Leading to Fall Hazards on Construction Sites. Int. J. Constr. Manag. 2022 , 22 , 3034–3044. [ Google Scholar ] [ CrossRef ]
• Kisaezehra; Farooq, M.U.; Bhutto, M.A.; Kazi, A.K. Real-Time Safety Helmet Detection Using Yolov5 at Construction Sites. Intell. Autom. Soft Comput. 2023 , 36 , 911–927. [ Google Scholar ] [ CrossRef ]
• Li, C.; Ding, L. Falling Objects Detection for near Miss Incidents Identification on Construction Site. In Proceedings of the ASCE International Conference on Computing in Civil Engineering, Atlanta, GA, USA, 17–19 June 2019; pp. 138–145. [ Google Scholar ] [ CrossRef ]
• Jeelani, I.; Ramshankar, H.; Han, K.; Albert, A.; Asadi, K. Real-Time Hazard Proximity Detection—Localization of Workers Using Visual Data. In Proceedings of the ASCE International Conference on Computing in Civil Engineering, Atlanta, GA, USA, 17–19 June 2019; pp. 281–289. [ Google Scholar ] [ CrossRef ]
• Lim, T.-K.; Park, S.-M.; Lee, H.-C.; Lee, D.-E. Artificial Neural Network–Based Slip-Trip Classifier Using Smart Sensor for Construction Workplace. J. Constr. Eng. Manag. 2015 , 142 , 04015065. [ Google Scholar ] [ CrossRef ]
• Yang, K.; Jebelli, H.; Ahn, C.R.; Vuran, M.C. Threshold-Based Approach to Detect Near-Miss Falls of Iron Workers Using Inertial Measurement Units. In Proceedings of the 2015 International Workshop on Computing in Civil Engineering, Austin, TX, USA, 21–23 June 2015; 2015; 2015, pp. 148–155. [ Google Scholar ] [ CrossRef ]
• Yang, K.; Ahn, C.R.; Vuran, M.C.; Aria, S.S. Semi-Supervised near-Miss Fall Detection for Ironworkers with a Wearable Inertial Measurement Unit. Autom. Constr. 2016 , 68 , 194–202. [ Google Scholar ] [ CrossRef ]
• Raviv, G.; Shapira, A.; Fishbain, B. AHP-Based Analysis of the Risk Potential of Safety Incidents: Case Study of Cranes in the Construction Industry. Saf. Sci. 2017 , 91 , 298–309. [ Google Scholar ] [ CrossRef ]
• Saurin, T.A.; Formoso, C.T.; Reck, R.; Beck da Silva Etges, B.M.; Ribeiro JL, D. Findings from the Analysis of Incident-Reporting Systems of Construction Companies. J. Constr. Eng. Manag. 2015 , 141 , 05015007. [ Google Scholar ] [ CrossRef ]
• Williams, E.; Sherratt, F.; Norton, E. Exploring the Value in near Miss Reporting for Construction Safety. In Proceedings of the 37th Annual Conference, Virtual Event, 6–10 December 2021; pp. 319–328. [ Google Scholar ]
• Baker, H.; Smith, S.; Masterton, G.; Hewlett, B. Data-Led Learning: Using Natural Language Processing (NLP) and Machine Learning to Learn from Construction Site Safety Failures. In Proceedings of the 36th Annual ARCOM Conference, Online, 7–8 September 2020; pp. 356–365. [ Google Scholar ]
• Jin, R.; Wang, F.; Liu, D. Dynamic Probabilistic Analysis of Accidents in Construction Projects by Combining Precursor Data and Expert Judgments. Adv. Eng. Inform. 2020 , 44 , 101062. [ Google Scholar ] [ CrossRef ]
• Zhou, Z.; Li, C.; Mi, C.; Qian, L. Exploring the Potential Use of Near-Miss Information to Improve Construction Safety Performance. Sustainability 2019 , 11 , 1264. [ Google Scholar ] [ CrossRef ]
• Boateng, E.B.; Pillay, M.; Davis, P. Predicting the Level of Safety Performance Using an Artificial Neural Network. Adv. Intell. Syst. Comput. 2019 , 876 , 705–710. [ Google Scholar ] [ CrossRef ]
• Zhang, M.; Cao, T.; Zhao, X. Using Smartphones to Detect and Identify Construction Workers’ Near-Miss Falls Based on ANN. J. Constr. Eng. Manag. 2018 , 145 , 04018120. [ Google Scholar ] [ CrossRef ]
• Gadekar, H.; Bugalia, N. Automatic Classification of Construction Safety Reports Using Semi-Supervised YAKE-Guided LDA Approach. Adv. Eng. Inform. 2023 , 56 , 101929. [ Google Scholar ] [ CrossRef ]
• Zhu, Y.; Liao, H.; Huang, D. Using Text Mining and Multilevel Association Rules to Process and Analyze Incident Reports in China. Accid. Anal. Prev. 2023 , 191 , 107224. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Li, M.; Lin, Q.; Jin, H. Research on Near-Miss Incidents Monitoring and Early Warning System for Building Construction Sites Based on Blockchain Technology. J. Constr. Eng. Manag. 2023 , 149 , 04023124. [ Google Scholar ] [ CrossRef ]
• Chung, W.W.S.; Tariq, S.; Mohandes, S.R.; Zayed, T. IoT-Based Application for Construction Site Safety Monitoring. Int. J. Constr. Manag. 2020 , 23 , 58–74. [ Google Scholar ] [ CrossRef ]
• Liu, X.; Xu, F.; Zhang, Z.; Sun, K. Fall-Portent Detection for Construction Sites Based on Computer Vision and Machine Learning. Eng. Constr. Archit. Manag. 2023; ahead-of-print . [ Google Scholar ] [ CrossRef ]
• Abbasi, H.; Guerrieri, A.; Lee, J.; Yang, K. Mobile Device-Based Struck-By Hazard Recognition in Construction Using a High-Frequency Sound. Sensors 2022 , 22 , 3482. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Wang, F.; Li, H.; Dong, C. Understanding Near-Miss Count Data on Construction Sites Using Greedy D-Vine Copula Marginal Regression. Reliab. Eng. Syst. Saf. 2021 , 213 , 107687. [ Google Scholar ] [ CrossRef ]
• Bugalia, N.; Tarani, V.; Student, G.; Kedia, J.; Gadekar, H. Machine Learning-Based Automated Classification Of Worker-Reported Safety Reports In Construction. J. Inf. Technol. Constr. 2022 , 27 , 926–950. [ Google Scholar ] [ CrossRef ]
• Chen, S.; Xi, J.; Chen, Y.; Zhao, J. Association Mining of Near Misses in Hydropower Engineering Construction Based on Convolutional Neural Network Text Classification. Comput. Intell. Neurosci. 2022 , 2022 , 4851615. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Tang, S.; Golparvar-Fard, M.; Naphade, M.; Gopalakrishna, M.M. Video-Based Activity Forecasting for Construction Safety Monitoring Use Cases. In Proceedings of the ASCE International Conference on Computing in Civil Engineering, Atlanta, GA, USA, 17–19 June 2019; pp. 204–210. [ Google Scholar ] [ CrossRef ]
• Rashid, K.M.; Behzadan, A.H. Risk Behavior-Based Trajectory Prediction for Construction Site Safety Monitoring. J. Constr. Eng. Manag. 2018 , 144 , 04017106. [ Google Scholar ] [ CrossRef ]
• Shen, X.; Marks, E. Near-Miss Information Visualization Tool in BIM for Construction Safety. J. Constr. Eng. Manag. 2016 , 142 , 04015100. [ Google Scholar ] [ CrossRef ]
• Erusta, N.E.; Sertyesilisik, B. An Investigation into Improving Occupational Health and Safety Performance of Construction Projects through Usage of BIM for Lean Management. In Communications in Computer and Information Science (CCIS) ; Springer: Berlin/Heidelberg, Germany, 2020; Volume 1188, pp. 91–100. [ Google Scholar ] [ CrossRef ]
• Coffland, M.M.; Kim, A.; Sadatsafavi, H.; Uber, M.M. Improved Data Storage for Better Safety Analysis and Decision Making in Large Construction Management Firms. Available online: https://www.researchgate.net/publication/320474383_Improved_Data_Storage_for_Better_Safety_Analysis_and_Decision_Making_in_Large_Construction_Management_Firms (accessed on 12 June 2024).
• Zhou, Z.; Li, Q.; Wu, W. Developing a Versatile Subway Construction Incident Database for Safety Management. J. Constr. Eng. Manag. 2011 , 138 , 1169–1180. [ Google Scholar ] [ CrossRef ]
• Wu, W.; Yang, H.; Li, Q.; Chew, D. An Integrated Information Management Model for Proactive Prevention of Struck-by-Falling-Object Accidents on Construction Sites. Autom. Constr. 2013 , 34 , 67–74. [ Google Scholar ] [ CrossRef ]
• Hoła, B. Identification and Evaluation of Processes in a Construction Enterprise. Arch. Civ. Mech. Eng. 2015 , 15 , 419–426. [ Google Scholar ] [ CrossRef ]
• Zhou, C.; Ding, L.; Skibniewski, M.J.; Luo, H.; Jiang, S. Characterizing Time Series of Near-Miss Accidents in Metro Construction via Complex Network Theory. Saf. Sci. 2017 , 98 , 145–158. [ Google Scholar ] [ CrossRef ]
• Woźniak, Z.; Hoła, B. Time Series Analysis of Hazardous Events Based on Data Recorded in a Polish Construction Company. Arch. Civ. Eng. 2024; in process . [ Google Scholar ]
• Drozd, W. Characteristics of Construction Site in Terms of Occupational Safety. J. Civ. Eng. Environ. Archit. 2016 , 63 , 165–172. [ Google Scholar ]
• Meliá, J.L.; Mearns, K.; Silva, S.A.; Lima, M.L. Safety Climate Responses and the Perceived Risk of Accidents in the Construction Industry. Saf. Sci. 2008 , 46 , 949–958. [ Google Scholar ] [ CrossRef ]
• Bugalia, N.; Maemura, Y.; Ozawa, K. A System Dynamics Model for Near-Miss Reporting in Complex Systems. Saf. Sci. 2021 , 142 , 105368. [ Google Scholar ] [ CrossRef ]
• Gyi, D.E.; Gibb, A.G.F.; Haslam, R.A. The Quality of Accident and Health Data in the Construction Industry: Interviews with Senior Managers. Constr. Manag. Econ. 1999 , 17 , 197–204. [ Google Scholar ] [ CrossRef ]
• Menzies, J. Structural Safety: Learning and Warnings. Struct. Eng. 2002 , 80 , 15–16. [ Google Scholar ]
• Fullerton, C.E.; Allread, B.S.; Teizer, J. Pro-Active-Real-Time Personnel Warning System. In Proceedings of the Construction Research Congress 2009: Building a Sustainable Future, Seattle, WA, USA, 5–7 April 2009; pp. 31–40. [ Google Scholar ] [ CrossRef ]
• Marks, E.D.; Wetherford, J.E.; Teizer, J.; Yabuki, N. Potential of Leading Indicator Data Collection and Analysis for Proximity Detection and Alert Technology in Construction. In Proceedings of the 30th ISARC—International Symposium on Automation and Robotics in Construction Conference, Montreal, QC, Canada, 11–15 August 2013; pp. 1029–1036. [ Google Scholar ] [ CrossRef ]
• Martin, H.; Lewis, T.M. Pinpointing Safety Leadership Factors for Safe Construction Sites in Trinidad and Tobago. J. Constr. Eng. Manag. 2014 , 140 , 04013046. [ Google Scholar ] [ CrossRef ]
• Hobson, P.; Emery, D.; Brown, L.; Bashford, R.; Gill, J. People–Plant Interface Training: Targeting an Industry Fatal Risk. Proc. Inst. Civ. Eng. Civ. Eng. 2014 , 167 , 138–144. [ Google Scholar ] [ CrossRef ]
• Marks, E.; Mckay, B.; Awolusi, I. Using near Misses to Enhance Safety Performance in Construction. In Proceedings of the ASSE Professional Development Conference and Exposition, Dallas, TX, USA, 7–10 June 2015. [ Google Scholar ]
• Popp, J.D.; Scarborough, M.S. Investigations of near Miss Incidents—New Facility Construction and Commissioning Activities. IEEE Trans. Ind. Appl. 2016 , 53 , 615–621. [ Google Scholar ] [ CrossRef ]
• Nickel, P.; Lungfiel, A.; Trabold, R.J. Reconstruction of near Misses and Accidents for Analyses from Virtual Reality Usability Study. In Lecture Notes in Computer Science ; Springer: Berlin/Heidelberg, Germany, 2017; Volume 10700, pp. 182–191. [ Google Scholar ] [ CrossRef ]
• Gambatese, J.A.; Pestana, C.; Lee, H.W. Alignment between Lean Principles and Practices and Worker Safety Behavior. J. Constr. Eng. Manag. 2017 , 143 , 04016083. [ Google Scholar ] [ CrossRef ]
• Van Voorhis, S.; Korman, R. Reading Signs of Trouble. Eng. News-Rec. 2017 , 278 , 14–17. [ Google Scholar ]
• Doan, D.R. Investigation of a near-miss shock incident. IEEE Trans. Ind. Appl. 2016 , 52 , 560–561. [ Google Scholar ] [ CrossRef ]
• Oswald, D.; Sherratt, F.; Smith, S. Problems with safety observation reporting: A construction industry case study. Saf. Sci. 2018 , 107 , 35–45. [ Google Scholar ] [ CrossRef ]
• Raviv, G.; Shapira, A. Systematic approach to crane-related near-miss analysis in the construction industry. Int. J. Constr. Manag. 2018 , 18 , 310–320. [ Google Scholar ] [ CrossRef ]
• Whiteoak, J.; Appleby, J. Mate, that was bloody close! A case history of a nearmiss program in the Australian construction industry. J. Health Saf. Environ. 2019 , 35 , 31–43. [ Google Scholar ]
• Duryan, M.; Smyth, H.; Roberts, A.; Rowlinson, S.; Sherratt, F. Knowledge transfer for occupational health and safety: Cultivating health and safety learning culture in construction firms. Accid. Anal. Prev. 2020 , 139 , 105496. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Shaikh, A.Y.; Osei-Kyei, R.; Hardie, M. A critical analysis of safety performance indicators in construction. Int. J. Build. Pathol. Adapt. 2020 , 39 , 547–580. [ Google Scholar ] [ CrossRef ]
• Martin, H.; Mohan, N.; Ellis, L.; Dunne, S. Exploring the Role of PPE Knowledge, Attitude, and Correct Practices in Safety Outcomes on Construction Sites. J. Archit. Eng. 2021 , 27 , 05021011. [ Google Scholar ] [ CrossRef ]
• Qin, Z.; Wu, S. A simulation model of engineering construction near-miss event disclosure strategy based on evolutionary game theory. In Proceedings of the 2021 4th International Conference on Information Systems and Computer Aided Education, Dalian, China, 24–26 September 2021; pp. 2572–2577. [ Google Scholar ] [ CrossRef ]
• Alamoudi, M. The Integration of NOSACQ-50 with Importance-Performance Analysis Technique to Evaluate and Analyze Safety Climate Dimensions in the Construction Sector in Saudi Arabia. Buildings 2022 , 12 , 1855. [ Google Scholar ] [ CrossRef ]
• Herrmann, A.W. Development of CROSS in the United States. In Proceedings of the Forensic Engineering 2022: Elevating Forensic Engineering—Selected Papers from the 9th Congress on Forensic Engineering, Denver, Colorado, 4–7 November 2022; Volume 2, pp. 40–43. [ Google Scholar ] [ CrossRef ]
• Al Shaaili, M.; Al Alawi, M.; Ekyalimpa, R.; Al Mawli, B.; Al-Mamun, A.; Al Shahri, M. Near-miss accidents data analysis and knowledge dissemination in water construction projects in Oman. Heliyon 2023 , 9 , e21607. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Agnusdei, G.P.; Gnoni, M.G.; Tornese, F.; De Merich, D.; Guglielmi, A.; Pellicci, M. Application of Near-Miss Management Systems: An Exploratory Field Analysis in the Italian Industrial Sector. Safety 2023 , 9 , 47. [ Google Scholar ] [ CrossRef ]
• Duan, P.; Zhou, J. A science mapping approach-based review of near-miss research in construction. Eng. Constr. Archit. Manag. 2023 , 30 , 2582–2601. [ Google Scholar ] [ CrossRef ]

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Share and Cite

Woźniak, Z.; Hoła, B. Analysing Near-Miss Incidents in Construction: A Systematic Literature Review. Appl. Sci. 2024 , 14 , 7260. https://doi.org/10.3390/app14167260

Woźniak Z, Hoła B. Analysing Near-Miss Incidents in Construction: A Systematic Literature Review. Applied Sciences . 2024; 14(16):7260. https://doi.org/10.3390/app14167260

Woźniak, Zuzanna, and Bożena Hoła. 2024. "Analysing Near-Miss Incidents in Construction: A Systematic Literature Review" Applied Sciences 14, no. 16: 7260. https://doi.org/10.3390/app14167260

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