problem solving regression analysis

Understanding regression analysis: overview and key uses

Last updated

22 August 2024

Reviewed by

Miroslav Damyanov

Regression analysis is a fundamental statistical method that helps us predict and understand how different factors (aka independent variables) influence a specific outcome (aka dependent variable). 

Imagine you're trying to predict the value of a house. Regression analysis can help you create a formula to estimate the house's value by looking at variables like the home's size and the neighborhood's average income. This method is crucial because it allows us to predict and analyze trends based on data. 

While that example is straightforward, the technique can be applied to more complex situations, offering valuable insights into fields such as economics, healthcare, marketing, and more.

• 3 uses for regression analysis in business

Businesses can use regression analysis to improve nearly every aspect of their operations. When used correctly, it's a powerful tool for learning how adjusting variables can improve outcomes. Here are three applications:

1. Prediction and forecasting

Predicting future scenarios can give businesses significant advantages. No method can guarantee absolute certainty, but regression analysis offers a reliable framework for forecasting future trends based on past data. Companies can apply this method to anticipate future sales for financial planning purposes and predict inventory requirements for more efficient space and cost management. Similarly, an insurance company can employ regression analysis to predict the likelihood of claims for more accurate underwriting. 

2. Identifying inefficiencies and opportunities

Regression analysis can help us understand how the relationships between different business processes affect outcomes. Its ability to model complex relationships means that regression analysis can accurately highlight variables that lead to inefficiencies, which intuition alone may not do. Regression analysis allows businesses to improve performance significantly through targeted interventions. For instance, a manufacturing plant experiencing production delays, machine downtime, or labor shortages can use regression analysis to determine the underlying causes of these issues.

3. Making data-driven decisions

Regression analysis can enhance decision-making for any situation that relies on dependent variables. For example, a company can analyze the impact of various price points on sales volume to find the best pricing strategy for its products. Understanding buying behavior factors can help segment customers into buyer personas for improved targeting and messaging.

• Types of regression models

There are several types of regression models, each suited to a particular purpose. Picking the right one is vital to getting the correct results. 

Simple linear regression analysis is the simplest form of regression analysis. It examines the relationship between exactly one dependent variable and one independent variable, fitting a straight line to the data points on a graph.

Multiple regression analysis examines how two or more independent variables affect a single dependent variable. It extends simple linear regression and requires a more complex algorithm.

Multivariate linear regression is suitable for multiple dependent variables. It allows the analysis of how independent variables influence multiple outcomes.

Logistic regression is relevant when the dependent variable is categorical, such as binary outcomes (e.g., true/false or yes/no). Logistic regression estimates the probability of a category based on the independent variables.

• 6 mistakes people make with regression analysis

Ignoring key variables is a common mistake when working with regression analysis. Here are a few more pitfalls to try and avoid:

1. Overfitting the model

If a model is too complex, it can become overly powerful and lead to a problem known as overfitting. This mistake is an especially significant problem when the independent variables don't impact the dependent data, though it can happen whenever the model over-adjusts to fit all the variables. In such cases, the model starts memorizing noise rather than meaningful data. When this happens, the model’s results will fit the training data perfectly but fail to generalize to new, unseen data, rendering the model ineffective for prediction or inference.  

2. Underfitting the model

A less complex model is unlikely to draw false conclusions mistakenly. However, if the model is too simplistic, it will face the opposite problem: underfitting. In this case, the model will fail to capture the underlying patterns in the data, meaning it won't perform well on either the training or new, unseen data. This lack of complexity prevents the model from making accurate predictions or drawing meaningful inferences. 

3. Neglecting model validation

Model validation is how you can be sure that a model isn't overfitting or underfitting. Imagine teaching a child to read. If you always read the same book to the child, they might memorize it and recite it perfectly, making it seem like they’ve learned to read. However, if you give them a new book, they might struggle and be unable to read it.

This scenario is similar to a model that performs well on its training data but fails with new data. Model validation involves testing the model with data it hasn’t seen before. If the model performs well on this new data, it indicates having truly learned to generalize. On the other hand, if the model only performs well on the training data and poorly on new data, it has overfitted to the training data, much like the child who can only recite the memorized book.

4. Multicollinearity

Regression analysis works best when the independent variables are genuinely independent. However, sometimes, two or more variables are highly correlated. This multicollinearity can make it hard for the model to accurately determine each variable's impact. 

If a model gives poor results, checking for correlated variables may reveal the issue. You can fix it by removing one or more correlated variables or using a principal component analysis (PCA) technique, which transforms the correlated variables into a set of uncorrelated components.

5. Misinterpreting coefficients

Errors are not always due to the model itself; human error is common. These mistakes often involve misinterpreting the results. For example, someone might misunderstand the units of measure and draw incorrect conclusions. Another frequent issue in scientific analysis is confusing correlation and causation. Regression analysis can only provide insights into correlation, not causation.

6. Poor data quality

The adage “garbage in, garbage out” strongly applies to regression analysis. When low-quality data is input into a model, it analyzes noise rather than meaningful patterns. Poor data quality can manifest as missing values, unrepresentative data, outliers, and measurement errors. Additionally, the model may have excluded essential variables significantly impacting the results. All these issues can distort the relationships between variables and lead to misleading results. 

• What are the assumptions that must hold for regression models?

To correctly interpret the output of a regression model, the following key assumptions about the underlying data process must hold:

The relationship between variables is linear.

There must be homoscedasticity, meaning the variance of the variables and the error term must remain constant.

All explanatory variables are independent of one another.

All variables are normally distributed.

• Real-life examples of regression analysis

Let's turn our attention to examining how a few industries use the regression analysis to improve their outcomes:

Regression analysis has many applications in healthcare, but two of the most common are improving patient outcomes and optimizing resources. 

Hospitals need to use resources effectively to ensure the best patient outcomes. Regression models can help forecast patient admissions, equipment and supply usage, and more. These models allow hospitals to plan and maximize their resources. 

Predicting stock prices, economic trends, and financial risks benefits the finance industry. Regression analysis can help finance professionals make informed decisions about these topics. 

For example, analysts often use regression analysis to assess how changes to GDP, interest rates, and unemployment rates impact stock prices. Armed with this information, they can make more informed portfolio decisions. 

The banking industry also uses regression analysis. When a loan underwriter determines whether to grant a loan, regression analysis allows them to calculate the probability that a potential lender will repay the loan.

Imagine how much more effective a company's marketing efforts could be if they could predict customer behavior. Regression analysis allows them to do so with a degree of accuracy. For example, marketers can analyze how price, advertising spend, and product features (combined) influence sales. Once they've identified key sales drivers, they can adjust their strategy to maximize revenue. They may approach this analysis in stages. 

For instance, if they determine that ad spend is the biggest driver, they can apply regression analysis to data specific to advertising efforts. Doing so allows them to improve the ROI of ads. The opposite may also be true. If ad spending has little to no impact on sales, something is wrong that regression analysis might help identify. 

• Regression analysis tools and software

Regression analysis by hand isn't practical. The process requires large numbers and complex calculations. Computers make even the most complex regression analysis possible. Even the most complicated AI algorithms can be considered fancy regression calculations. Many tools exist to help users create these regressions.

Another programming language—while MATLAB is a commercial tool, the open-source project Octave aims to implement much of the functionality. These languages are for complex mathematical operations, including regression analysis. Its tools for computation and visualization have made it very popular in academia, engineering, and industry for calculating regression and displaying the results. MATLAB integrates with other toolboxes so developers can extend its functionality and allow for application-specific solutions.

Python is a more general programming language than the previous examples, but many libraries are available that extend its functionality. For regression analysis, packages like Scikit-Learn and StatsModels provide the computational tools necessary for the job. In contrast, packages like Pandas and Matplotlib can handle large amounts of data and display the results. Python is a simple-to-learn, easy-to-read programming language, which can give it a leg up over the more dedicated math and statistics languages. 

SAS (Statistical Analysis System) is a commercial software suite for advanced analytics, multivariate analysis, business intelligence, and data management. It includes a procedure called PROC REG that allows users to efficiently perform regression analysis on their data. The software is well-known for its data-handling capabilities, extensive documentation, and technical support. These factors make it a common choice for large-scale enterprise use and industries requiring rigorous statistical analysis. 

Stata is another statistical software package. It provides an integrated data analysis, management, and graphics environment. The tool includes tools for performing a range of regression analysis tasks. This tool's popularity is due to its ease of use, reproducibility, and ability to handle complex datasets intuitively. The extensive documentation helps beginners get started quickly. Stata is widely used in academic research, economics, sociology, and political science.

Most people know Excel , but you might not know that Microsoft's spreadsheet software has an add-in called Analysis ToolPak that can perform basic linear regression and visualize the results. Excel is not an excellent choice for more complex regression or very large datasets. But for those with basic needs who only want to analyze smaller datasets quickly, it's a convenient option already in many tech stacks. 

SPSS (Statistical Package for the Social Sciences) is a versatile statistical analysis software widely used in social science, business, and health. It offers tools for various analyses, including regression, making it accessible to users through its user-friendly interface. SPSS enables users to manage and visualize data, perform complex analyses, and generate reports without coding. Its extensive documentation and support make it popular in academia and industry, allowing for efficient handling of large datasets and reliable results.

What is a regression analysis in simple terms?

Regression analysis is a statistical method used to estimate and quantify the relationship between a dependent variable and one or more independent variables. It helps determine the strength and direction of these relationships, allowing predictions about the dependent variable based on the independent variables and providing insights into how each independent variable impacts the dependent variable.

What are the main types of variables used in regression analysis?

Dependent variables : typically continuous (e.g., house price) or binary (e.g., yes/no outcomes).

Independent variables : can be continuous, categorical, binary, or ordinal.

What does a regression analysis tell you?

Regression analysis identifies the relationships between a dependent variable and one or more independent variables. It quantifies the strength and direction of these relationships, allowing you to predict the dependent variable based on the independent variables and understand the impact of each independent variable on the dependent variable.

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Regression Analysis Tutorial and Examples

Topics: Regression Analysis

This tutorial covers many aspects of regression analysis including: choosing the type of regression analysis to use, specifying the model, interpreting the results, determining how well the model fits, making predictions, and checking the assumptions. At the end, I include examples of different types of regression analyses.

Why Choose Regression and the Hallmarks of a Good Regression Analysis

Before we begin the regression analysis tutorial, there are several important questions to answer.

Why should we choose regression at all? What are the common mistakes that even experts make when it comes to regression analysis? And, how do you distinguish a good regression analysis from a less rigorous regression analysis? Read these posts to find out:

• Tribute to Regression Analysis : See why regression is my favorite! Sure, regression generates an equation that describes the relationship between one or more predictor variables and the response variable. But, there’s much more to it than just that.
• Four Tips on How to Perform a Regression Analysis that Avoids Common Problems : Keep these tips in mind through out all stages of this tutorial to ensure a top-quality regression analysis.
• Sample Size Guidelines : These guidelines help ensure that you have sufficient power to detect a relationship and provide a reasonably precise estimate of the strength of that relationship.

Tutorial: How to Choose the Correct Type of Regression Analysis

• Giving Thanks for the Regression Menu : regression choices using a yummy Thanksgiving context!
• Linear or Nonlinear Regression : How to determine when you should use one or the other.
• What is the Difference between Linear and Nonlinear Equations : Both types of equations can model curvature, so what is the difference between them?

Tutorial: How to Specify Your Regression Model

Specifying a regression model is an iterative process. The interpretation and assumption verification sections of this regression tutorial show you how to confirm that you’ve specified the model correctly and how to adjust your model based on the results.

• How to Choose the Best Regression Model : some common statistical methods, complications you may face, and provide some practical advice.
• Stepwise and Best Subsets Regression : Minitab provides two automatic tools that help identify useful predictors during the exploratory stages of model building.
• Curve Fitting with Linear and Nonlinear Regression : Sometimes your data just don’t follow a straight line and you need to fit a curved relationship.
• Interaction effects : interactions using Ketchup and Soy Sauce.
• Overfitting the model : Overly complex models can produce misleading results. Learn about overfit models and how to detect and avoid them.
• Hierarchical models : I review reasons to fit, or not fit, a hierarchical model. A hierarchical model contains all lower-order terms that comprise the higher-order terms that also appear in the model.
• Standardizing the variables : In certain cases, standardizing the variables in your regression model can reveal statistically significant findings that you might otherwise miss.
• Five reasons why your R-squared can be too high : If you specify the wrong regression model, or use the wrong model fitting process, the R-squared can be too high.

Tutorial: How to Interpret your Regression Results

So, you’ve chosen the correct type of regression and specified the model. Now, you want to interpret the results. The following topics in the regression tutorial show you how to interpret the results and effectively present them:

• Regression coefficients and p-values
• Regression Constant (Y intercept)
• How to statistically test the difference between regression slopes and constants
• R-squared and the goodness-of-fit
• How high should R-squared be?
• How to interpret a model with a low R-squared
• Adjusted R-squared and Predicted R-squared
• S, the standard error of the regression
• F-test of overall significance
• How to Compare Regression Slopes
• Present Your Regression Results to Avoid Costly Mistakes : Research shows that presentation affects the number of interpretation mistakes.
• Identify the Most Important Predictor Variables : After you've settled on a model, it’s common to ask, “Which variable is most important?”

Tutorial: How to Use Regression to Make Predictions

• How to Predict with Minitab : A prediction guide that uses BMI to predict body fat percentage.
• Predicted R-squared : This statistic indicates how well a regression model predicts responses for new observations rather than just the original data set.
• Prediction intervals : See how presenting prediction intervals is better than presenting only the regression equation and predicted values.
• Prediction intervals versus other intervals : I compare prediction intervals to confidence and tolerance intervals so you’ll know when to use each type of interval.

Tutorial: How to Check the Regression Assumptions and Fix Problems

• Residual plots : What they should look like and reasons why they might not!
• How important are normal residuals : If you have a large enough sample, nonnormal residuals may not be a problem.
• Multicollinearity : Highly correlated predictors can be a problem, but not always!
• Heteroscedasticity : You want the residuals to have a constant variance (homoscedasticity), but what if they don’t?
• Box-Cox transformation : If you can’t resolve the underlying problem, Cody Steele shows how easy it can be to transform the problem away!

Examples of Different Types of Regression Analyses

The final part of the regression tutorial contains examples of the different types of regression analysis that Minitab can perform. Many of these regression examples include the data sets so you can try it yourself!

• Linear Model Features in Minitab
• Multiple regression with response optimization : Highlights features in the Minitab Assistant.
• Orthogonal regression : how orthogonal regression (a.k.a. Deming Regression) can test the equivalence of different instruments.
• Partial least squares (PLS) regression : using PLS to successfully analyze a very small and highly multicollinear data set.

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What is Regression Analysis?

• Regression Analysis – Linear Model Assumptions
• Regression Analysis – Simple Linear Regression
• Regression Analysis – Multiple Linear Regression

Regression Analysis in Finance

Regression tools, additional resources, regression analysis.

The estimation of relationships between a dependent variable and one or more independent variables

Regression analysis is a set of statistical methods used for the estimation of relationships between a dependent variable and one or more independent variables . It can be utilized to assess the strength of the relationship between variables and for modeling the future relationship between them.

Regression analysis includes several variations, such as linear, multiple linear, and nonlinear. The most common models are simple linear and multiple linear. Nonlinear regression analysis is commonly used for more complicated data sets in which the dependent and independent variables show a nonlinear relationship.

Regression analysis offers numerous applications in various disciplines, including finance .

Regression Analysis – Linear Model Assumptions

Linear regression analysis is based on six fundamental assumptions:

• The dependent and independent variables show a linear relationship between the slope and the intercept.
• The independent variable is not random.
• The value of the residual (error) is zero.
• The value of the residual (error) is constant across all observations.
• The value of the residual (error) is not correlated across all observations.
• The residual (error) values follow the normal distribution.

Regression Analysis – Simple Linear Regression

Simple linear regression is a model that assesses the relationship between a dependent variable and an independent variable. The simple linear model is expressed using the following equation:

Y = a + bX + ϵ

• Y – Dependent variable
• X – Independent (explanatory) variable
• a – Intercept
• b – Slope
• ϵ – Residual (error)

Check out the following video to learn more about simple linear regression:

Regression Analysis – Multiple Linear Regression

Multiple linear regression analysis is essentially similar to the simple linear model, with the exception that multiple independent variables are used in the model. The mathematical representation of multiple linear regression is:

Y = a + b X 1  + c X 2  + d X 3 + ϵ

• X 1 , X 2 , X 3  – Independent (explanatory) variables
• b, c, d – Slopes

Multiple linear regression follows the same conditions as the simple linear model. However, since there are several independent variables in multiple linear analysis, there is another mandatory condition for the model:

• Non-collinearity: Independent variables should show a minimum correlation with each other. If the independent variables are highly correlated with each other, it will be difficult to assess the true relationships between the dependent and independent variables.

Regression analysis comes with several applications in finance. For example, the statistical method is fundamental to the Capital Asset Pricing Model (CAPM) . Essentially, the CAPM equation is a model that determines the relationship between the expected return of an asset and the market risk premium.

The analysis is also used to forecast the returns of securities, based on different factors, or to forecast the performance of a business. Learn more forecasting methods in CFI’s Budgeting and Forecasting Course !

1. Beta and CAPM

In finance, regression analysis is used to calculate the Beta (volatility of returns relative to the overall market) for a stock. It can be done in Excel using the Slope function .

Download CFI’s free beta calculator !

2. Forecasting Revenues and Expenses

When forecasting financial statements for a company, it may be useful to do a multiple regression analysis to determine how changes in certain assumptions or drivers of the business will impact revenue or expenses in the future. For example, there may be a very high correlation between the number of salespeople employed by a company, the number of stores they operate, and the revenue the business generates.

The above example shows how to use the Forecast function in Excel to calculate a company’s revenue, based on the number of ads it runs.

Learn more forecasting methods in CFI’s Budgeting and Forecasting Course !

Excel remains a popular tool to conduct basic regression analysis in finance, however, there are many more advanced statistical tools that can be used.

Python and R are both powerful coding languages that have become popular for all types of financial modeling, including regression. These techniques form a core part of data science and machine learning, where models are trained to detect these relationships in data.

Learn more about regression analysis, Python, and Machine Learning in CFI’s Business Intelligence & Data Analysis certification.

To learn more about related topics, check out the following free CFI resources:

• Cost Behavior Analysis
• Forecasting Methods
• Joseph Effect
• Variance Inflation Factor (VIF)
• High Low Method vs. Regression Analysis
• See all data science resources
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What Is Regression Analysis in Business Analytics?

• 14 Dec 2021

Countless factors impact every facet of business. How can you consider those factors and know their true impact?

Imagine you seek to understand the factors that influence people’s decision to buy your company’s product. They range from customers’ physical locations to satisfaction levels among sales representatives to your competitors' Black Friday sales.

Understanding the relationships between each factor and product sales can enable you to pinpoint areas for improvement, helping you drive more sales.

To learn how each factor influences sales, you need to use a statistical analysis method called regression analysis .

If you aren’t a business or data analyst, you may not run regressions yourself, but knowing how analysis works can provide important insight into which factors impact product sales and, thus, which are worth improving.

Access your free e-book today.

Foundational Concepts for Regression Analysis

Before diving into regression analysis, you need to build foundational knowledge of statistical concepts and relationships.

Independent and Dependent Variables

Start with the basics. What relationship are you aiming to explore? Try formatting your answer like this: “I want to understand the impact of [the independent variable] on [the dependent variable].”

The independent variable is the factor that could impact the dependent variable . For example, “I want to understand the impact of employee satisfaction on product sales.”

In this case, employee satisfaction is the independent variable, and product sales is the dependent variable. Identifying the dependent and independent variables is the first step toward regression analysis.

Correlation vs. Causation

One of the cardinal rules of statistically exploring relationships is to never assume correlation implies causation. In other words, just because two variables move in the same direction doesn’t mean one caused the other to occur.

If two or more variables are correlated , their directional movements are related. If two variables are positively correlated , it means that as one goes up or down, so does the other. Alternatively, if two variables are negatively correlated , one goes up while the other goes down.

A correlation’s strength can be quantified by calculating the correlation coefficient , sometimes represented by r . The correlation coefficient falls between negative one and positive one.

r = -1 indicates a perfect negative correlation.

r = 1 indicates a perfect positive correlation.

r = 0 indicates no correlation.

Causation means that one variable caused the other to occur. Proving a causal relationship between variables requires a true experiment with a control group (which doesn’t receive the independent variable) and an experimental group (which receives the independent variable).

While regression analysis provides insights into relationships between variables, it doesn’t prove causation. It can be tempting to assume that one variable caused the other—especially if you want it to be true—which is why you need to keep this in mind any time you run regressions or analyze relationships between variables.

With the basics under your belt, here’s a deeper explanation of regression analysis so you can leverage it to drive strategic planning and decision-making.

Related: How to Learn Business Analytics without a Business Background

What Is Regression Analysis?

Regression analysis is the statistical method used to determine the structure of a relationship between two variables (single linear regression) or three or more variables (multiple regression).

According to the Harvard Business School Online course Business Analytics , regression is used for two primary purposes:

• To study the magnitude and structure of the relationship between variables
• To forecast a variable based on its relationship with another variable

Both of these insights can inform strategic business decisions.

“Regression allows us to gain insights into the structure of that relationship and provides measures of how well the data fit that relationship,” says HBS Professor Jan Hammond, who teaches Business Analytics, one of three courses that comprise the Credential of Readiness (CORe) program . “Such insights can prove extremely valuable for analyzing historical trends and developing forecasts.”

One way to think of regression is by visualizing a scatter plot of your data with the independent variable on the X-axis and the dependent variable on the Y-axis. The regression line is the line that best fits the scatter plot data. The regression equation represents the line’s slope and the relationship between the two variables, along with an estimation of error.

Physically creating this scatter plot can be a natural starting point for parsing out the relationships between variables.

Types of Regression Analysis

There are two types of regression analysis: single variable linear regression and multiple regression.

Single variable linear regression is used to determine the relationship between two variables: the independent and dependent. The equation for a single variable linear regression looks like this:

In the equation:

• ŷ is the expected value of Y (the dependent variable) for a given value of X (the independent variable).
• x is the independent variable.
• α is the Y-intercept, the point at which the regression line intersects with the vertical axis.
• β is the slope of the regression line, or the average change in the dependent variable as the independent variable increases by one.
• ε is the error term, equal to Y – ŷ, or the difference between the actual value of the dependent variable and its expected value.

Multiple regression , on the other hand, is used to determine the relationship between three or more variables: the dependent variable and at least two independent variables. The multiple regression equation looks complex but is similar to the single variable linear regression equation:

Each component of this equation represents the same thing as in the previous equation, with the addition of the subscript k, which is the total number of independent variables being examined. For each independent variable you include in the regression, multiply the slope of the regression line by the value of the independent variable, and add it to the rest of the equation.

How to Run Regressions

You can use a host of statistical programs—such as Microsoft Excel, SPSS, and STATA—to run both single variable linear and multiple regressions. If you’re interested in hands-on practice with this skill, Business Analytics teaches learners how to create scatter plots and run regressions in Microsoft Excel, as well as make sense of the output and use it to drive business decisions.

Calculating Confidence and Accounting for Error

It’s important to note: This overview of regression analysis is introductory and doesn’t delve into calculations of confidence level, significance, variance, and error. When working in a statistical program, these calculations may be provided or require that you implement a function. When conducting regression analysis, these metrics are important for gauging how significant your results are and how much importance to place on them.

Why Use Regression Analysis?

Once you’ve generated a regression equation for a set of variables, you effectively have a roadmap for the relationship between your independent and dependent variables. If you input a specific X value into the equation, you can see the expected Y value.

This can be critical for predicting the outcome of potential changes, allowing you to ask, “What would happen if this factor changed by a specific amount?”

Returning to the earlier example, running a regression analysis could allow you to find the equation representing the relationship between employee satisfaction and product sales. You could input a higher level of employee satisfaction and see how sales might change accordingly. This information could lead to improved working conditions for employees, backed by data that shows the tie between high employee satisfaction and sales.

Whether predicting future outcomes, determining areas for improvement, or identifying relationships between seemingly unconnected variables, understanding regression analysis can enable you to craft data-driven strategies and determine the best course of action with all factors in mind.

Do you want to become a data-driven professional? Explore our eight-week Business Analytics course and our three-course Credential of Readiness (CORe) program to deepen your analytical skills and apply them to real-world business problems.

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Statistics By Jim

Making statistics intuitive

Five Regression Analysis Tips to Avoid Common Problems

By Jim Frost 18 Comments

In this post, I offer five tips that will not only help you avoid common problems but also make the modeling process easier. I’ll close by showing you the difference between the modeling process that a top analyst uses versus the procedure of a less rigorous analyst.

Tip 1: Conduct A Lot of Research Before Starting

Before you begin the regression analysis, you should review the literature to develop an understanding of the relevant variables, their relationships, and the expected coefficient signs and effect magnitudes. Developing your knowledge base helps you gather the correct data in the first place, and it allows you to specify the best regression equation without resorting to data mining.

Regrettably, large data bases stuffed with handy data combined with automated model building procedures have pushed analysts away from this knowledge based approach. Data mining procedures can build a misleading model that has significant variables and a good R-squared using randomly generated data!

In my blog post, Using Data Mining to Select Regression Model Can Create Serious Problems , I show this in action. The output below is a model that stepwise regression built from entirely random data. In the final step, the R-squared is decently high, and all of the variables have very low p-values!

Automated model building procedures can have a place in the exploratory phase. However, you can’t expect them to produce the correct model precisely. For more information, read my Guide to Stepwise Regression and Best Subsets Regression .

Tip 2: Use a Simple Model When Possible

It seems that complex problems should require complicated regression equations. However, studies show that simplification usually produces more precise models.* How simple should the models be? In many cases, three independent variables are sufficient for complex problems.

The tip is to start with a simple a model and then make it more complicated only when it is truly needed. If you make a model more complex, confirm that the prediction intervals are more precise (narrower). When you have several models with comparable predictive abilities, choose the simplest because it is likely to be the best model. Another benefit is that simpler models are easier to understand and explain to others!

As you make a model more elaborate, the R-squared increases, but it becomes more likely that you are customizing it to fit the vagaries of your specific dataset rather than actual relationships in the population . This overfitting reduces generalizability and produces results that you can’t trust.

Learn how both adjusted R-squared and predicted R-squared can help you include the correct number of variables and avoid overfitting.

Related post : Overfitting Regression Models: Problems, Detection, and Avoidance

Tip 3: Correlation Does Not Imply Causation . . . Even in Regression

Correlation does not imply causation. Statistics classes have burned this familiar mantra into the brains of all statistics students! It seems simple enough. However, analysts can forget this important rule while performing regression analysis. As you build a model that has significant variables and a high R-squared, it’s easy to forget that you might only be revealing correlation. Causation is an entirely different matter. Typically, to establish causation, you need to perform a designed experiment with randomization. If you’re using regression to analyze data that weren’t collected in such an experiment, you can’t be certain about causation.

Fortunately, correlation can be just fine in some cases. For instance, if you want to predict the outcome , you don’t always need variables that have causal relationships with the dependent variable. If you measure a variable that is related to changes in the outcome but doesn’t influence the outcome, you can still obtain good predictions. Sometimes it is easier to measure these proxy variables. However, if your goal is to affect the outcome by setting the values of the input variables, you must identify variables with truly causal relationships.

For example, if vitamin consumption is only correlated with improved health but does not cause good health, then altering vitamin use won’t improve your health. There must be a causal relationship between two variables for changes in one to cause changes in the other.

Related posts : Causation versus Correlation in Statistics

Tip 4: Include Graphs, Confidence, and Prediction Intervals in the Results

This tip focuses on the fact that how you present your results can influence how people interpret them. The information can be the same, but the presentation style can prompt different reactions. For instance, confidence intervals and statistical significance provide consistent information . When a p-value is less than the 0.05 significance level , the corresponding 95% confidence interval will always exclude zero. However, the impact on the reader is very different.

A study by Cumming* finds that statistical reports which refer only to statistical significance bring about correct interpretations only 40% of the time. When the results also include confidence intervals, the percentage rises to 95%! Other research by Soyer and Hogarth* show dramatic increases in correct interpretations when you include graphs in regression analysis reports . In general, you want to make the statistical results as intuitively understandable as possible.

Related post : Confidence Intervals vs Prediction Intervals vs Tolerance Intervals .

Tip 5: Check Your Residual Plots!

For more information, read my post: Check Your Residual Plots to Ensure Trustworthy Regression Results!

Differences Between a Top Analyst and a Less Rigorous Analyst

Top analysts tend to do the following:

• Conducts research to understand the study area before starting.
• Uses large quantities of reliable data and a few independent variables with well established relationships.
• Uses sound reasoning to determine which variables to include in the regression model.
• Combines different lines of research as needed.
• Presents the results using charts, prediction intervals, and confidence intervals in a lucid manner that ensures the appropriate interpretation by others.

On the other hand, a less rigorous analyst tends to do the following:

• Does not do the research to understand the research area and similar studies.
• Uses regression outside of designed experiments to hunt for causal relationships.
• Uses data-mining to rummage for relationships because databases provide a lot of convenient data.
• Includes variables in the model based mainly on statistical significance.
• Uses a complicated model to increase R-squared.
• Reports only the basic statistics of coefficients, p-values , and R-squared values.

I hope these regression analysis tips have helped you out! Do you have any tips of your own to share? For more information about how to choose the best model, read my post: Model Specification: Choosing the Correct Regression Model .

If you’re learning regression, check out my Regression Tutorial !

Armstrong J., Illusions in Regression Analysis, International Journal of Forecasting , 2012 (3), 689-694.

Ord, K. (2012), The Illusion of Predictability: A call to action, International Journal of Forecasting, March 5, 2012 .

Soyer, E. and Robin M. Hogarth, The illusion of predictability: How regression statistics mislead experts, International Journal of Forecasting , Volume 28, Issue 3, July–September 2012, Pages 695-711.

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Reader Interactions

April 17, 2022 at 11:26 pm

Hello Jim, How can we use regression if we have a equation of total income and sources of income. I.e. total income = wages + rent + profit + gambling

This data is deterministic, I am unable to do regression.

Is there some way to use some othe variable and do regression.

Also how can we use regression to show before and after covid 19 effect in regression ?

August 30, 2021 at 7:45 pm

Thanks for the great insights here and elsewhere. I’m trying to find the best model building method within the limitations of Excel to teach my intro business stats students. (These are not stats students, so simpler is always better.)

I’m assuming best subsets is not feasible with more than two or three independent variables. So my first question is which method would you recommend assuming Excel is our only software?

Forgive my second question, which will no doubt reveal my lack of theoretical experience with regression, but I’d like to understand, when building the best model, why adding another variable can increase the adjusted r squared even when the added variable itself has an insignificant p-value? In other words, what takes precedence, the adjusted r squared or the variable’s p-value? In the example I’m working on (a textbook example!), I started with two independent variables ( p = 0.00, p = 0.03) and an adjusted r squared of 0.45. In adding a third variable, my adjusted r squared jumps to 0.47, but now two of the three variables have insignificant p-values. In adding a fourth variable, my r squared jumps to 0.55 and all four variables now have significant p-values. How can this be best explained and which of the models should we use?

(I have just purchased your book on regression, so perhaps the answers are already there, but I have yet to find an intro stats textbook that addresses regression statistics that appear to contradict one another.)

Thanks for your time!

August 31, 2021 at 2:19 am

Hi Bradley,

I think the best approach for Excel is the manual model reduction approach. As far as I know, Excel doesn’t have any automated model fitting process. But, you can fit the full model and then one-by-one remove any variables that are not significant. For example, if you have multiple non-significant variables, remove the one with the highest p-value but leave the other non-significant variables in and refit. Repeat until there are no insignificant variables. Of course, as with any automated method, check to make sure that the final model and the signs and magnitudes of the coefficients make theoretical sense. By the way, I have written a post about using Excel to fit regression models that you might be interested in.

For your second question, that’s a good one that I’m willing to be that most don’t know the answer to! When the t-value is greater than 1, the adjusted R-squared increases. That’s just a byproduct baked into its calculations. However, for statistical significant, the t-value needs to be ~1.96, depending on the DF. Consequently, there’s a range from t = 1 to 1.96 where the adjusted R-squared will increase but the variable is not significant.

Unfortunately, it’s impossible for me to answer your question definitively about which model to use given your example. It’s not just about the statistics involved, and they can point you in different directions, as you’ve seen! It also involves subject-area knowledge about which variables should and should not be include and an evaluation of the coefficient signs and magnitude to see if they make theoretical sense. Of course, you also need to check the residuals. Patterns in residuals would tell you the model needs fixing regardless of what the various statistics say! I do include a discussion about model specification in my regression book. It’s the entirety of Chapter 7. I talk about all the issues I mention here along with others.

Off hand, I’d lean towards the model with four variables assuming it passes everything I mention here given that it has the highest adjusted R-squared and all the variables are significant. Unless you have many fewer than 40 observations, in which case you might be overfitting your model. But, the differences in your adjusted R-squared really aren’t that large, so dropping some variables if need wouldn’t be problematic.

I hope that helps!

February 1, 2020 at 6:42 pm

Dear Sir, Thanks a ton for your patience with me, I know I am taking most of your time. Its just this last thing. Refer to this post ‘https://statisticsbyjim.com/regression/standard-error-regression-vs-r-squared/’, there is fitted line plot graph there. This graph basically shows S measuring the precision of the model’s predictions. Consequently, it uses S to obtain a rough estimate of the 95% prediction interval. I want to know what are there data points you are referring to?? Are these the residuals?? How do I get the values of these?? I am gonna be using the standard 95% of the data points should fall within a range that extends from +/- 2 * standard error of the regression from the fitted line only but I want to make this fitted line graph to explain and check whether do my data points fall or not. So my confusion is how to make this graph, can I make it on minitab? Also I have found prediction data as per this post, ‘https://statisticsbyjim.com/regression/prediction-precision-applied-regression/’, does any of the output will help me make this graph? Sir I am not talking about what percentage my predictions should be useful at, I am still not well familiar with regression so I cant decide with that on my own so I will take what you have mentioned in your post ‘95% of the data points should fall within a range that extends from +/- 2 * standard error of the regression from the fitted line’, the problem is how do I get this the fitted line plot graph for multiple regression. Also can I directly make this graph on minitab?

Thanks a tom for your patience with me.

Best Regards, Anshum Saran

February 4, 2020 at 5:04 pm

You can find the dataset for creating that graph you’re asking about in my post about making predictions with regression analysis . In that post, you’ll find the link to the dataset. Take that dataset and then use the fitted line plot feature in Minitab. Have Minitab display the prediction intervals, which is one of the options.

You can use fitted line plots only when you have one predictor. That’s because you need one axis for the predictor and one for the response. If you have more predictors, you’d need extra dimensions! While you can’t use a fitted line plot with multiple regression, I show you how to use Minitab’s prediction feature to calculate prediction intervals when you have multiple predictors. That’s what you’ll need to do with your data.

Again, you should buy my regression ebook which covers this in more detail.

February 1, 2020 at 10:30 am

Sir, I read your reply and I am confused. I also read both the posts as well. I did find out where to check the S value, but I’m not sure what am i comparing it to. I cannot find any study material about this online as well. Can you help make me understand. I am very new to this and I am really trying my best to understand it. The post regarding the precision in predictions just discuss one random prediction, While the other posts shows a graph. I want to make that graph how do I do it? what will be on x axis and what will be on the y axis. How will I make my regression line. Is there a way to make the graph on minitab? I have this software which im using for my analysis.

Thanks and appreciate your help and time you have taken out for replying back.

Regards, Anshum

February 1, 2020 at 4:20 pm

Ah, I see where the confusion lies now. As the analyst and subject-area expert, you need to supply the value for comparison. For your predictions, how much precision do you need for them to be useful. I talk about that in the post about the standard error of the regression. The model doesn’t make predictions that are precise enough to be useful. What is considered useful varies by subject-matter and application. There is no statistical measure for determining how much precision is required. So, you’ll need to use your subject-area knowledge and standards of the field to figure out how precise you need the predictions for them to be useful. Then, compare your S to the S required for sufficiently precise prediction. You can also use prediction intervals for the same purpose.

I’ve made all my graphs in Minitab. I’m not sure which graphs you’re referring to. I have multiple graphs in multiple posts. Please specify precisely which graphs in which posts.

February 1, 2020 at 2:46 am

Good Day Sir, I appreciate your reply, I am a bit confused about the part where you talk about ‘You can use S in your output (24236.7) for a good estimate of the precision. 95% of new observations will fall within +/- 2*S from the predicted value’ I was unable to understand how do I prove this?? I did read your posts related to this but unfortunately it only talks about it using linear regression. Can this be used in multiple regression as well?? I an using minitab and it doesnt have this feature. Can you please help me with this last question.

Please explain how do I use my Std Error in verifying my model.

Thanks and Regards, Anshum

February 1, 2020 at 3:08 am

I had to remove your data and output because it was rather long. However, you need to use the standard error of the regression (S), rather than the standardized residuals. In Minitab, you’ll find S listed in Model Summary table, which is the same section as R-squared. For more information about this statistic, read my post about the standard error of the regression . Multiple regression is linear regression. So, yes, you can use it multiple regression. You’ll find it right there in the output. 🙂

To see precision used in a multiple regression context, read my post about precision in predictions . One of the examples uses multiple predictors. That post focuses on using prediction intervals, but you’ll see S in the example output in the Model Summary table.

January 20, 2020 at 4:44 am

Good Day Sir, I had written to you earlier as well, I haven’t got a reply yet. I am trying to do a multiple regression forecasting, but I am unable to interpret the results from the ANOVA table. I have used the obtained equation for forecasting my values and have found some promising results. I request you to please have a look at my analysis and help me understand and interpret the same. Sir If you look at my p value for the regression equation, it is within the significance, but the p value for the other variables (independent) are above the value of significance. I am not sure how to interpret it also the value of t and f in the analysis. Also if this is a success and it can be modelled and used for my forecasting. I would love to hear from you at your soonest convenience as I have to submit my thesis by the end of this month and have to run it by my supervisor. Thanks in advance for your valuable time and feedback.

Regression Equation:

1 Year Timecharter Rate Capesiz = 120984 – 6.90 Average Haul Iron Ore and Coal – 0.00203 Capesize Bulkcarrier Demolition

Model Summary

S R-sq R-sq(adj) R-sq(pred) 24236.7 55.78% 47.74% 35.23%

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value Regression 2 8151898785 4075949392 6.94 0.011 Average Haul Iron Ore and Coal 1 2095391492 2095391492 3.57 0.086 Capesize Bulkcarrier Demolition 1 840607035 840607035 1.43 0.257 Error 11 6461589869 587417261 Total 13 14613488654

Thanks and Best Regards, Anshum

January 20, 2020 at 10:56 am

First off, given that your thesis depends on regression analysis and the extensive nature of your questions, I highly recommend that you get my ebook about regression analysis .

Your overall F-test of significance says that the model is statistically significant but there’s not enough evidence to suggest that any of the individual predictors are significant. I write about this condition in my post about the overall F-test . I think part of the problem is that you have very few observations, which lowers the power of the analysis.

Typically, you don’t interpret the t and F-values directly. Those are the test statistics which the analysis uses to calculate the p-values. So, you can just interpret the p-values. Read my post for more information about how to interpret the coefficients and p-values .

In terms of using the model to make predictions, you’d need to first check the residual plots to be sure that the model provides a good fit for the data. Otherwise, the predictions might be biased. You can’t make that determination from the numeric output. Assuming the residual plots look good, I see one additional problem. Your R-squared and particularly predicted R-squared are low. While R-squared is often overrated, a fairly high R-squared is important when you need to make precise predictions. Your predictions are likely to be imprecise. You can use S in your output (24236.7) for a good estimate of the precision. 95% of new observations will fall within +/- 2*S from the predicted value. For more information about these concepts, read the following posts: Making Predictions with Regression Analysis : pay particular attention to the sections on precision Understand Precision to Avoid Costly Mistakes : again, focus on precision S vs R-squared : More about how S is better than R-squared when it comes to precision

In a nutshell, given the small sample size, lack of significant predictors, and low R-squared (and particularly the low predicted R-squared), your model doesn’t provide much explanatory power. Predictions based on the model are likely to be too imprecise to be useful (although you can assess the precision using information I provided to make the determination).

Best of luck with your thesis!

November 1, 2017 at 5:25 pm

Very helpful! Thanks!

November 1, 2017 at 5:54 pm

Thank you, Vivian!

October 30, 2017 at 1:43 pm

Thank You Jim for spreading knowledge. I am working on a research paper where I want develop a regression model for which the variables haven’t been used before in any literature. I have two simple questions . 1. If i have 4 independent variables how will i Show them in mathematical form secondly if my independent variables have correlation can i include them in my model

October 30, 2017 at 1:59 pm

Hi, thank you for writing. I’m not sure that I understand your first question. After you fit the model, you’ll see the regression equation in the output. That’s how to write the mathematical form. As for correlated independent variables, or multicollinearity as it is called, yes, some correlation is OK. You need to check the VIFs. If they are less than 5, you should be good. I write a blog post about multicollinearity that you should read.

I hope this helps! Jim

September 17, 2017 at 1:43 am

Thank you very much for informative posts. I am goning to conduct a 3 levels ordered logistic regression analysis on the world value survey data using stata and I´m not sure what should I check before building models? would you plz kindly explain

September 18, 2017 at 2:12 pm

Hi Toktam, the very first thing I’d do is research what others have done in this area. Maybe others have even used the same data for the same reason? At the very least, you want to learn about the area, see what others have found, and see what variables should be related to your dependent variable. This process helps you with identifying candidate variables and determining whether your results make sense. Check out my blog post about model specification for more ideas.

September 21, 2017 at 6:22 am

Thank you for prompt reply. can I ask more detailed questions about particular issues in multilevel analysis (using stata) ?

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Home » Regression Analysis – Methods, Types and Examples

Regression Analysis – Methods, Types and Examples

Table of Contents

Regression Analysis

Regression analysis is a set of statistical processes for estimating the relationships among variables . It includes many techniques for modeling and analyzing several variables when the focus is on the relationship between a dependent variable and one or more independent variables (or ‘predictors’).

Regression Analysis Methodology

Here is a general methodology for performing regression analysis:

• Define the research question: Clearly state the research question or hypothesis you want to investigate. Identify the dependent variable (also called the response variable or outcome variable) and the independent variables (also called predictor variables or explanatory variables) that you believe are related to the dependent variable.
• Collect data: Gather the data for the dependent variable and independent variables. Ensure that the data is relevant, accurate, and representative of the population or phenomenon you are studying.
• Explore the data: Perform exploratory data analysis to understand the characteristics of the data, identify any missing values or outliers, and assess the relationships between variables through scatter plots, histograms, or summary statistics.
• Choose the regression model: Select an appropriate regression model based on the nature of the variables and the research question. Common regression models include linear regression, multiple regression, logistic regression, polynomial regression, and time series regression, among others.
• Assess assumptions: Check the assumptions of the regression model. Some common assumptions include linearity (the relationship between variables is linear), independence of errors, homoscedasticity (constant variance of errors), and normality of errors. Violation of these assumptions may require additional steps or alternative models.
• Estimate the model: Use a suitable method to estimate the parameters of the regression model. The most common method is ordinary least squares (OLS), which minimizes the sum of squared differences between the observed and predicted values of the dependent variable.
• I nterpret the results: Analyze the estimated coefficients, p-values, confidence intervals, and goodness-of-fit measures (e.g., R-squared) to interpret the results. Determine the significance and direction of the relationships between the independent variables and the dependent variable.
• Evaluate model performance: Assess the overall performance of the regression model using appropriate measures, such as R-squared, adjusted R-squared, and root mean squared error (RMSE). These measures indicate how well the model fits the data and how much of the variation in the dependent variable is explained by the independent variables.
• Test assumptions and diagnose problems: Check the residuals (the differences between observed and predicted values) for any patterns or deviations from assumptions. Conduct diagnostic tests, such as examining residual plots, testing for multicollinearity among independent variables, and assessing heteroscedasticity or autocorrelation, if applicable.
• Make predictions and draw conclusions: Once you have a satisfactory model, use it to make predictions on new or unseen data. Draw conclusions based on the results of the analysis, considering the limitations and potential implications of the findings.

Types of Regression Analysis

Types of Regression Analysis are as follows:

Linear Regression

Linear regression is the most basic and widely used form of regression analysis. It models the linear relationship between a dependent variable and one or more independent variables. The goal is to find the best-fitting line that minimizes the sum of squared differences between observed and predicted values.

Multiple Regression

Multiple regression extends linear regression by incorporating two or more independent variables to predict the dependent variable. It allows for examining the simultaneous effects of multiple predictors on the outcome variable.

Polynomial Regression

Polynomial regression models non-linear relationships between variables by adding polynomial terms (e.g., squared or cubic terms) to the regression equation. It can capture curved or nonlinear patterns in the data.

Logistic Regression

Logistic regression is used when the dependent variable is binary or categorical. It models the probability of the occurrence of a certain event or outcome based on the independent variables. Logistic regression estimates the coefficients using the logistic function, which transforms the linear combination of predictors into a probability.

Ridge Regression and Lasso Regression

Ridge regression and Lasso regression are techniques used for addressing multicollinearity (high correlation between independent variables) and variable selection. Both methods introduce a penalty term to the regression equation to shrink or eliminate less important variables. Ridge regression uses L2 regularization, while Lasso regression uses L1 regularization.

Time Series Regression

Time series regression analyzes the relationship between a dependent variable and independent variables when the data is collected over time. It accounts for autocorrelation and trends in the data and is used in forecasting and studying temporal relationships.

Nonlinear Regression

Nonlinear regression models are used when the relationship between the dependent variable and independent variables is not linear. These models can take various functional forms and require estimation techniques different from those used in linear regression.

Poisson Regression

Poisson regression is employed when the dependent variable represents count data. It models the relationship between the independent variables and the expected count, assuming a Poisson distribution for the dependent variable.

Generalized Linear Models (GLM)

GLMs are a flexible class of regression models that extend the linear regression framework to handle different types of dependent variables, including binary, count, and continuous variables. GLMs incorporate various probability distributions and link functions.

Regression Analysis Formulas

Regression analysis involves estimating the parameters of a regression model to describe the relationship between the dependent variable (Y) and one or more independent variables (X). Here are the basic formulas for linear regression, multiple regression, and logistic regression:

Linear Regression:

Simple Linear Regression Model: Y = β0 + β1X + ε

Multiple Linear Regression Model: Y = β0 + β1X1 + β2X2 + … + βnXn + ε

In both formulas:

• Y represents the dependent variable (response variable).
• X represents the independent variable(s) (predictor variable(s)).
• β0, β1, β2, …, βn are the regression coefficients or parameters that need to be estimated.
• ε represents the error term or residual (the difference between the observed and predicted values).

Multiple Regression:

Multiple regression extends the concept of simple linear regression by including multiple independent variables.

Multiple Regression Model: Y = β0 + β1X1 + β2X2 + … + βnXn + ε

The formulas are similar to those in linear regression, with the addition of more independent variables.

Logistic Regression:

Logistic regression is used when the dependent variable is binary or categorical. The logistic regression model applies a logistic or sigmoid function to the linear combination of the independent variables.

Logistic Regression Model: p = 1 / (1 + e^-(β0 + β1X1 + β2X2 + … + βnXn))

In the formula:

• p represents the probability of the event occurring (e.g., the probability of success or belonging to a certain category).
• X1, X2, …, Xn represent the independent variables.
• e is the base of the natural logarithm.

The logistic function ensures that the predicted probabilities lie between 0 and 1, allowing for binary classification.

Regression Analysis Examples

Regression Analysis Examples are as follows:

• Stock Market Prediction: Regression analysis can be used to predict stock prices based on various factors such as historical prices, trading volume, news sentiment, and economic indicators. Traders and investors can use this analysis to make informed decisions about buying or selling stocks.
• Demand Forecasting: In retail and e-commerce, real-time It can help forecast demand for products. By analyzing historical sales data along with real-time data such as website traffic, promotional activities, and market trends, businesses can adjust their inventory levels and production schedules to meet customer demand more effectively.
• Energy Load Forecasting: Utility companies often use real-time regression analysis to forecast electricity demand. By analyzing historical energy consumption data, weather conditions, and other relevant factors, they can predict future energy loads. This information helps them optimize power generation and distribution, ensuring a stable and efficient energy supply.
• Online Advertising Performance: It can be used to assess the performance of online advertising campaigns. By analyzing real-time data on ad impressions, click-through rates, conversion rates, and other metrics, advertisers can adjust their targeting, messaging, and ad placement strategies to maximize their return on investment.
• Predictive Maintenance: Regression analysis can be applied to predict equipment failures or maintenance needs. By continuously monitoring sensor data from machines or vehicles, regression models can identify patterns or anomalies that indicate potential failures. This enables proactive maintenance, reducing downtime and optimizing maintenance schedules.
• Financial Risk Assessment: Real-time regression analysis can help financial institutions assess the risk associated with lending or investment decisions. By analyzing real-time data on factors such as borrower financials, market conditions, and macroeconomic indicators, regression models can estimate the likelihood of default or assess the risk-return tradeoff for investment portfolios.

Importance of Regression Analysis

Importance of Regression Analysis is as follows:

• Relationship Identification: Regression analysis helps in identifying and quantifying the relationship between a dependent variable and one or more independent variables. It allows us to determine how changes in independent variables impact the dependent variable. This information is crucial for decision-making, planning, and forecasting.
• Prediction and Forecasting: Regression analysis enables us to make predictions and forecasts based on the relationships identified. By estimating the values of the dependent variable using known values of independent variables, regression models can provide valuable insights into future outcomes. This is particularly useful in business, economics, finance, and other fields where forecasting is vital for planning and strategy development.
• Causality Assessment: While correlation does not imply causation, regression analysis provides a framework for assessing causality by considering the direction and strength of the relationship between variables. It allows researchers to control for other factors and assess the impact of a specific independent variable on the dependent variable. This helps in determining the causal effect and identifying significant factors that influence outcomes.
• Model Building and Variable Selection: Regression analysis aids in model building by determining the most appropriate functional form of the relationship between variables. It helps researchers select relevant independent variables and eliminate irrelevant ones, reducing complexity and improving model accuracy. This process is crucial for creating robust and interpretable models.
• Hypothesis Testing: Regression analysis provides a statistical framework for hypothesis testing. Researchers can test the significance of individual coefficients, assess the overall model fit, and determine if the relationship between variables is statistically significant. This allows for rigorous analysis and validation of research hypotheses.
• Policy Evaluation and Decision-Making: Regression analysis plays a vital role in policy evaluation and decision-making processes. By analyzing historical data, researchers can evaluate the effectiveness of policy interventions and identify the key factors contributing to certain outcomes. This information helps policymakers make informed decisions, allocate resources effectively, and optimize policy implementation.
• Risk Assessment and Control: Regression analysis can be used for risk assessment and control purposes. By analyzing historical data, organizations can identify risk factors and develop models that predict the likelihood of certain outcomes, such as defaults, accidents, or failures. This enables proactive risk management, allowing organizations to take preventive measures and mitigate potential risks.

When to Use Regression Analysis

• Prediction : Regression analysis is often employed to predict the value of the dependent variable based on the values of independent variables. For example, you might use regression to predict sales based on advertising expenditure, or to predict a student’s academic performance based on variables like study time, attendance, and previous grades.
• Relationship analysis: Regression can help determine the strength and direction of the relationship between variables. It can be used to examine whether there is a linear association between variables, identify which independent variables have a significant impact on the dependent variable, and quantify the magnitude of those effects.
• Causal inference: Regression analysis can be used to explore cause-and-effect relationships by controlling for other variables. For example, in a medical study, you might use regression to determine the impact of a specific treatment while accounting for other factors like age, gender, and lifestyle.
• Forecasting : Regression models can be utilized to forecast future trends or outcomes. By fitting a regression model to historical data, you can make predictions about future values of the dependent variable based on changes in the independent variables.
• Model evaluation: Regression analysis can be used to evaluate the performance of a model or test the significance of variables. You can assess how well the model fits the data, determine if additional variables improve the model’s predictive power, or test the statistical significance of coefficients.
• Data exploration : Regression analysis can help uncover patterns and insights in the data. By examining the relationships between variables, you can gain a deeper understanding of the data set and identify potential patterns, outliers, or influential observations.

Applications of Regression Analysis

Here are some common applications of regression analysis:

• Economic Forecasting: Regression analysis is frequently employed in economics to forecast variables such as GDP growth, inflation rates, or stock market performance. By analyzing historical data and identifying the underlying relationships, economists can make predictions about future economic conditions.
• Financial Analysis: Regression analysis plays a crucial role in financial analysis, such as predicting stock prices or evaluating the impact of financial factors on company performance. It helps analysts understand how variables like interest rates, company earnings, or market indices influence financial outcomes.
• Marketing Research: Regression analysis helps marketers understand consumer behavior and make data-driven decisions. It can be used to predict sales based on advertising expenditures, pricing strategies, or demographic variables. Regression models provide insights into which marketing efforts are most effective and help optimize marketing campaigns.
• Health Sciences: Regression analysis is extensively used in medical research and public health studies. It helps examine the relationship between risk factors and health outcomes, such as the impact of smoking on lung cancer or the relationship between diet and heart disease. Regression analysis also helps in predicting health outcomes based on various factors like age, genetic markers, or lifestyle choices.
• Social Sciences: Regression analysis is widely used in social sciences like sociology, psychology, and education research. Researchers can investigate the impact of variables like income, education level, or social factors on various outcomes such as crime rates, academic performance, or job satisfaction.
• Operations Research: Regression analysis is applied in operations research to optimize processes and improve efficiency. For example, it can be used to predict demand based on historical sales data, determine the factors influencing production output, or optimize supply chain logistics.
• Environmental Studies: Regression analysis helps in understanding and predicting environmental phenomena. It can be used to analyze the impact of factors like temperature, pollution levels, or land use patterns on phenomena such as species diversity, water quality, or climate change.
• Sports Analytics: Regression analysis is increasingly used in sports analytics to gain insights into player performance, team strategies, and game outcomes. It helps analyze the relationship between various factors like player statistics, coaching strategies, or environmental conditions and their impact on game outcomes.

Advantages and Disadvantages of Regression Analysis

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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The complete guide to regression analysis.

19 min read What is regression analysis and why is it useful? While most of us have heard the term, understanding regression analysis in detail may be something you need to brush up on. Here’s what you need to know about this popular method of analysis.

When you rely on data to drive and guide business decisions, as well as predict market trends, just gathering and analyzing what you find isn’t enough — you need to ensure it’s relevant and valuable.

The challenge, however, is that so many variables can influence business data: market conditions, economic disruption, even the weather! As such, it’s essential you know which variables are affecting your data and forecasts, and what data you can discard.

And one of the most effective ways to determine data value and monitor trends (and the relationships between them) is to use regression analysis, a set of statistical methods used for the estimation of relationships between independent and dependent variables.

In this guide, we’ll cover the fundamentals of regression analysis, from what it is and how it works to its benefits and practical applications.

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What is regression analysis?

Regression analysis is a statistical method. It’s used for analyzing different factors that might influence an objective – such as the success of a product launch, business growth, a new marketing campaign – and determining which factors are important and which ones can be ignored.

Regression analysis can also help leaders understand how different variables impact each other and what the outcomes are. For example, when forecasting financial performance, regression analysis can help leaders determine how changes in the business can influence revenue or expenses in the future.

Running an analysis of this kind, you might find that there’s a high correlation between the number of marketers employed by the company, the leads generated, and the opportunities closed.

This seems to suggest that a high number of marketers and a high number of leads generated influences sales success. But do you need both factors to close those sales? By analyzing the effects of these variables on your outcome,  you might learn that when leads increase but the number of marketers employed stays constant, there is no impact on the number of opportunities closed, but if the number of marketers increases, leads and closed opportunities both rise.

Regression analysis can help you tease out these complex relationships so you can determine which areas you need to focus on in order to get your desired results, and avoid wasting time with those that have little or no impact. In this example, that might mean hiring more marketers rather than trying to increase leads generated.

How does regression analysis work?

Regression analysis starts with variables that are categorized into two types: dependent and independent variables. The variables you select depend on the outcomes you’re analyzing.

Understanding variables:

1. dependent variable.

This is the main variable that you want to analyze and predict. For example, operational (O) data such as your quarterly or annual sales, or experience (X) data such as your net promoter score (NPS) or customer satisfaction score (CSAT) .

These variables are also called response variables, outcome variables, or left-hand-side variables (because they appear on the left-hand side of a regression equation).

There are three easy ways to identify them:

• Is the variable measured as an outcome of the study?
• Does the variable depend on another in the study?
• Do you measure the variable only after other variables are altered?

2. Independent variable

Independent variables are the factors that could affect your dependent variables. For example, a price rise in the second quarter could make an impact on your sales figures.

You can identify independent variables with the following list of questions:

• Is the variable manipulated, controlled, or used as a subject grouping method by the researcher?
• Does this variable come before the other variable in time?
• Are you trying to understand whether or how this variable affects another?

Independent variables are often referred to differently in regression depending on the purpose of the analysis. You might hear them called:

Explanatory variables

Explanatory variables are those which explain an event or an outcome in your study. For example, explaining why your sales dropped or increased.

Predictor variables

Predictor variables are used to predict the value of the dependent variable. For example, predicting how much sales will increase when new product features are rolled out .

Experimental variables

These are variables that can be manipulated or changed directly by researchers to assess the impact. For example, assessing how different product pricing ($10 vs $15 vs $20) will impact the likelihood to purchase.

Subject variables (also called fixed effects)

Subject variables can’t be changed directly, but vary across the sample. For example, age, gender, or income of consumers.

Unlike experimental variables, you can’t randomly assign or change subject variables, but you can design your regression analysis to determine the different outcomes of groups of participants with the same characteristics. For example, ‘how do price rises impact sales based on income?’

Carrying out regression analysis

So regression is about the relationships between dependent and independent variables. But how exactly do you do it?

Assuming you have your data collection done already, the first and foremost thing you need to do is plot your results on a graph. Doing this makes interpreting regression analysis results much easier as you can clearly see the correlations between dependent and independent variables.

Let’s say you want to carry out a regression analysis to understand the relationship between the number of ads placed and revenue generated.

On the Y-axis, you place the revenue generated. On the X-axis, the number of digital ads. By plotting the information on the graph, and drawing a line (called the regression line) through the middle of the data, you can see the relationship between the number of digital ads placed and revenue generated.

This regression line is the line that provides the best description of the relationship between your independent variables and your dependent variable. In this example, we’ve used a simple linear regression model.

Statistical analysis software can draw this line for you and precisely calculate the regression line. The software then provides a formula for the slope of the line, adding further context to the relationship between your dependent and independent variables.

Simple linear regression analysis

A simple linear model uses a single straight line to determine the relationship between a single independent variable and a dependent variable.

This regression model is mostly used when you want to determine the relationship between two variables (like price increases and sales) or the value of the dependent variable at certain points of the independent variable (for example the sales levels at a certain price rise).

While linear regression is useful, it does require you to make some assumptions.

For example, it requires you to assume that:

• the data was collected using a statistically valid sample collection method that is representative of the target population
• The observed relationship between the variables can’t be explained by a ‘hidden’ third variable – in other words, there are no spurious correlations.
• the relationship between the independent variable and dependent variable is linear – meaning that the best fit along the data points is a straight line and not a curved one

Multiple regression analysis

As the name suggests, multiple regression analysis is a type of regression that uses multiple variables. It uses multiple independent variables to predict the outcome of a single dependent variable. Of the various kinds of multiple regression, multiple linear regression is one of the best-known.

Multiple linear regression is a close relative of the simple linear regression model in that it looks at the impact of several independent variables on one dependent variable. However, like simple linear regression, multiple regression analysis also requires you to make some basic assumptions.

For example, you will be assuming that:

• there is a linear relationship between the dependent and independent variables (it creates a straight line and not a curve through the data points)
• the independent variables aren’t highly correlated in their own right

An example of multiple linear regression would be an analysis of how marketing spend, revenue growth, and general market sentiment affect the share price of a company.

With multiple linear regression models you can estimate how these variables will influence the share price, and to what extent.

Multivariate linear regression

Multivariate linear regression involves more than one dependent variable as well as multiple independent variables, making it more complicated than linear or multiple linear regressions. However, this also makes it much more powerful and capable of making predictions about complex real-world situations.

For example, if an organization wants to establish or estimate how the COVID-19 pandemic has affected employees in its different markets, it can use multivariate linear regression, with the different geographical regions as dependent variables and the different facets of the pandemic as independent variables (such as mental health self-rating scores, proportion of employees working at home, lockdown durations and employee sick days).

Through multivariate linear regression, you can look at relationships between variables in a holistic way and quantify the relationships between them. As you can clearly visualize those relationships, you can make adjustments to dependent and independent variables to see which conditions influence them. Overall, multivariate linear regression provides a more realistic picture than looking at a single variable.

However, because multivariate techniques are complex, they involve high-level mathematics that require a statistical program to analyze the data.

Logistic regression

Logistic regression models the probability of a binary outcome based on independent variables.

So, what is a binary outcome? It’s when there are only two possible scenarios, either the event happens (1) or it doesn’t (0). e.g. yes/no outcomes, pass/fail outcomes, and so on. In other words, if the outcome can be described as being in either one of two categories.

Logistic regression makes predictions based on independent variables that are assumed or known to have an influence on the outcome. For example, the probability of a sports team winning their game might be affected by independent variables like weather, day of the week, whether they are playing at home or away and how they fared in previous matches.

What are some common mistakes with regression analysis?

Across the globe, businesses are increasingly relying on quality data and insights to drive decision-making — but to make accurate decisions, it’s important that the data collected and statistical methods used to analyze it are reliable and accurate.

Using the wrong data or the wrong assumptions can result in poor decision-making, lead to missed opportunities to improve efficiency and savings, and — ultimately — damage your business long term.

• Assumptions

When running regression analysis, be it a simple linear or multiple regression, it’s really important to check that the assumptions your chosen method requires have been met. If your data points don’t conform to a straight line of best fit, for example, you need to apply additional statistical modifications to accommodate the non-linear data. For example, if you are looking at income data, which scales on a logarithmic distribution, you should take the Natural Log of Income as your variable then adjust the outcome after the model is created.

• Correlation vs. causation

It’s a well-worn phrase that bears repeating – correlation does not equal causation. While variables that are linked by causality will always show correlation, the reverse is not always true. Moreover, there is no statistic that can determine causality (although the design of your study overall can).

If you observe a correlation in your results, such as in the first example we gave in this article where there was a correlation between leads and sales, you can’t assume that one thing has influenced the other. Instead, you should use it as a starting point for investigating the relationship between the variables in more depth.

• Choosing the wrong variables to analyze

Before you use any kind of statistical method, it’s important to understand the subject you’re researching in detail. Doing so means you’re making informed choices of variables and you’re not overlooking something important that might have a significant bearing on your dependent variable.

• Model building The variables you include in your analysis are just as important as the variables you choose to exclude. That’s because the strength of each independent variable is influenced by the other variables in the model. Other techniques, such as Key Drivers Analysis, are able to account for these variable interdependencies.

Benefits of using regression analysis

There are several benefits to using regression analysis to judge how changing variables will affect your business and to ensure you focus on the right things when forecasting.

Here are just a few of those benefits:

Make accurate predictions

Regression analysis is commonly used when forecasting and forward planning for a business. For example, when predicting sales for the year ahead, a number of different variables will come into play to determine the eventual result.

Regression analysis can help you determine which of these variables are likely to have the biggest impact based on previous events and help you make more accurate forecasts and predictions.

Identify inefficiencies

Using a regression equation a business can identify areas for improvement when it comes to efficiency, either in terms of people, processes, or equipment.

For example, regression analysis can help a car manufacturer determine order numbers based on external factors like the economy or environment.

Using the initial regression equation, they can use it to determine how many members of staff and how much equipment they need to meet orders.

Drive better decisions

Improving processes or business outcomes is always on the minds of owners and business leaders, but without actionable data, they’re simply relying on instinct, and this doesn’t always work out.

This is particularly true when it comes to issues of price. For example, to what extent will raising the price (and to what level) affect next quarter’s sales?

There’s no way to know this without data analysis. Regression analysis can help provide insights into the correlation between price rises and sales based on historical data.

How do businesses use regression? A real-life example

Marketing and advertising spending are common topics for regression analysis. Companies use regression when trying to assess the value of ad spend and marketing spend on revenue.

A typical example is using a regression equation to assess the correlation between ad costs and conversions of new customers. In this instance,

• our dependent variable (the factor we’re trying to assess the outcomes of) will be our conversions
• the independent variable (the factor we’ll change to assess how it changes the outcome) will be the daily ad spend
• the regression equation will try to determine whether an increase in ad spend has a direct correlation with the number of conversions we have

The analysis is relatively straightforward — using historical data from an ad account, we can use daily data to judge ad spend vs conversions and how changes to the spend alter the conversions.

By assessing this data over time, we can make predictions not only on whether increasing ad spend will lead to increased conversions but also what level of spending will lead to what increase in conversions. This can help to optimize campaign spend and ensure marketing delivers good ROI.

This is an example of a simple linear model. If you wanted to carry out a more complex regression equation, we could also factor in other independent variables such as seasonality, GDP, and the current reach of our chosen advertising networks.

By increasing the number of independent variables, we can get a better understanding of whether ad spend is resulting in an increase in conversions, whether it’s exerting an influence in combination with another set of variables, or if we’re dealing with a correlation with no causal impact – which might be useful for predictions anyway, but isn’t a lever we can use to increase sales.

Using this predicted value of each independent variable, we can more accurately predict how spend will change the conversion rate of advertising.

Regression analysis tools

Regression analysis is an important tool when it comes to better decision-making and improved business outcomes. To get the best out of it, you need to invest in the right kind of statistical analysis software.

The best option is likely to be one that sits at the intersection of powerful statistical analysis and intuitive ease of use, as this will empower everyone from beginners to expert analysts to uncover meaning from data, identify hidden trends and produce predictive models without statistical training being required.

To help prevent costly errors, choose a tool that automatically runs the right statistical tests and visualizations and then translates the results into simple language that anyone can put into action.

With software that’s both powerful and user-friendly, you can isolate key experience drivers, understand what influences the business, apply the most appropriate regression methods, identify data issues, and much more.

With Qualtrics’ Stats iQ™, you don’t have to worry about the regression equation because our statistical software will run the appropriate equation for you automatically based on the variable type you want to monitor. You can also use several equations, including linear regression and logistic regression, to gain deeper insights into business outcomes and make more accurate, data-driven decisions.

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Teach yourself statistics

Linear Regression Example

In this lesson, we apply regression analysis to some fictitious data, and we show how to interpret the results of our analysis.

Note: Your browser does not support HTML5 video. If you view this web page on a different browser (e.g., a recent version of Edge, Chrome, Firefox, or Opera), you can watch a video treatment of this lesson.

Note: Regression computations are usually handled by a software package or a graphing calculator. For this example, however, we will do the computations "manually", since the gory details have educational value.

Problem Statement

Last year, five randomly selected students took a math aptitude test before they began their statistics course. The Statistics Department has three questions.

• What linear regression equation best predicts statistics performance, based on math aptitude scores?
• If a student made an 80 on the aptitude test, what grade would we expect her to make in statistics?
• How well does the regression equation fit the data?

How to Find the Regression Equation

In the table below, the x i column shows scores on the aptitude test. Similarly, the y i column shows statistics grades. The last two columns show deviations scores - the difference between the student's score and the average score on each measurement. The last two rows show sums and mean scores that we will use to conduct the regression analysis.

And for each student, we also need to compute the squares of the deviation scores (the last two columns in the table below).

And finally, for each student, we need to compute the product of the deviation scores (the last column in the table below).

The regression equation is a linear equation of the form: ŷ = b 0 + b 1 x . To conduct a regression analysis, we need to solve for b 0 and b 1 . Computations are shown below. Notice that all of our inputs for the regression analysis come from the above three tables.

First, we solve for the regression coefficient (b 1 ):

b 1 = Σ [ (x i - x )(y i - y ) ] / Σ [ (x i - x ) 2 ]

b 1 = 470/730

b 1 = 0.644

Once we know the value of the regression coefficient (b 1 ), we can solve for the regression slope (b 0 ):

b 0 = y - b 1 * x

b 0 = 77 - (0.644)(78)

b 0 = 26.768

Therefore, the regression equation is: ŷ = 26.768 + 0.644x .

How to Use the Regression Equation

Once you have the regression equation, using it is a snap. Choose a value for the independent variable ( x ), perform the computation, and you have an estimated value (ŷ) for the dependent variable.

In our example, the independent variable is the student's score on the aptitude test. The dependent variable is the student's statistics grade. If a student made an 80 on the aptitude test, the estimated statistics grade (ŷ) would be:

ŷ = b 0 + b 1 x

ŷ = 26.768 + 0.644x = 26.768 + 0.644 * 80

ŷ = 26.768 + 51.52 = 78.288

Warning: When you use a regression equation, do not use values for the independent variable that are outside the range of values used to create the equation. That is called extrapolation , and it can produce unreasonable estimates.

In this example, the aptitude test scores used to create the regression equation ranged from 60 to 95. Therefore, only use values inside that range to estimate statistics grades. Using values outside that range (less than 60 or greater than 95) is problematic.

How to Find the Coefficient of Determination

Whenever you use a regression equation, you should ask how well the equation fits the data. One way to assess fit is to check the coefficient of determination , which can be computed from the following formula.

R 2 = { ( 1 / N ) * Σ [ (x i - x ) * (y i - y ) ] / (σ x * σ y ) } 2

where N is the number of observations used to fit the model, Σ is the summation symbol, x i is the x value for observation i, x is the mean x value, y i is the y value for observation i, y is the mean y value, σ x is the standard deviation of x, and σ y is the standard deviation of y.

Computations for the sample problem of this lesson are shown below. We begin by computing the standard deviation of x (σ x ):

σ x = sqrt [ Σ ( x i - x ) 2 / N ]

σ x = sqrt( 730/5 ) = sqrt(146) = 12.083

Next, we find the standard deviation of y, (σ y ):

σ y = sqrt [ Σ ( y i - y ) 2 / N ]

σ y = sqrt( 630/5 ) = sqrt(126) = 11.225

R 2 = [ ( 1/5 ) * 470 / ( 12.083 * 11.225 ) ] 2

R 2 = ( 94 / 135.632 ) 2 = ( 0.693 ) 2 = 0.48

A coefficient of determination equal to 0.48 indicates that about 48% of the variation in statistics grades (the dependent variable ) can be explained by the relationship to math aptitude scores (the independent variable ). This would be considered a good fit to the data, in the sense that it would substantially improve an educator's ability to predict student performance in statistics class.

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What Is Regression Analysis? Types, Importance, and Benefits

In this post

Regression analysis basics

• How does regression analysis work?
• Types of regression analysis

When is regression analysis used?

Benefits of regression analysis, applications of regression analysis, top statistical analysis software.

Businesses collect data to make better decisions.

But when you count on data for building strategies, simplifying processes, and improving customer experience, more than collecting it, you need to understand and analyze it to be able to draw valuable insights. Analyzing data helps you study what’s already happened and predict what may happen in the future. 

Data analysis has many components, and while some can be easy to understand and perform, others are rather complex. The good news is that many statistical analysis software offer meaningful insights from data in a few steps.

You have to understand the fundamentals before using or relying on a statistical program to give accurate results because even though generating results is easy, interpreting them is another ballgame. 

While interpreting data, considering the factors that affect the data becomes essential. Regression analysis helps you do just that. With the assistance of this statistical analysis method , you can find the most important and least important factors in any data set and understand how they relate. 

This guide covers the fundamentals of regression analysis, its process, benefits, and applications.

What is regression analysis? 

Regression analysis is a statistical process that helps assess the relationships between a dependent variable and one or more independent variables.

The primary purpose of regression analysis is to describe the relationship between variables, but it can also be used to:

• Estimate the value of one variable using the known values of other variables.
• Predict results and shifts in a variable based on its relationship with other variables. 
• Control the influence of variables while exploring the relationship between variables.  

To understand regression analysis comprehensively, you must build foundational knowledge of the statistical concepts.

Regression analysis helps identify the factors that impact data insights. You can use it to understand which factors play a role in creating an outcome and how significant they are. These factors are called variables.

You need to grasp two main types of variables.

• The main factor you're focusing on is the dependent variable . This variable is often measured as an outcome of analyses and depends on one or more other variables.
• The factors or variables that impact your dependent variable are called independent variables . Variables like these are often altered for analysis. They’re also called explanatory variables or predictor variables.

Correlation vs. causation 

Causation indicates that one variable is the result of the occurrence of the other variable. Correlation suggests a connection between variables. Correlation and causation can coexist, but correlation does not imply causation. 

Overfitting

Overfitting is a statistical modeling error that occurs when a function lines up with a limited set of data points and makes predictions based on those instead of exploring new data points. As a result, the model can only be used as a reference to its initial data set and not to any other data sets.

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How does regression analysis work .

For a minute, let's imagine that you own an ice cream stand. In this case, we can consider “revenue” and “temperature” to be the two factors under analysis. The first step toward conducting a successful regression statistical analysis is gathering data on the variables. 

You collect all your monthly sales numbers for the past two years and any data on the independent variables or explanatory variables you’re analyzing. In this case, it’s the average monthly temperature for the past two years.

To begin to understand whether there’s a relationship between these two variables, you need to plot these data points on a graph that looks like the following theoretical example of a scatter plot:

The amount of sales is represented on the y-axis (vertical axis), and temperature is represented on the x-axis (horizontal axis). The dots represent one month's data – the average temperature and sales in that same month.

Observing this data shows that sales are higher on days when the temperature increases. But by how much? If the temperature goes higher, how much do you sell? And what if the temperature drops? 

Drawing a regression line roughly in the middle of all the data points helps you figure out how much you typically sell when it’s a specific temperature. Let’s use a theoretical scatter plot to depict a regression line: 

The regression line explains the relationship between the predicted values and dependent variables. It can be created using statistical analysis software or Microsoft Excel. 

Your regression analysis tool must also display a formula that defines the slope of the line. For example: 

y = 100 + 2x + error term

On observing the formula, you can conclude that when there is no x , y equals 100, which means that when the temperature is very low, you can make an average of 100 sales. Provided the other variables remain constant, you can use this to predict the future of sales. For every rise in the temperature, you make an average of two more sales.

A regression line always has an error term because an independent variable cannot be a perfect predictor of a dependent variable. Deciding whether this variable is worth your attention depends on the error term – the larger the error term, the less certain the regression line. 

Types of regression analysis 

Various types of regression analysis are at your disposal, but the five mentioned below are the most commonly used.

Linear regression

A linear regression model is defined as a straight line that attempts to predict the relationship between variables. It’s mainly classified into two types: simple and multiple linear regression. 

We’ll discuss those in a moment, but let’s first cover the five fundamental assumptions made in the linear regression model. 

• The dependent and independent variables display a linear relationship.
• The value of the residual is zero.
• The value of the residual is constant and not correlated across all observations.
• The residual is normally distributed.
• Residual errors are homoscedastic – they have a constant variance.

Simple linear regression analysis 

Linear regression analysis helps predict a variable's value (dependent variable) based on the known value of one other variable (independent variable).

Linear regression fits a straight line, so a simple linear model attempts to define the relationship between two variables by estimating the coefficients of the linear equation.

Simple linear regression equation:

Y = a + bX + ϵ

Where, Y – Dependent variable (response variable) X – Independent variable (predictor variable) a – Intercept (y-intercept) b – Slope ϵ – Residual (error)

I n such a linear regression model, a response variable has a single corresponding predictor variable that impacts its value. For example, consider the linear regression formula:

  y = 5x + 4  

If the value of x is defined as 3, only one possible outcome of y is possible.

Multiple linear regression analysis

In most cases, simple linear regression analysis can't explain the connections between data. As the connection becomes more complex, the relationship between data is better explained using more than one variable. 

Multiple regression analysis describes a response variable using more than one predictor variable. It is used when a strong correlation between each independent variable has the ability to affect the dependent variable. 

Multiple linear regression equation: 

Y = a + bX1 + cX2 + dX3 + ϵ

Where, Y – Dependent variable X1, X2, X3 – Independent variables a – Intercept (y-intercept) b, c, d – Slopes ϵ – Residual (error)

Ordinary least squares

Ordinary Least Squares regression estimates the unknown parameters in a model. It estimates the coefficients of a linear regression equation by minimizing the sum of the squared errors between the actual and predicted values configured as a straight line.

Polynomial regression

A linear regression algorithm only works when the relationship between the data is linear. What if the data distribution was more complex, as shown in the figure below?  

As seen above, the data is nonlinear. A linear model can't be used to fit nonlinear data because it can't sufficiently define the patterns in the data.

Polynomial regression is a type of multiple linear regression used when data points are present in a nonlinear manner. It can determine the curvilinear relationship between independent and dependent variables having a nonlinear relationship.

Polynomial regression equation: 

y = b0+b1x1+ b2x1^2+ b2x1^3+...... bnx1^n

Logistic regression

Logistic regression models the probability of a dependent variable as a function of independent variables. The values of a dependent variable can take one of a limited set of binary values (0 and 1) since the outcome is a probability. 

Logistic regression is often used when binary data (yes or no; pass or fail) needs to be analyzed. In other words, using the logistic regression method to analyze your data is recommended if your dependent variable can have either one of two binary values.

Let’s say you need to determine whether an email is spam. We need to set up a threshold based on which the classification can be done. Using logistic regression here makes sense as the outcome is strictly bound to 0 (spam) or 1 (not spam) values.  

Bayesian linear regression

In other regression methods, the output is derived from one or more attributes. But what if those attributes are unavailable? 

The bayesian regression method is used when the dataset that needs to be analyzed has less or poorly distributed data because its output is derived from a probability distribution instead of point estimates. When data is absent, you can place a prior on the regression coefficients to substitute the data. As we add more data points, the accuracy of the regression model improves. 

Imagine a company launches a new product and wants to predict its sales. Due to the lack of available data, we can’t use a simple regression analysis model. But Bayesian regression analysis lets you set up a prior and calculate future projections.

Additionally, once new data from the new product sales come in, the prior is immediately updated. As a result, the forecast for the future is influenced by the latest and previous data. 

The Bayesian technique is mathematically robust. Because of this, it doesn’t require you to have any prior knowledge of the dataset during usage. However, its complexity means it takes time to draw inferences from the model, and using it doesn't make sense when you have too much data.

Quantile regression analysis

The linear regression method estimates a variable's mean based on the values of other predictor variables. But we don’t always need to calculate the conditional mean. In most situations, we only need the median, the 0.25 quantile, and so on. In cases like this, we can use quantile regression. 

Quantile regression defines the relationship between one or more predictor variables and specific percentiles or quantiles of a response variable. It resists the influence of outlying observations. No assumptions about the distribution of the dependent variable are made in quantile regression, so you can use it when linear regression doesn’t satisfy its assumptions. 

Let's consider two students who have taken an Olympiad exam open for all age groups. Student A scored 650, while student B scored 425. This data shows that student A has performed better than student B. 

But quantile regression helps remind us that since the exam was open for all age groups, we have to factor in r the student's age to determine the correct outcome in their individual conditional quantile spaces. 

We know the variable causing such a difference in the data distribution. As a result, the scores of the students are compared for the same age groups.

What is regularization? 

Regularization is a technique that prevents a regression model from overfitting by including extra information. It’s implemented by adding a penalty term to the data model. It allows you to keep the same number of features by reducing the magnitude of the variables. It reduces the magnitude of the coefficient of features toward zero.

The two types of regularization techniques are L1 and L2. A regression model using the L1 regularization technique is known as Lasso regression, and the one using the L2 regularization technique is called Ridge regression.

Ridge regression

Ridge regression is a regularization technique you would use to eliminate the correlations between independent variables (multicollinearity) or when the number of independent variables in a set exceeds the number of observations. 

Ridge regression performs L2 regularization. In such a regularization, the formula used to make predictions is the same for ordinary least squares, but a penalty is added to the square of the magnitude of regression coefficients. This is done so that each feature has as little effect on the outcome as possible. 

Lasso regression

Lasso stands for Least Absolute Shrinkage and Selection Operator. 

Lasso regression is a regularized linear regression that uses an L1 penalty that pushes some regression coefficient values to become closer to zero. By setting features to zero, it automatically chooses the required feature and avoids overfitting.

So if the dataset has high correlation, high levels of multicollinearity, or when specific features such as variable selection or parameter elimination need to be automated, you can use lasso regression.

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Regression analysis is a powerful tool used to derive statistical inferences for the future using observations from the past . It identifies the connections between variables occurring in a dataset and determines the magnitude of these associations and their significance on outcomes.

Across industries, it’s a useful statistical analysis tool because it provides exceptional flexibility. So the next time someone at work proposes a plan that depends on multiple factors, perform a regression analysis to predict an accurate outcome. 

In the real world, various factors determine how a business grows. Often these factors are interrelated, and a change in one can positively or negatively affect the other. 

Using regression analysis to judge how changing variables will affect your business has two primary benefits.

• Making data-driven decisions: Businesses use regression analysis when planning for the future because it helps determine which variables have the most significant impact on the outcome according to previous results. Companies can better focus on the right things when forecasting and making data-backed predictions.
• Recognizing opportunities to improve: Since regression analysis shows the relations between two variables, businesses can use it to identify areas of improvement in terms of people, strategies, or tools by observing their interactions. For example, increasing the number of people on a project might positively impact revenue growth . 

Both small and large industries are loaded with an enormous amount of data. To make better decisions and eliminate guesswork, many are now adopting regression analysis because it offers a scientific approach to management.

Using regression analysis, professionals can observe and evaluate the relationship between various variables and subsequently predict this relationship's future characteristics. 

Companies can utilize regression analysis in numerous forms. Some of them:

• Many finance professionals use regression analysis to forecast future opportunities and risks . The capital asset pricing model (CAPM) that decides the relationship between an asset's expected return and the associated market risk premium is an often-used regression model in finance for pricing assets and discovering capital costs. Regression analysis is also used to calculate beta (β), which is described as the volatility of returns while considering the overall market for a stock.
• Insurance firms use regression analysis to forecast the creditworthiness of a policyholder . It can also help choose the number of claims that may be raised in a specific period.
• Sales forecasting uses regression analysis to predict sales based on past performance. It can give you a sense of what has worked before, what kind of impact it has created, and what can improve to provide more accurate and beneficial future results. 
• Another critical use of regression models is the optimization of business processes . Today, managers consider regression an indispensable tool for highlighting the areas that have the maximum impact on operational efficiency and revenues, deriving new insights, and correcting process errors. 

Businesses with a data-driven culture use regression analysis to draw actionable insights from large datasets. For many leading industries with extensive data catalogs, it proves to be a valuable asset. As the data size increases, further executives lean into regression analysis to make informed business decisions with statistical significance. 

While Microsoft Excel remains a popular tool for conducting fundamental regression data analysis, many more advanced statistical tools today drive more accurate and faster results. Check out the top statistical analysis software in 2023 here. 

To be included in this category, the regression analysis software product must be able to:

• Execute a simple linear regression or a complex multiple regression analysis for various data sets.
• Provide graphical tools to study model estimation, multicollinearity, model fits, line of best fit, and other aspects typical of the type of regression.
• Possess a clean, intuitive, and user-friendly user interface (UI) design

*Below are the top 5 leading statistical analysis software solutions from G2’s Winter 2023 Grid® Report. Some reviews may be edited for clarity.

1. IBM SPSS statistics

IBM SPSS Statistics allows you to predict the outcomes and apply various nonlinear regression procedures that can be used for business and analysis projects where standard regression techniques are limiting or inappropriate. With IBM SPSS Statistics, you can specify multiple regression models in a single command to observe the correlation between independent and dependent variables and expand regression analysis capabilities on a dataset.

What users like best :

"I have used a couple of different statistical softwares. IBM SPSS is an amazing software, a one-stop shop for all statistics-related analysis. The graphical user interface is elegantly built for ease. I was quickly able to learn and use it"

- IBM SPSS Statistics Review , Haince Denis P.

What users dislike:

"Some of the interfaces could be more intuitive. Thankfully much information is available from various sources online to help the user learn how to set up tests."

- IBM SPSS Statistics Review , David I.

To make data science more intuitive and collaborative, Posit provides users across key industries with R and Python-based tools, enabling them to leverage powerful analytics and gather valuable insights.

What users like best:

"Straightforward syntax, excellent built-in functions, and powerful libraries for everything else. Building anything from simple mathematical functions to complicated machine learning models is a breeze."

- Posit Review , Brodie G.

"Its GUI could be more intuitive and user-friendly. One needs a lot of time to understand and implement it. Including a package manager would be a good idea, as it has become common in many modern IDEs. There must be an option to save console commands, which is currently unavailable."

- Posit Review , Tanishq G.

JMP is a data analysis software that helps make sense of your data using cutting-edge and modern statistical methods. Its products are intuitively interactive, visually compelling, and statistically profound. 

"The instructional videos on the website are great; I had no clue what I was doing before I watched them. The videos make the application very user-friendly."

- JMP Review , Ashanti B.

"Help function can be brief in terms of what the functionality entails, and that's disappointing because the way the software is set up to communicate data visually and intuitively suggests the presence of a logical and explainable scientific thought process, including an explanation of the "why.” The graph builder could also use more intuitive means to change layout features."

- JMP Review , Zeban K.

4. Minitab statistical software

Minitab Statistical Software is a data and statistical analysis tool used to help businesses understand their data and make better decisions. It allows companies to tap into the power of regression analysis by analyzing new and old data to discover trends, predict patterns, uncover hidden relationships between variables, and create stunning visualizations. 

"The greatest program for learning and analyzing as it allows you to improve the settings with incredibly accurate graphs and regression charts. This platform allows you to analyze the outcomes or data with their ideal values."

- Minitab Statistical Software Review , Pratibha M.

"The software price is steep, and licensing is troublesome. You are required to be online or connected to the company VPN for licensing, especially for corporate use. So without an internet connection, you cannot use it at all. Also, if you are in the middle of doing an analysis and happen to lose your internet connection, you will risk losing the project or the study you are working on."

- Minitab Statistical Software Review , Siew Kheong W.

EViews offers user-friendly tools to perform data modeling and forecasting. It operates with an innovative, easy-to-use object-oriented interface used by researchers, financial institutions, government agencies, and educators.

"As an economist, this software is handy as it assists me in conducting advanced research, analyzing data, and interpreting results for policy recommendations. I just cannot do without EViews. I like its recent updates that have also enhanced the UI."

- EViews Review , T homas M.

"In my experience, importing data from Excel is not easy using EViews compared to other statistical software. One needs to develop expertise while importing data into EViews from different formats. Moreover, the price of the software is very high."

 - EViews Review , Md. Zahid H.

Collecting data gathers no moss.

Data collection has become easy in the modern world, but more than just gathering is required. Businesses must know how to get the most value from this data. Analysis helps companies to understand the available information, derive actionable insights, and make informed decisions. Businesses should thoroughly know the data analysis process inside and out to refine operations, improve customer service, and track performance. 

Learn more ab out the various stages of data analysis and implement them to drive success. 

Devyani Mehta

Devyani Mehta is a content marketing specialist at G2. She has worked with several SaaS startups in India, which has helped her gain diverse industry experience. At G2, she shares her insights on complex cybersecurity concepts like web application firewalls, RASP, and SSPM. Outside work, she enjoys traveling, cafe hopping, and volunteering in the education sector. Connect with her on LinkedIn.

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Simple Linear Regression Examples

Many of simple  linear regression examples (problems and solutions) from the real life can be given to help you understand the core meaning.

On this page:

• Simple linear regression examples: problems with solutions .
• Infographic in PDF

In our previous post linear regression models , we explained in details what is simple and multiple linear regression. Here, we concentrate on the examples of linear regression from the real life.

Simple Linear Regression Examples, Problems, and Solutions

Simple linear regression allows us to study the correlation between only two variables:

• One variable (X) is called independent variable or predictor.
• The other variable (Y), is known as dependent variable or outcome.

and the simple linear regression equation is:

Y = Β 0  + Β 1 X

X  – the value of the independent variable, Y  – the value of the dependent variable. Β 0  – is a constant (shows the value of Y when the value of X=0) Β 1  – the regression coefficient (shows how much Y changes for each unit change in X)

You have to study the relationship between the monthly e-commerce sales and the online advertising costs. You have the survey results for 7 online stores for the last year.

Your task is to find the equation of the straight line that fits the data best.

The following table represents the survey results from the 7 online stores.

We can see that there is a positive relationship between the monthly e-commerce sales (Y) and online advertising costs (X).

The positive correlation means that the values of the dependent variable (y) increase when the values of the independent variable (x) rise.

So, if we want to predict the monthly e-commerce sales from the online advertising costs, the higher the value of advertising costs, the higher our prediction of sales.

We will use the above data to build our Scatter diagram.

Now, let’ see how the Scatter diagram looks like:

The Scatter plot shows how much one variable affects another. In our example, above Scatter plot shows how much online advertising costs affect the monthly e-commerce sales. It shows their correlation.

Let’s see the simple linear regression equation.

Y = 125.8 + 171.5*X

Note : You can find easily the values for Β 0   and Β 1 with the help of paid or free statistical software, online linear regression calculators or Excel. All you need are the values for the independent (x) and dependent (y) variables (as those in the above table).

Now, we have to see our regression line:

Graph of the Regression Line:

Linear regression aims to find the best-fitting straight line through the points. The best-fitting line is known as the regression line.

If data points are closer when plotted to making a straight line, it means the correlation between the two variables is higher. In our example, the relationship is strong.

The orange diagonal line in diagram 2 is the regression line and shows the predicted score on e-commerce sales for each possible value of the online advertising costs.

Interpretation of the results:

The slope of 171.5 shows that each increase of one unit in X, we predict the average of Y to increase by an estimated 171.5 units.

The formula estimates that for each increase of 1 dollar in online advertising costs, the expected monthly e-commerce sales are predicted to increase by $171.5.

This was a simple linear regression example for a positive relationship in business. Let’s see an example of the negative relationship.

You have to examine the relationship between the age and price for used cars sold in the last year by a car dealership company.

Here is the table of the data:

Now, we see that we have a negative relationship between the car price (Y) and car age(X) – as car age increases, price decreases.

When we use the simple linear regression equation, we have the following results:

Y = 7836 – 502.4*X

Let’s use the data from the table and create our Scatter plot and linear regression line:

The above 3 diagrams are made with  Meta Chart .

Result Interpretation:

With an estimated slope of – 502.4, we can conclude that the average car price decreases $502.2 for each year a car increases in age.

If you need more examples in the field of statistics and data analysis or more data visualization types , our posts “ descriptive statistics examples ” and “ binomial distribution examples ” might be useful to you.

Download the following infographic in PDF with the simple linear regression examples:

About The Author

Silvia Valcheva

Silvia Valcheva is a digital marketer with over a decade of experience creating content for the tech industry. She has a strong passion for writing about emerging software and technologies such as big data, AI (Artificial Intelligence), IoT (Internet of Things), process automation, etc.

Hi. I really enjoy your article, seems to me that it can help to many students in order to improve their skills. Thanks,

solved perfectly, great article

Thanks Silvia for your articles. They are quite informative and easy to understand.

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Regression Analysis: A Comprehensive Guide to Quantitative Forecasting

Regression analysis stands as a cornerstone in the realm of quantitative forecasting, offering an extensive suite of methods for researchers and analysts who seek to understand and predict relationships among variables. It is an indispensable statistical tool that aids decision-making across fields as varied as economics, medicine, and environmental studies. At its core, regression analysis is utilized to discern patterns in data, forecast future trends, optimize business strategies, and support scientific research.

This academic exposé delves into the intricacies of regression analysis, highlighting its multifaceted uses, strengths, and limitations. We begin by establishing a sound foundation on the topic and thereafter explore the types, methodology, outputs, applications, recent developments, and lastly provide a summation of its crucial role in today’s data-driven landscape.

Introduction to Regression Analysis

Definition of regression analysis.

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. The primary goal is to understand how the typical value of the dependent variable changes when any one of the independent variables is varied while the other independent variables are held fixed.

In simpler terms, it attempts to explain the variation in a variable of interest, such as sales or growth, by breaking it down into the effect of various factors.

Brief History and Evolution of Regression Analysis

Historically, regression analysis has its origins in the 19th century with Sir Francis Galton's work on heredity, coining the term "regression" to describe the phenomenon he observed that the heights of descendants of tall ancestors tended to regress down towards a normal average.

Over the years, the technique has evolved significantly, absorbing contributions from mathematicians and statisticians to become more sophisticated and applicable across various scientific disciplines.

Importance and Application of Regression Analysis in Various Fields

Unlocking the power of statistics with a probability-based approach.

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Unlocking the Secrets of Da Vinci: A Comprehensive Guide

Today, regression analysis is crucial across myriad fields for making informed decisions. In finance, it predicts stock prices, in marketing it analyzes consumer behavior, and in healthcare, it assesses treatment effectiveness. Its applications are not limited to these areas and its versatility is what makes it an essential analytical tool for professionals and researchers alike.

Types of Regression Analysis

Explanation of simple linear regression.

Simple linear regression is the most basic form of regression that involves predicting a quantitative response based on a single predictor variable. It is represented by the equation Y = a + bX + e, where Y is the dependent variable, X is the independent variable, a is the y-intercept, b is the slope, and e is the error term. The method assumes a straight-line relationship between the two variables.

Understanding Multiple Linear Regression

When more than one independent variable is present, multiple linear regression is employed. This method is capable of handling numerous predictors and gauging the influence of each on the dependent variable. It extends the simple linear regression model by incorporating multiple coefficients, one for each variable. This allows for a multi-dimensional analysis of data.

Unveiling the Concept of Polynomial Regression

Polynomial regression steps beyond the straight-line relationship and involves an equation where the power of the independent variable is greater than one. It is particularly useful when the relationship between variables is curvilinear. This type of regression can model a wider range of curves and can thus fit complex datasets more flexibly than simple or multiple linear regressions.

Overview of Ridge Regression

Ridge Regression is a technique used when data suffer from multicollinearity, where predictor variables are highly correlated. Unlike standard least squares regression, which can have significant problems in the presence of multicollinearity, Ridge Regression adds a degree of bias to the regression estimates, which serves to reduce the standard errors.

Understanding Lasso Regression

The Lasso Regression is similar to Ridge Regression but has the ability to reduce the coefficient estimates for the least important variables all the way to zero. This acts as a form of automatic variable selection and thus produces simpler and more interpretable models, which is particularly beneficial in the context of large datasets with many features.

Insights into Logistic Regression

Unlike the previously mentioned methods that predict quantitative outcomes, Logistic Regression is used for categorical dependent variables, particularly for binary classification. It estimates the probability that a certain event occurs, such as pass/fail, win/lose, alive/dead, based on an underlying linear relationship between the logits of the probabilities and the predictors.

Highlights on Stepwise Regression

Stepwise Regression is a method of fitting regression models in which the choice of predictive variables is carried out by an automated procedure. During this process, variables are added or subtracted from the multivariable model based on their statistical significance, often using F-tests or t-tests.

Other Types of Regression (Brief Overview)

There are numerous other types of regression analysis techniques available to address specific circumstances and datasets including Quantile Regression, Cox Regression, and Elastic Net Regression. Each carries its own assumptions, application contexts, and considerations, offering diverse tools for robust analysis of complex data patterns.

Steps in Conducting a Regression Analysis

Problem definition.

The first crucial step in conducting a regression analysis is defining the problem. This involves understanding the context and outlining the specific question or hypothesis that the regression model aims to address. A clear problem statement guides the direction of the analysis and ensures that the right type of regression analysis is employed.

Data Collection

Data collection comes as the second step. This phase involves gathering adequate and relevant data to work with. The quality and quantity of data collected directly influence the reliability of the analytical results. The researcher must pay attention to the sources, nature, and integrity of the data to mitigate any potential biases or errors.

Variables Identification

Once data is collected, identifying and classifying variables into independent and dependent categories is imperative. This process requires a thorough understanding of the dataset and the hypothesized relationships. Proper identification ensures that the appropriate modeling techniques are applied and that the findings from the analysis will be valid.

Model Specification

Model specification involves choosing the suitable regression model based upon the nature of the dependent variable and the shape of the relationship between the variables. Here, the researcher decides whether to use simple linear, multiple linear, or another type of regression and defines how the variables will be included in the model.

Model Fit and Assumptions Checking

Once the model is specified, fitting the model to the data is the next step. This includes estimating the regression coefficients. Additionally, checking the underlying assumptions of the selected regression model, such as linearity, independence of errors, homoscedasticity, and normality of error distributions, is critical to ensure accuracy and reliability of the results.

Interpretation of Results

The last step is interpreting the results obtained from the analysis. Coefficients need to be examined to understand the relationship between the independent variables and the dependent variable, the error term to check the model’s predictive power, and the significance levels to determine the reliability of the predictions. It's essential to report and interpret these findings in a manner that's comprehensible and actionable for the intended audience.

Understanding Regression Analysis Outputs

Deciphering coefficients.

In regression analysis, coefficients represent the magnitude and direction of the relationship between an independent variable and the dependent variable. Deciphering these values allows researchers to understand how much the dependent variable is expected to change with a one-unit change in the independent variable, holding all other variables constant.

Recognizing Error Term

The error term in a regression equation is indicative of the variation in the dependent variable that cannot be explained by the independent variables in the model. It represents the distance between the actual data points and the predicted values by the model, often reflecting information that was not accounted for in the model.

Understanding R-squared and Adjusted R-squared

R-squared, also known as the coefficient of determination, is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variables. Meanwhile, the Adjusted R-squared adjusts the R-squared value based on the number of predictors in the model, providing a more accurate reflection of the model's explanatory power, especially in the context of multiple regression.

Making Sense of Confidence Intervals and Significance Levels

Confidence intervals and significance levels are crucial in assessing the reliability of the regression estimates. Confidence intervals offer a range of values within which the true population parameter is likely to fall, while significance levels (often denoted by p-values) inform whether the observed relationship between the variables is statistically significant, not likely due to random chance.

Advantages and Limitations of Regression Analysis

Highlighting the strengths of regression analysis.

The major advantages of regression analysis include its ability to infer relationships, predict future values, and control for various confounding variables. It enables analysts to quantify the impact of changes in predictor variables on the outcome, making it an essential tool for data-driven decision-making.

Identifying the Weaknesses and Pitfalls

However, regression analysis is not without its limitations. The accuracy of the results depends heavily on the appropriateness of the selected model and underlying assumptions. Misinterpretation of results can occur if these conditions are not properly checked or understood. Influential points, multicollinearity, or autocorrelation can also distort the outcome, and it's critical to be aware of these potential pitfalls.

Practical Application of Regression Analysis

Regression analysis in business and economics.

In the realms of business and economics, regression analysis is frequently employed for demand forecasting, risk management, and optimizing operational efficiencies. Example usages include predicting sales based on advertising spends or assessing the impact of economic variables on market trends.

Role of Regression Analysis in Healthcare and Medicine

Healthcare and medicine leverage regression analysis to analyze patient outcomes, the efficacy of new drugs, and to calculate risk scores for diseases. It helps in building models that can predict health events or responses to treatments, contributing immensely to patient care and public health policies.

Use of Regression Analysis in Social Sciences

In the social sciences, regression analysis provides insights into the factors that influence human behavior and social phenomena. It's instrumental in fields such as psychology, sociology, and political science, where researchers can isolate and examine the effects of socioeconomic variables on various outcomes.

Regression Analysis in Environmental Studies

Environmental studies utilize regression analysis to model ecological processes and forecast environmental changes. For instance, understanding the factors that influence pollution levels or the impact of climate variables on species distributions.

Regression Analysis in Engineering

Engineers apply regression analysis for quality control, product design, and optimization. It assists in understanding how various design parameters affect the performance or reliability of engineered systems, leading to better and more efficient designs.

Recent Developments and Future Trends in Regression Analysis

Venture into machine learning integration with regression.

The interfacing of regression analysis with machine learning signifies a significant development, as it enhances predictive modeling with algorithms that can learn patterns from large datasets. Techniques such as regularized regression, like Lasso and Ridge mentioned earlier, are at the forefront of this overlap.

Overview of Big Data and Regression Analysis

As we stride deeper into the age of Big Data, regression analysis techniques adapt and evolve to handle the immense volume and complexity of data. Big data analytics often require sophisticated forms of regression that can process high-dimensional datasets efficiently and effectively.

Predictions and Future Trends in the Field

The future of regression analysis promises to unfurl with the continued integration of new computational techniques and the adoption of more robust statistical methodologies to accommodate evolving data trends. The omnipresence of data and the drive towards precision in predictions assure that regression analysis will persist as a linchpin in quantitative analysis for years to come.

Concluding Thoughts on Regression Analysis

Recap of key points of regression analysis.

From the simple linear models to complex machine learning integrations, regression analysis encompasses an expansive spectrum of methods tailored to interpret the past, illuminate the present, and predict the future. It provides a robust framework for quantitative forecasting and decision-making across a variety of domains.

Importance of Regression Analysis in Decision Making and Policy Formulation

The ability of regression analysis to distill insights from raw data and identify cause-and-effect relationships underpins its significant role in guiding policy formulation and strategic decision-making. Its structured approach enables stakeholders to make data-driven choices with increased confidence.

Encouraging Further Study and Application of Regression Analysis

The persistent evolution of analytical methods, alongside the increasing volume and variety of available data, underscores the importance of continuous learning and application of regression analysis techniques. Individuals and organizations are encouraged to invest in problem solving training courses and online certificate course offerings, broadening their analytical repertoire and enhancing their ability to harness the full potential of regression analysis in the data-rich world that lies ahead.

What are the primary assumptions made in regression analysis and why are they vital for accurate forecasting?

Understanding regression analysis.

Regression analysis stands as a statistical tool. It models relationships between variables. Forecasters and researchers rely on it heavily. For accurate forecasting, assumptions must hold true. Proper understanding of these assumptions ensures robust models.

Linearity Assumption

The linearity assumption is fundamental. It posits a linear relationship between predictor and outcome variables. When this assumption is violated, predictions become unreliable. Linearity can be checked with scatter plots or residual plots. Non-linear relationships require alternative modeling approaches.

Independence Assumption

Independence assumes observations are not correlated. When they are, we encounter autocorrelation . Autocorrelation distorts standard errors. This leads to incorrect statistical tests. Time series data often violate this assumption. Thus, special care is necessary in such analyses.

Homoscedasticity Assumption

Homoscedasticity implies constant variance of errors. Unequal variances, or heteroscedasticity , affect confidence intervals and hypothesis tests. This assumption can be scrutinized through residual plots. Corrective measures include transformations or robust standard errors.

Normality Assumption

Errors should distribute normally for precise hypothesis testing. Non-normality signals potential model issues. These may include incorrect specification or outliers. The normality assumption mainly affects small sample sizes.

No Multicollinearity Assumption

Multicollinearity exists when predictors correlate strongly. This complicates the interpretation of individual coefficients. Variance inflation factor (VIF) helps detect multicollinearity. High VIF values suggest a need to reconsider the model.

Why These Assumptions Matter

Assumptions in regression are not arbitrary. They cement the foundation for reliable results. Valid inference on coefficients depends on these. Accurate forecasting does too.

- Predictive Accuracy : Correct assumptions guide toward accurate predictions.

- Correct Inference : Meeting assumptions leads to valid hypothesis tests.

- Confidence in Results : Adhering to assumptions builds confidence in findings.

- Tool Selection : Awareness of assumptions guides the choice of statistical tools.

These conditions interlink to ensure that the regression models crafted produce outcomes close to reality. It is this adherence that transforms raw data into insightful, actionable forecasts. For those keen on extracting truth from numbers, the journey begins and ends with meeting these assumptions.

How is multicollinearity detected in regression analysis and what strategies can be used to address it?

Multicollinearity detection.

Detecting multicollinearity involves several statistical methods. Analysts often start with correlation matrices . Strong correlations suggest multicollinearity. Correlations close to 1 or -1 are red flags. Correlation coefficients represent the strength and direction of linear relationships. They range from -1 to 1. High absolute values indicate potential problems.

Variance Inflation Factor

Another key tool is the Variance Inflation Factor (VIF) . VIF quantifies multicollinearity severity. It measures how much variance increases for estimated regression coefficients. VIF values above 5 or 10 indicate high multicollinearity. Some experts accept a lower threshold. They consider VIF above 2.5 as problematic.

Tolerance Levels

VIF relates inversely to tolerance . Tolerance measures how well a model predicts without a predictor. Low tolerance values suggest multicollinearity. Values below 0.1 often warrant further investigation. They can signal that the independent variable has multicollinearity issues.

Eigenvalue Analysis

Eigenvalue analysis offers deeper insight. It involves decomposing the matrix. Small eigenvalues can show multicollinearity presence. Analysts compare them to the condition index. A condition index over 30 suggests serious multicollinearity.

Condition Index

The condition index is crucial. It measures matrix sensitivity to minor changes. High values can indicate numerical problems. They often flag high multicollinearity.

Addressing Multicollinearity

Omit variables.

One strategy is to omit variables . Multicollinear variables may not all be necessary. Removing one can solve the problem. Depth in understanding the data guides this choice. It involves model simplification .

Combine Variables

Another method is to combine variables . This can involve creating indices or scores. It reduces the number of predictors. It combines related information into a single predictor.

Principal Component Analysis

Principal Component Analysis (PCA) is more complex. It creates uncorrelated predictors. PCA transforms the data into principal components. These components help maintain the information. They do so without multicollinearity.

Regularization Techniques

Regularization techniques like Ridge regression adjust coefficients. They shrink them towards zero. This can reduce multicollinearity impacts. It ensures better generalization for the model.

Increase Sample Size

Lastly, increasing the sample size can help. More data provides more information. It can reduce variance in the estimates. It also lowers the chances of finding false relationships.

Understanding and addressing multicollinearity strengthens regression analysis. It ensures valid, reliable, and interpretable models. Analysts must detect and remedy this issue to ensure clear conclusions. We can better understand how variables really relate to each other. With this insight, we make more accurate predictions and better decisions.

How are outliers identified and treated in regression analysis to ensure reliability of the forecast?

Outliers in regression analysis, defining outliers.

Outliers present significant challenges in regression analysis. These are atypical observations. They deviate markedly from other data points. Analysts often spot them during preliminary data analysis. Outliers can distort predictions. They can affect the regression equation disproportionately. Accurate identification is crucial for reliable forecasting.

Identifying Outliers

Several methods aid outlier detection. Visual approaches include scatter plots. They allow quick outlier identification. Histograms and boxplots also serve this purpose. Statistical tests offer more precision. The Z-score method detects data points far from the mean. Grubbs' test identifies the most extreme outlier.

Standardizing Data

d -values standardize the difference between values. The interquartile range (IQR) method detects values beyond a threshold. Usually, these are 1.5 times the IQR above the third quartile. Or below the first quartile.

Treatment of Outliers

Once identified, several treatment options exist. Simplest is removal. This option suits clear errors or irrelevant data. Another approach involves transformation. It reduces the impact of extreme values. Logarithmic transformation is one example.

Advanced Methods

Robust regression techniques downplay outliers. They weigh them less in the analysis. This method maintains outlier inclusion while reducing influence. Winsorizing is another technique. It replaces extreme values. It uses the nearest value within the acceptable range.

Addressing Influential Points

Influential points affect regression results significantly. These outliers can skew regression lines dramatically. Cook’s Distance is a measure of influence. Analysts use it to assess each point's impact on the regression coefficients.

Testing and Validation

After outlier treatment, model reevaluation is necessary. One must check for improvement in model fit. Adjustments continue until the model shows robust predictive power. Cross-validation can assess the regression's reliability.

Outliers have major effects on regression analyses. Identifying and addressing them is key. Proper treatment ensures reliable and accurate forecasting. Analysts must balance outlier detection and treatment. This balance ensures the integrity of their models. It also prevents overfitting and maintains model validity.

He is a content producer who specializes in blog content. He has a master's degree in business administration and he lives in the Netherlands.

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Maths - Solved Example Problems for Regression Analysis | 11th Business Mathematics and Statistics(EMS) : Chapter 9 : Correlation and Regression analysis

Chapter: 11th business mathematics and statistics(ems) : chapter 9 : correlation and regression analysis, solved example problems for regression analysis.

Example 9.9

Calculate the regression coefficient and obtain the lines of regression for the following data

Regression coefficient of  X  on  Y

(i) Regression equation of  X  on  Y

(ii) Regression coefficient  of   Y  on  X

(iii) Regression equation of  Y  on  X

 Y = 0.929 X –3.716+11

 = 0.929 X +7.284

The regression equation of  Y  on  X  is  Y = 0.929 X  + 7.284

Example 9.10

Calculate the two regression equations of  X  on  Y  and  Y  on  X  from the data given below, taking deviations from a actual means of  X  and  Y .

Estimate the likely demand when the price is Rs.20.

Calculation of Regression equation

(ii) Regression Equation of Y on X

When  X  is 20,  Y  will be

             = –0.25 (20)+44.25

             = –5+44.25

= 39.25 (when the price is Rs. 20, the likely demand is 39.25)

Example 9.11

Obtain regression equation of  Y  on  X  and estimate  Y  when  X =55 from the following

(i) Regression coefficients of  Y  on  X

(ii)  Regression equation of Y on X

Y –51.57 = 0.942( X –48.29 )

Y = 0.942 X –45.49+51.57=0.942 #–45.49+51.57

Y =  0.942X+6.08

The regression equation of  Y  on  X  is  Y = 0.942 X +6.08 Estimation of  Y  when  X = 55

Y = 0.942(55)+6.08=57.89

Example 9.12

Find the means of  X  and  Y  variables and the coefficient of correlation between them from the following two regression equations:

2 Y –X–50 = 0

3Y–2X–10 = 0.

We are given                 

2Y–X–50    = 0    ... (1)

3Y–2X–10  = 0    ... (2)

Solving equation (1) and (2)   

We get        Y = 90       

Putting the value of  Y  in equation (1)

We get        X       = 130

Calculating correlation coefficient

Let us assume equation (1) be the regression equation of  Y  on  X

2 Y  =  X +50

It may be noted that in the above problem one of the regression coefficient is greater than 1 and the other is less than 1. Therefore our assumption on given equations are correct.

Example 9.13

4X–5Y+33 = 0

20X–9Y–107 = 0

4X–5Y+33 = 0    ... (1)

20X–9Y–107       = 0    ... (2)

Solving equation (1) and (2)

We get        Y = 17

Let us assume equation (1) be the regression equation of  X  on  Y

Let us assume equation (2) be the regression equation of  Y  on  X

But this  is not possible because both the regression coefficient are greater than

So our above assumption is wrong. Therefore treating equation (1) has regression equation of  Y  on  X  and equation (2) has regression equation of  X  on  Y  . So we get

Example 9.14

The following table shows the sales and advertisement expenditure of a form

Coefficient of correlation  r = 0.9. Estimate the likely sales for a proposed advertisement expenditure of Rs. 10 crores.

When advertisement expenditure is 10 crores i.e.,  Y =10 then sales  X =6(10)+4=64 which implies sales is 64.

Example 9.15

There are two series of index numbers  P  for price index and  S  for stock of the commodity. The mean and standard deviation of  P  are 100 and 8 and of S are 103 and 4 respectively. The correlation coefficient between the two series is 0.4. With these data obtain the regression lines of  P  on  S  and  S  on  P .

Let us consider  X  for price  P  and  Y  for stock  S . Then the mean and  SD  for  P  is considered as X-Bar  = 100 and σ x =8. respectively and the mean and SD of S is considered as Y -Bar  =103 and σ y =4. The correlation coefficient between the series is  r ( X , Y )=0.4

Let the regression line  X  on  Y  be

Example 9.16

For 5 pairs of observations the following results are obtained ∑ X =15, ∑ Y =25, ∑ X 2 =55, ∑ Y 2 =135, ∑ XY =83 Find the equation of the lines of regression and estimate the value of  X  on the first line when  Y =12 and value of  Y  on the second line if  X =8.

Y –5 = 0.8(X–3)

             = 0.8 X +2.6

When  X =8 the value of  Y  is estimated as

             = 0.8(8)+2.6

Example 9.17

The two regression lines are 3 X +2 Y =26 and 6 X +3 Y =31. Find the correlation coefficient.

Let the regression equation of  Y  on  X  be

3 X +2 Y  = 26

Example 9.18

In a laboratory experiment on correlation research study the equation of the two regression lines were found to be 2X– Y +1=0 and 3 X –2 Y +7=0 . Find the means of  X  and  Y . Also work out the values of the regression coefficient and correlation between the two   variables  X  and  Y .

Solving the two regression equations we get mean values of  X  and  Y

Example 9.19

For the given lines of regression 3 X –2 Y =5and X–4 Y =7. Find

(i) Regression coefficients

(ii) Coefficient of correlation

(i) First convert the given equations  Y  on  X  and  X  on  Y  in standard form and find their regression coefficients respectively.

Given regression lines are

3X–2Y = 5 ... (1)

X–4Y = 7   ... (2)

Let the line of regression of X on Y is       

Coefficient of correlation

Since the two regression coefficients are positive then the correlation coefficient is also positive and it is given by

Exercise 9.2

1. From the data given below

Find (a) The two regression equations, (b) The coefficient of correlation between marks in Economics and statistics, (c) The mostly likely marks in Statistics when the marks in Economics is 30.

2. The heights ( in cm.) of a group of fathers and sons are given below

Find the lines of regression and estimate the height of son when the height of the father is 164 cm.

3. The following data give the height in inches ( X ) and the weight in lb. ( Y ) of a random sample of 10 students from a large group of students of age 17 years:

Estimate weight of the student of a height 69 inches.

4. Obtain the two regression lines from the following data  N =20, ∑ X =80, ∑ Y =40, ∑ X 2=1680, ∑ Y 2=320 and ∑ XY =480

5. Given the following data, what will be the possible yield when the rainfall is 29₹₹

Coefficient of correlation between rainfall and production is 0.8

6. The following data relate to advertisement expenditure(in lakh of rupees) and their corresponding sales( in crores of rupees)

Estimate the sales corresponding to advertising expenditure of Rs. 30 lakh.

7. You are given the following data:

If the Correlation coefficient between  X  and  Y  is 0.66, then find (i) the two regression coefficients, (ii) the most likely value of  Y  when  X =10

8. Find the equation of the regression line of  Y  on  X , if the observations (  X i ,  Y i ) are the following (1,4) (2,8) (3,2) ( 4,12) ( 5, 10) ( 6, 14) ( 7, 16) ( 8, 6) (9, 18)

9. A survey was conducted to study the relationship between expenditure on accommodation ( X ) and expenditure on Food and Entertainment ( Y ) and the following results were obtained:

Write down the regression equation and estimate the expenditure on Food and Entertainment, if the expenditure on accommodation is Rs. 200.

10. For 5 observations of pairs of ( X ,  Y ) of variables  X  and  Y  the following results are obtained. ∑ X =15, ∑ Y =25, ∑X 2 =55, ∑ Y 2 =135, ∑ XY =83. Find the equation of the lines of regression and estimate the values of  X  and  Y  if  Y =8 ;  X =12.

11. The two regression lines were found to be 4 X –5 Y +33=0 and 20 X –9 Y –107=0 . Find the mean values and coefficient of correlation between  X  and  Y .

12. The equations of two lines of regression obtained in a correlation analysis are the following 2 X =8–3 Y  and 2 Y =5– X  . Obtain the value of the regression coefficients and correlation coefficient.

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• How to Overcome Challenges in Statistics Homework

How to Understand and Overcome Challenges in Statistics Homework

Solving your statistics homework can be a daunting task, but with the right approach and techniques, you can solve them effectively. This guide will provide you with a step-by-step approach to solving homework similar to the examples provided, focusing on regression analysis, descriptive statistics, and value assessment. These tips and techniques will help you navigate through any statistics homework with confidence.

1. Understanding the Problem Statement

The first step in solving any statistics homework is to thoroughly understand the problem statement. Break down the homework into smaller, manageable parts and identify the key objectives. For example, in the given homework, you might need to:

• Analyze data to make informed decisions.
• Develop regression models to predict outcomes.
• Summarize data using descriptive statistics.
• Interpret statistical results to draw meaningful conclusions.

2. Data Preparation

Before diving into the analysis, it is crucial to prepare your data. This includes:

• Cleaning the data: Remove any missing or inconsistent values.
• Structuring the data: Organize the data into a format suitable for analysis (e.g., data frames in R or Python).
• Understanding the variables: Identify the types of variables (categorical or continuous) and their relationships.

3. Descriptive Statistics

Descriptive statistics provide a summary of the data and are the foundation of any statistical analysis. Here’s how you can approach it:

• Central Tendency: Calculate the mean, median, and mode to understand the central point of the data.
• Dispersion: Measure the spread of the data using range, variance, and standard deviation.
• Visualization: Use graphs like histograms, box plots, and scatter plots to visualize the data distribution and detect patterns or outliers.

Example: In the homework regarding Fortune magazine’s 100 best companies to work for, you could:

• Calculate the mean and median salaries for salaried and hourly employees.
• Compare salary distributions across different company sizes using box plots.

For students seeking help with descriptive statistics homework , consider utilizing resources or tools that provide step-by-step guidance on these calculations. This can include statistical software, online tutorials, or consulting with a tutor for personalized assistance.

4. Regression Analysis

Regression analysis is a powerful tool for predicting outcomes and understanding relationships between variables. and if you need further assistance, consider the following steps:

• Simple Linear Regression: Start with a basic model to predict a dependent variable using one independent variable.
• Multiple Regression: Incorporate multiple independent variables to improve the accuracy of your predictions.
• Dummy Variables: Use dummy variables to include categorical data in your regression models.

Steps to develop a regression model:

• Formulate the model: Identify the dependent variable (what you want to predict) and independent variables (predictors).
• Estimate the coefficients: Use statistical software to estimate the coefficients of the regression equation.
• Evaluate the model: Check the goodness-of-fit (R-squared value) and perform hypothesis tests (t-tests) to assess the significance of the predictors.
• Interpret the results: Understand the implications of the regression coefficients and the overall model.

Example: In the homework involving Environment Canada’s Fuel Economy Guide, you could:

• Develop a regression model to predict highway MPG based on engine displacement, type of fuel, and type of drive.
• Interpret the coefficients to understand the impact of each variable on fuel efficiency.

5. Hypothesis Testing

Hypothesis testing helps determine whether the relationships observed in the data are statistically significant. Common tests include:

• T-tests: Compare the means of two groups.
• ANOVA: Compare the means of three or more groups.
• Chi-square tests: Test relationships between categorical variables.

Example: In the car value assessment homework, you might:

• Test whether the type of car (small, family, upscale) significantly affects the cost per mile.
• Determine if smaller cars provide better value than larger cars using regression analysis and hypothesis testing.

6. Model Refinement

After developing your initial models, refine them by:

• Removing insignificant variables: Use p-values to identify and remove predictors that do not significantly contribute to the model.
• Checking for multicollinearity: Ensure that independent variables are not highly correlated, which can distort the results.
• Cross-validation: Validate the model using different subsets of the data to ensure its robustness.

7. Reporting and Interpretation

Finally, present your findings in a clear and concise manner. Your report should include:

• Introduction: Briefly describe the purpose of the analysis and the data used.
• Methodology: Explain the steps taken to prepare the data, conduct the analysis, and develop the models.
• Results: Present the key findings, including descriptive statistics, regression coefficients, and hypothesis test results.
• Discussion: Interpret the results, discuss their implications, and make recommendations based on your analysis.
• Conclusion: Summarize the main points and suggest areas for further research or analysis.

Example: In the car value assessment homework, your report might conclude that smaller cars generally offer better value than larger cars, supported by statistical evidence from your regression models and hypothesis tests.

Tools and Software

Leverage statistical software such as R , Python (with libraries like Pandas and Statsmodels), SPSS , or Excel for your analysis. These tools provide powerful functions for data manipulation, visualization, and statistical modeling.

By following these steps, you can effectively solve complex statistics homework and provide meaningful insights from your analysis. Remember to stay organized, validate your models, and clearly communicate your findings. With practice, you’ll become proficient in tackling any statistics homework that comes your way.

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