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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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A Refresher on Regression Analysis

Understanding one of the most important types of data analysis.

You probably know by now that whenever possible you should be making data-driven decisions at work . But do you know how to parse through all the data available to you? The good news is that you probably don’t need to do the number crunching yourself (hallelujah!) but you do need to correctly understand and interpret the analysis created by your colleagues. One of the most important types of data analysis is called regression analysis.

• Amy Gallo is a contributing editor at Harvard Business Review, cohost of the Women at Work podcast , and the author of two books: Getting Along: How to Work with Anyone (Even Difficult People) and the HBR Guide to Dealing with Conflict . She writes and speaks about workplace dynamics. Watch her TEDx talk on conflict and follow her on LinkedIn . amyegallo

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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

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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? 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.

Want to learn more about Statistical Analysis Software? Explore Statistical Analysis products.

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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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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What Is Regression?

Understanding regression, calculating regression, the bottom line.

• Macroeconomics

Regression: Definition, Analysis, Calculation, and Example

Regression is a statistical method used in finance, investing, and other disciplines that attempts to determine the strength and character of the relationship between a dependent variable and one or more independent variables.

Linear regression is the most common form of this technique. Also called simple regression or ordinary least squares (OLS), linear regression establishes the linear relationship between two variables.

Linear regression is graphically depicted using a straight line of best fit with the slope defining how the change in one variable impacts a change in the other. The y-intercept of a linear regression relationship represents the value of the dependent variable when the value of the independent variable is zero. Nonlinear regression models also exist, but are far more complex.

Key Takeaways

• Regression is a statistical technique that relates a dependent variable to one or more independent variables.
• A regression model is able to show whether changes observed in the dependent variable are associated with changes in one or more of the independent variables.
• It does this by essentially determining a best-fit line and seeing how the data is dispersed around this line.
• Regression helps economists and financial analysts in things ranging from asset valuation to making predictions.
• For regression results to be properly interpreted, several assumptions about the data and the model itself must hold.

In economics, regression is used to help investment managers value assets and understand the relationships between factors such as commodity prices and the stocks of businesses dealing in those commodities.

While a powerful tool for uncovering the associations between variables observed in data, it cannot easily indicate causation. Regression as a statistical technique should not be confused with the concept of regression to the mean, also known as mean reversion .

Joules Garcia / Investopedia

Regression captures the correlation between variables observed in a data set and quantifies whether those correlations are statistically significant or not.

The two basic types of regression are simple linear regression and  multiple linear regression , although there are nonlinear regression methods for more complicated data and analysis. Simple linear regression uses one independent variable to explain or predict the outcome of the dependent variable Y, while multiple linear regression uses two or more independent variables to predict the outcome. Analysts can use stepwise regression to examine each independent variable contained in the linear regression model.

Regression can help finance and investment professionals. For instance, a company might use it to predict sales based on weather, previous sales, gross domestic product (GDP) growth, or other types of conditions. The capital asset pricing model (CAPM) is an often-used regression model in finance for pricing assets and discovering the costs of capital.

Regression and Econometrics

Econometrics is a set of statistical techniques used to analyze data in finance and economics. An example of the application of econometrics is to study the income effect using observable data. An economist may, for example, hypothesize that as a person increases their income , their spending will also increase.

If the data show that such an association is present, a regression analysis can then be conducted to understand the strength of the relationship between income and consumption and whether or not that relationship is statistically significant.

Note that you can have several independent variables in an analysis—for example, changes to GDP and inflation in addition to unemployment in explaining stock market prices. When more than one independent variable is used, it is referred to as  multiple linear regression . This is the most commonly used tool in econometrics.

Econometrics is sometimes criticized for relying too heavily on the interpretation of regression output without linking it to economic theory or looking for causal mechanisms. It is crucial that the findings revealed in the data are able to be adequately explained by a theory.

Linear regression models often use a least-squares approach to determine the line of best fit. The least-squares technique is determined by minimizing the sum of squares created by a mathematical function. A square is, in turn, determined by squaring the distance between a data point and the regression line or mean value of the data set.

Once this process has been completed (usually done today with software), a regression model is constructed. The general form of each type of regression model is:

Simple linear regression:

Y = a + b X + u \begin{aligned}&Y = a + bX + u \\\end{aligned} ​ Y = a + b X + u ​

Multiple linear regression:

Y = a + b 1 X 1 + b 2 X 2 + b 3 X 3 + . . . + b t X t + u where: Y = The dependent variable you are trying to predict or explain X = The explanatory (independent) variable(s) you are  using to predict or associate with Y a = The y-intercept b = (beta coefficient) is the slope of the explanatory variable(s) u = The regression residual or error term \begin{aligned}&Y = a + b_1X_1 + b_2X_2 + b_3X_3 + ... + b_tX_t + u \\&\textbf{where:} \\&Y = \text{The dependent variable you are trying to predict} \\&\text{or explain} \\&X = \text{The explanatory (independent) variable(s) you are } \\&\text{using to predict or associate with Y} \\&a = \text{The y-intercept} \\&b = \text{(beta coefficient) is the slope of the explanatory} \\&\text{variable(s)} \\&u = \text{The regression residual or error term} \\\end{aligned} ​ Y = a + b 1 ​ X 1 ​ + b 2 ​ X 2 ​ + b 3 ​ X 3 ​ + ... + b t ​ X t ​ + u where: Y = The dependent variable you are trying to predict or explain X = The explanatory (independent) variable(s) you are  using to predict or associate with Y a = The y-intercept b = (beta coefficient) is the slope of the explanatory variable(s) u = The regression residual or error term ​

Example of How Regression Analysis Is Used in Finance

Regression is often used to determine how specific factors—such as the price of a commodity, interest rates, particular industries, or sectors—influence the price movement of an asset. The aforementioned CAPM is based on regression, and it's utilized to project the expected returns for stocks and to generate costs of capital. A stock’s returns are regressed against the returns of a broader index, such as the S&P 500, to generate a beta for the particular stock.

Beta is the stock’s risk in relation to the market or index and is reflected as the slope in the CAPM. The return for the stock in question would be the dependent variable Y, while the independent variable X would be the market risk premium.

Additional variables such as the market capitalization of a stock, valuation ratios, and recent returns can be added to the CAPM to get better estimates for returns. These additional factors are known as the Fama-French factors, named after the professors who developed the multiple linear regression model to better explain asset returns.

Why Is It Called Regression?

Although there is some debate about the origins of the name, the statistical technique described above most likely was termed “regression” by Sir Francis Galton in the 19th century to describe the statistical feature of biological data (such as heights of people in a population) to regress to some mean level. In other words, while there are shorter and taller people, only outliers are very tall or short, and most people cluster somewhere around (or “regress” to) the average.

What Is the Purpose of Regression?

In statistical analysis, regression is used to identify the associations between variables occurring in some data. It can show the magnitude of such an association and determine its statistical significance. Regression is a powerful tool for statistical inference and has been used to try to predict future outcomes based on past observations.

How Do You Interpret a Regression Model?

A regression model output may be in the form of Y = 1.0 + (3.2) X 1 - 2.0( X 2 ) + 0.21.

Here we have a multiple linear regression that relates some variable Y with two explanatory variables X 1 and X 2 . We would interpret the model as the value of Y changes by 3.2× for every one-unit change in X 1 (if X 1 goes up by 2, Y goes up by 6.4, etc.) holding all else constant. That means controlling for X 2 , X 1 has this observed relationship. Likewise, holding X1 constant, every one unit increase in X 2 is associated with a 2× decrease in Y. We can also note the y-intercept of 1.0, meaning that Y = 1 when X 1 and X 2 are both zero. The error term (residual) is 0.21.

What Are the Assumptions That Must Hold for Regression Models?

To properly interpret the output of a regression model, the following main assumptions about the underlying data process of what you are analyzing must hold:

• The relationship between variables is linear;
• There must be homoskedasticity , or the variance of the variables and error term must remain constant;
• All explanatory variables are independent of one another;
• All variables are normally distributed .

Regression is a statistical method that tries to determine the strength and character of the relationship between one dependent variable and a series of other variables. It is used in finance, investing, and other disciplines.

Regression analysis uncovers the associations between variables observed in data, but cannot easily indicate causation.

Margo Bergman. “ Quantitative Analysis for Business: 12. Simple Linear Regression and Correlation .” University of Washington Pressbooks, 2022.

Margo Bergman. “ Quantitative Analysis for Business: 13. Multiple Linear Regression .” University of Washington Pressbooks, 2022.

Fama, Eugene F., and Kenneth R. French, via Wiley Online Library. “ The Cross-Section of Expected Stock Returns .” The Journal of Finance , vol. 47, no. 2, June 1992, pp. 427–465.

Stanton, Jeffrey M., via Taylor & Francis Online. “ Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors .” Journal of Statistics Education , vol. 9, no. 3, 2001.

CFA Institute. “ Basics of Multiple Regression and Underlying Assumptions .”

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Regression Analysis

Regression analysis is a quantitative research method which is used when the study involves modelling and analysing several variables, where the relationship includes a dependent variable and one or more independent variables. In simple terms, regression analysis is a quantitative method used to test the nature of relationships between a dependent variable and one or more independent variables.

The basic form of regression models includes unknown parameters (β), independent variables (X), and the dependent variable (Y).

Regression model, basically, specifies the relation of dependent variable (Y) to a function combination of independent variables (X) and unknown parameters (β)

                                    Y  ≈  f (X, β)   

Regression equation can be used to predict the values of ‘y’, if the value of ‘x’ is given, and both ‘y’ and ‘x’ are the two sets of measures of a sample size of ‘n’. The formulae for regression equation would be

Do not be intimidated by visual complexity of correlation and regression formulae above. You don’t have to apply the formula manually, and correlation and regression analyses can be run with the application of popular analytical software such as Microsoft Excel, Microsoft Access, SPSS and others.

Linear regression analysis is based on the following set of assumptions:

1. Assumption of linearity . There is a linear relationship between dependent and independent variables.

2. Assumption of homoscedasticity . Data values for dependent and independent variables have equal variances.

3. Assumption of absence of collinearity or multicollinearity . There is no correlation between two or more independent variables.

4. Assumption of normal distribution . The data for the independent variables and dependent variable are normally distributed

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Predict, analyze, optimize: regression techniques in data analytics.

Regression analysis is one of the most powerful tools in the data analyst’s toolkit. This statistical technique allows businesses to understand relationships between variables, make valuable predictions, and drive strategic decision-making. At CIAT, our data analytics programs recognize regression analysis as a fundamental and indispensable skill, equipping students with the power to uncover relationships, make predictions, and drive data-informed business decisions. Let’s dive into the world of regression analysis and explore its techniques and contributions to business data analytics.

What is Regression Analysis?

At its core, regression analysis is a set of statistical methods used to estimate relationships between variables. It helps businesses determine how changes in one or more independent variable affect a dependent variable. For example, a company might use regression analysis to understand how advertising spend, product pricing, and economic indicators impact sales revenue.

Key Techniques in Regression Analysis

• Simple Linear Regression: This basic form of regression examines the linear relationship between two independent and dependent variables. It’s useful for straightforward analyses, such as determining how temperature affects ice cream sales.
• Multiple Linear Regression: When multiple independent variables are involved, a multiple linear regression model comes into play. This technique allows businesses to assess the impact of several factors simultaneously, providing a more comprehensive understanding of complex relationships.
• Logistic Regression: Unlike a linear model, which predicts continuous outcomes, logistic regression is used for binary outcomes. It’s beneficial in predicting the probability of an event, such as whether a customer will make a purchase.
• Polynomial Regression: When relationships between variables exhibit non-linear patterns, a polynomial regression model can capture curved relationships, providing greater flexibility in data analysis.
• Ridge and Lasso Regression: These advanced techniques help handle multicollinearity (when independent variables are highly correlated) and perform feature selection, respectively. Lasso and Ridge regression are instrumental when dealing with large datasets with many variables.

Contributions and Importance in Business Data Analytics

Predictive modeling: .

Regression analysis enables businesses to create sophisticated forecasting models. For example, a retail company might use multiple regression to predict sales based on seasonality, economic indicators, and marketing spending. This allows for more accurate inventory management and resource allocation.

Decision Making: 

Regression analysis provides concrete data to support strategic decisions by quantifying the impact of various factors. A manufacturing company could use regression to determine which production factors most significantly impact output, informing decisions on where to invest in process improvements.

Performance Optimization: 

Regression helps pinpoint areas for improvement. An e-commerce platform might use regression to identify which website features strongly correlate with conversion rates, allowing them to focus development efforts on high-impact areas.

Risk Assessment: 

In finance, regression models can assess the risk of loan defaults by analyzing factors like credit score, income, and debt-to-income ratio. This enables more accurate risk pricing and informed lending decisions.

Marketing Effectiveness: 

Marketers use regression to analyze the performance of different channels. For instance, a company might use regression to determine how TV ads, social media campaigns, and email marketing contribute to sales, optimizing budget allocation across these channels.

Product Development: 

By analyzing customer survey data, feedback, and market trends through regression, companies can identify which product features are most valued by consumers. This insight guides R&D efforts toward creating products with higher market potential.

Quality Control:

In manufacturing, regression can identify which process variables most significantly impact product quality. This allows for proactive adjustments to maintain consistent quality standards.

Customer Behavior Analysis: 

Regression models can predict customer churn by analyzing usage patterns, customer service interactions, and billing history. This enables businesses to implement targeted retention strategies for at-risk customers.

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Challenges and considerations.

While regression analysis is powerful, it’s not without challenges. Analysts must be aware of potential pitfalls such as:

• Overfitting: When a model is too complex and fits the noise in the data rather than the underlying relationship.
• Assumption violations: Failing to meet key statistical assumptions can compromise the validity and reliability of regression results.
• Correlation vs. Causation: It’s crucial to remember that correlation doesn’t imply causation. Additional research and experimentation are often needed to establish causal relationships.

The Future of Regression in Business Analytics

As businesses continue to generate vast amounts of data, the importance of regression analysis in extracting actionable insights will only grow. Advanced machine learning techniques are expanding the capabilities of traditional regression methods, allowing for more complex and accurate models. Integrating real-time data and automated model updating also makes regression analysis more dynamic and responsive to changing business conditions.

Mastering Regression Analysis with CIAT

Given the critical role of regression analysis in business data analytics, professionals with strong skills in this area are in high demand. If you want to build a career in this exciting field, consider our Associate of Applied Science in Business Data Analytics program.

CIAT’s program provides hands-on experience with the latest data analytics tools and techniques, including regression analysis coverage. You’ll learn how to apply these powerful methods to real-world business problems, positioning yourself for success in the data-driven business world. With expert instructors and a curriculum designed to meet industry needs, CIAT’s program is your stepping stone to a rewarding career in business data analytics. Take the first step towards becoming a data analytics expert – explore CIAT’s Associate of Applied Science in Business Data Analytics program today!

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How to Use Regression Analysis to Forecast Sales: A Step-by-Step Guide

Updated: August 27, 2021

Published: December 21, 2020

There are various ways to understand the effectiveness of your sales team activity or how well sales teams are at driving sales to reach operational and financial goals. Sales forecasting , a method that predicts sales performance based on historical performance, is one way to get this understanding.

Sales forecasting is important because it can help you identify what is going right, as well as what areas of your current strategy need to be adapted and changed to ensure future success.

For example, if your team is consistently below quotas, sales forecasting can help determine where and why these issues are happening. Forecasting can also help you decide on future business endeavors, like when you’d have the revenue to invest in new products or expand your business.

Some forecasting methods involve doing basic math, like adding up month to month sales, and others are more in-depth. Regression analysis is one of these methods, and it requires in-depth statistical analysis.

If you’re anything like me and not at all mathematically inclined, conducting this type of forecast may seem daunting. Thankfully, this piece will give an easy to understand breakdown of regression analysis in sales and guide you through an easy to follow example using Google Sheets .

What is regression analysis?

In statistics, regression analysis is a mathematical method used to understand the relationship between a dependent variable and an independent variable. Results of this analysis demonstrate the strength of the relationship between the two variables and if the dependent variable is significantly impacted by the independent variable.

There are multiple different types of regression analysis, but the most basic and common form is simple linear regression that uses the following equation: Y = bX + a

That type of explanation isn’t really helpful, though, if you don’t already have a grasp of mathematical processes, which I certainly don’t. Let’s take a look at what regression analysis means, in layman’s terms, for sales forecasting.

What is regression analysis in sales?

In simple terms, sales regression analysis is used to understand how certain factors in your sales process affect sales performance and predict how sales would change over time if you continued the same strategy or pivoted to different methods.

Independent and dependent variables are still at play here, but the dependent variable is always the same: sales performance. Whether it’s total revenue or number of deals closed, your dependent variable will always be sales performance. The independent variable is the factor you are examining that will change sales performance, like the number of salespeople you have or how much money is spent on advertising.

Sales regression forecasting results help businesses understand how their sales teams are or are not succeeding and what the future could look like based on past sales performance. The results can also be used to predict future sales based on changes that haven’t yet been made, like if hiring more salespeople would increase business revenue.

So, what do these words mean, math wise? Like I said before, I’m not good at math. But, I did conduct a simple sales regression analysis that is easy to follow and didn’t require many calculations on my part. Let’s go over this example below.

How To Use Regression Analysis To Forecast Sales

Let’s say that you want to run a sales forecast to understand if having your salespeople make more sales calls will mean that they close more deals. To conduct this forecast, you need historical data that depicts the number of sales calls made over a certain period. So, mathematically, the number of sales calls is the independent variable, or X value, and the dependent variable is the number of deals closed per month, or Y value.

I made up the data set below to represent monthly sales calls, and a corresponding number of deals closed over a two year period.

So, the overall regression equation is Y = bX + a , where:

• X is the independent variable (number of sales calls)
• Y is the dependent variable (number of deals closed)
• b is the slope of the line
• a is the point of interception, or what Y equals when X is zero

Since we’re using Google Sheets, its built-in functions will do the math for us and we don’t need to try and calculate the values of these variables. We simply need to use the historical data table and select the correct graph to represent our data. The first step of the process is to highlight the numbers in the X and Y column and navigate to the toolbar, select Insert, and click Chart from the dropdown menu.

The default graph that appears isn’t what we need, so I clicked on the Chart editor tool and selected Scatter plot, as shown in the gif below.

After selecting the scatter plot, I clicked Customize, Series, and scrolled down to select the Trendline box (shown below).

After all of these customizations, I get the following scatter plot.

The Sheets tool did the math for me, but the line in the chart is the b variable from the regression equation, or slope, that creates the line of best fit . The blue dots are the y values, or the number of deals closed based on the number of sales calls.

values, or the number of deals closed based on the number of sales calls.

So, the scatter plot answers my overall question of whether having salespeople make more sales calls will close more deals. The answer is yes, and I know this because the line of best fit trendline is moving upwards, which indicates a positive relationship. Even though one month can have 20 sales calls and 10 deals and the next has 10 calls and 40 deals, the statistical analysis of the historical data in the table assumes that, on average, more sales calls means more deals closed.

I’m fine with this data. It means that simply having salespeople make more calls per-month than they have before will increase deal count. However, this scatter plot does not give us the specific forecast numbers that you’ll need to understand your future sales performance. Let’s use the same example to obtain that information.

Let’s say your boss tells you that they want to generate more quarterly revenue, which is directly related to sales activity. You can assume closing more deals means generating more revenue, but you still want the data to prove that having your salespeople make more calls would actually close more deals.

The built-in FORECAST.LINEAR equation in Sheets will help you understand this, based on the historical data in the first table .

I made the table below within the same sheet to create my forecast breakdown. In my Sheets document, this new table uses the same columns as the first (A, B, and C) and begins in row 26.

I went with 50 because the highest number of sales calls made in any given month from the original data table is 40 and we want to know what happens to deal totals if that number actually increases. I could’ve only used 50, but I increased the number by 10 each month to get an accurate forecast that is based on statistics, not a one-off occurrence.

After creating this chart, I followed this path within the Insert dropdown menu in the Sheets toolbar: Insert -> Function -> Statistical -> FORECAST.LINEAR .

This part gets a little bit technical, but it’s simpler than it looks. The instruction menu below tells me that I’ll obtain my forecasts by filling in the relevant column numbers for the target number of sales calls.

Here is the breakdown of what the elements of the FORECAST.LINEAR equation mean:

• x is the value on the x-axis (in the scatter plot) that we want to forecast, which is the target call volume.
• data_y uses the first and last row number in column C in the original table, 2 and 24.
• data_x uses the first and last row number in column B in the original table, 2 and 24.
• data_y goes before data_x because the dependent variable in column C changes because of the number in column B.

This equation, as the FORECAST.LINEAR instructions tell us, will calculate the expected y value (number of deals closed) for a specific x value based on a linear regression of the original data set. There are two ways to fill out the equation. The first option, shown below, is to manually input the x value for the number of target calls and repeat for each row.

=FORECAST.LINEAR(50, C2:C24, B2:B24)

The second option is to use the corresponding cell number for the first x value and drag the equation down to each subsequent cell. This is what the equation would look like if I used the cell number for 50 in the second data table:

=FORECAST.LINEAR(B27, C2:C24, B2:B24)

To reiterate, I use the number 50 because I want to be sure that making more sales calls results in more closed deals and more revenue, not just a random occurrence. This is what the number of deals closed would be, not rounded up to exact decimal points.

Overall, the results of this linear regression analysis and expected forecast tell me that the number of sales calls is directly related to the number of deals closed per month. If you ask your salespeople to make ten more calls per month than the previous month, the number of deals closed will increase, which will help your business generate more revenue.

While Google Sheets helped me do the math without any further calculations, other tools are available to streamline and simplify this process.

Sales Regression Forecasting Tools

A critical factor in conducting a successful regression analysis is having data and having enough data. While you can add and just use two numbers, regression requires enough data to determine if there is a significant relationship between your variables. Without enough data points, it will be challenging to run an accurate forecast. If you don’t yet have enough data, it may be best to wait until you have enough.

Once you have the data you need, the below list of tools that can help you through the process of collecting, storing, and exporting your sales data.

InsightSquared

InsightSquared is a revenue intelligence platform that uses AI to make accurate forecasting predictions.

While it can’t run a regression analysis, it can give you the data you need to conduct the regression on your own. Specifically, it provides data breakdowns of the teams, representatives, and sales activities that are driving the best results. You can use this insight to come up with further questions to ask in your regression analysis to better understand performance.

Since sorting through data is essential for beginning your analysis, MethodData is valuable tool. The service can create custom sales reports based on the variables you need for your specific regression, and the automated processes save you time. Instead digging through your data and cleaning it up enough to be usable, it happens automatically once you create your custom reports.

HubSpot Sales Hub

HubSpot’s Sales Hub automatically records and tracks all relevant sales and performance data related to your teams. Specific items collected include activity reports for sales calls, emails sent, and meetings taken with clients, but you can also create custom reports.

If you want an immediate overview of your sales forecast, the Sales Hub comes with a probability forecast report . It gives a breakdown of how likely it will be that you’ll meet your monthly or quarterly sales goals (shown in the image below). These projections can help you come up with further questions to analyze in your regression analysis to understand what is (or isn’t) going wrong.

Automate.io

If you’re a HubSpot Sales Hub user and you want to use Google Sheets to conduct your regression analysis as I did, Automate.io allows you to sync and export data to external apps, including Google Sheets, eliminating the risks that can sometimes come from a simple copy+paste.

Another factor that can affect your analysis is whether you’re even doing it right. Like I said before, I’m bad at math, so I used an online tool. If you feel confident enough, feel free to use a pen, paper, and a quality calculator to run your analysis by hand .

If you’re like me, using statistical analysis tools like Excel , Google Sheets, RStudio , and SPSS can help you through the process, no hard calculations required. Paired with one of the data export tools listed above, you’ll have a seamless strategy to clean and organize your data and run your linear regression analysis.

Regression Analysis Helps You Better Understand Sales Performance

A regression analysis will give you statistical insight into the factors that influence sales performance.

If you take the time to come up with a viable regression question that focuses on two business-specific variables and use the right data, you’ll be able to accurately forecast expected sales performance and understand what elements of your strategy can remain the same, or what needs to change to meet new business goals.

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Regression analysis: the ultimate guide.

19 min read 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.

When you rely on data to drive and guide business decisions, as well as predict market trends, just  gathering and analysing  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 dependent variables and independent 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  analysing 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 analszing 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 categorised into two types: dependent and independent variables. The variables you select depend on the outcomes you’re analysing.

Understanding variables:

1. dependent variable.

This is the main variable that you want to analyse 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 organisation 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 visualise 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 analyse 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 analyse 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 analyse

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 visualisations 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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Regression Analysis: Types, Importance and Limitations

Meaning of regression analysis.

Regression analysis refers to a statistical method used for studying the relationship in between dependent variables (target) and one or more independent variables (predictors). It enables in easily determining the strength of relationship among these 2 types of variable for modelling future relationship in between them. Regression analysis explains variations taking place in target in relation to changes in select predictors. It is mostly used in investment and finance disciplines. Finance and investment managers utilize this statistical technique for valuing assets, discovering capital costs and easy understanding of relationship among variables like commodity prices and businesses stocks dealing in such commodities.  Business also used regression analysis for predicting sales volume on the basis of previous growth, GDP growth, weather and many other factors.

Types of Regression Analysis

Logistic regression is one in which dependent variable is binary is nature. Independent variable either can be continuous or binary. It is a form of binomial regression that estimates parameters of logistic model. Data having two possible criterions are deal with using the logistic regression. 

Ridge regression is widely used when there is high correlation between the independent variables. In such multi collinear data, although least square estimates are unbiased but their variances are quite large that deviates observed value from true value. Ridge regression reduces the standard errors by adding a degree of bias to the estimates of regression.

Advantages of Regression Analysis

Disadvantages of regression analysis, related posts:, add commercemates to your homescreen.

• > Machine Learning

What is Regression Analysis? Types and Applications

• Ayush Singh Rawat
• Jun 07, 2021

Introduction

The field of Artificial Intelligence and machine learning is set to conquer most of the human disciplines; from art and literature to commerce and sociology; from computational biology and decision analysis to games and puzzles.” ~Anand Krish 

Regression analysis is a way to find trends in data. 

For example, you might guess that there’s a connection between how much you eat and how much you weigh; regression analysis can help you quantify that equation.

Regression analysis will provide you with an equation for a graph so that you can make predictions about your data. 

For example, if you’ve been putting on weight over the last few years, it can predict how much you’ll weigh in ten years time if you continue to put on weight at the same rate. 

It will also give you a slew of statistics (including a p-value and a correlation coefficient) to tell you how accurate your model is.

Introduction to Regression Analysis

Regression analysis is a statistical technique for analysing and comprehending the connection between two or more variables of interest. The methodology used to do regression analysis aids in understanding which elements are significant, which may be ignored, and how they interact with one another.

Regression is a statistical approach used in finance, investment, and other fields to identify the strength and type of a connection between one dependent variable (typically represented by Y) and a sequence of other variables (known as independent variables).

Regression is essentially the "best guess" at utilising a collection of data to generate some form of forecast. It is the process of fitting a set of points to a graph.

Regression analysis is a mathematical method for determining which of those factors has an effect. It provides answers to the following questions: 

Which factors are most important 

Which of these may we disregard

How do those elements interact with one another, and perhaps most significantly, how confident are we in all of these variables

These elements are referred to as variables in regression analysis. You have your dependent variable, which is the key aspect you're attempting to understand or forecast. Then there are your independent variables, which are the elements you assume have an effect on your dependent variable.

(Most related blog: 7 Types of Regression Techniques in ML )

Types of Regression Analysis

Types of regression analysis

Simple linear regression

The relationship between a dependent variable and a single independent variable is described using a basic linear regression methodology. A Simple Linear Regression model reveals a linear or slanted straight line relation, thus the name.

The simple linear model is expressed using the following equation:

Y = a + bX + ϵ

• Y – variable that is dependent
• X – Independent (explanatory) variable
• a – Intercept
• ϵ – Residual (error)

The dependent variable needs to be continuous/real, which is the most crucial component of Simple Linear Regression. On the other hand, the independent variable can be evaluated using either continuous or categorical values.

Multiple linear regression

Multiple linear regression (MLR), often known as multiple regression, is a statistical process that uses multiple explanatory factors to predict the outcome of a response variable. 

MLR is a method of representing the linear relationship between explanatory (independent) and response (dependent) variables.

The mathematical representation of multiple linear regression is:

y=ß0+ ß1 x1+ …………..ßn xn + ϵ

Where, y = the dependent variable’s predicted value

B0 = the y-intercept

B1X1= B1 is the coefficient for regression of the first independent variable X1 (The effect of increasing the independent variable's value on the projected y value is referred to as X1.)

… = Repeat for as many independent variables as you're testing.

BnXn = the last independent variable's regression coefficient

ϵ = model error (i.e. how much flexibility is there in our y estimate)

Multiple linear regression uses the same criteria as single linear regression. Due to the huge number of independent variables in multiple linear regression, there is an extra need for the model:

The absence of a link between two independent variables with a low correlation is referred to as non-collinearity. It would be hard to determine the true correlations between the dependent and independent variables if the independent variables were strongly correlated.

(Related blog: Pearson’s Correlation Coefficient ‘r’ )

Non-linear regression

A sort of regression analysis in which data is fitted to a model and then displayed numerically is known as nonlinear regression. 

Simple linear regression connects two variables (X and Y) in a straight line (y = mx + b), whereas nonlinear regression connects two variables (X and Y) in a nonlinear (curved) relationship.

The goal of the model is to minimise the sum of squares as much as possible. The sum of squares is a statistic that tracks how much Y observations differ from the nonlinear (curved) function that was used to anticipate Y.

In the same way that linear regression modelling aims to graphically trace a specific response from a set of factors, nonlinear regression modelling aims to do the same. 

Because the function is generated by a series of approximations (iterations) that may be dependent on trial-and-error, nonlinear models are more complex to develop than linear models. 

The Gauss-Newton methodology and the Levenberg-Marquardt approach are two well-known approaches used by mathematicians.

(Must check: Statistical Data Analysis )

What are applications of Regression Analysis ?

Most of the regression analysis is done to carry out processes in finances. So, here are 5 applications of Regression Analysis in the field of finance and others relating to it.

Applications of regression analysis

Forecasting:

The most common use of regression analysis in business is for forecasting future opportunities and threats. Demand analysis, for example, forecasts the amount of things a customer is likely to buy. 

When it comes to business, though, demand is not the only dependent variable. Regressive analysis can anticipate significantly more than just direct income. 

For example, we may predict the highest bid for an advertising by forecasting the number of consumers who would pass in front of a specific billboard. 

Insurance firms depend extensively on regression analysis to forecast policyholder creditworthiness and the amount of claims that might be filed in a particular time period.

The Capital Asset Pricing Model (CAPM), which establishes the link between an asset's projected return and the related market risk premium, relies on the linear regression model.

It is also frequently used in financial analysis by financial analysts to anticipate corporate returns and operational performance.

The beta coefficient of a stock is calculated using regression analysis. Beta is a measure of return volatility in relation to total market risk. 

Because it reflects the slope of the CAPM regression, we can rapidly calculate it in Excel using the SLOPE tool.

Comparing with competition:

It may be used to compare a company's financial performance to that of a certain counterpart.

It may also be used to determine the relationship between two firms' stock prices (this can be extended to find correlation between 2 competing companies, 2 companies operating in an unrelated industry etc).

It can assist the firm in determining which aspects are influencing their sales in contrast to the comparative firm. These techniques can assist small enterprises in achieving rapid success in a short amount of time.

Identifying problems:

Regression is useful not just for providing factual evidence for management choices, but also for detecting judgement mistakes. 

A retail store manager, for example, may assume that extending shopping hours will significantly boost sales. 

However, RA might suggest that the increase in income isn't enough to cover the increase in operational cost as a result of longer working hours (such as additional employee labour charges). 

As a result, this research may give quantitative backing for choices and help managers avoid making mistakes based on their intuitions.

Reliable source

Many businesses and their top executives are now adopting regression analysis (and other types of statistical analysis ) to make better business decisions and reduce guesswork and gut instinct. 

Regression enables firms to take a scientific approach to management. Both small and large enterprises are frequently bombarded with an excessive amount of data. 

Managers may use regression analysis to filter through data and choose the relevant factors to make the best decisions possible.

For a long time, regression analysis has been utilised extensively by enterprises to transform data into useful information, and it continues to be a valuable asset to many leading sectors.

The significance of regression analysis lies in the fact that it is all about data: data refers to the statistics and statistics that identify your company. 

The benefits of regression analysis are that it allows you to essentially crunch the data to assist you make better business decisions now and in the future.

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Regression analysis for business

Regression models are the first step into Machine Learning.

To understand linear regression, we must first understand regression with a simple example. Let’s say you have a construction business. A simple linear regression could help you find a relationship between revenue and temperature, with revenue as the dependent variable. If there are multiple variables, then you can use logistic regression, which helps you find the relationship between temperature, pricing and number of workers affecting the revenue. Thus, regression analysis can analyze the impact of various factors on sales and profit. Implementing regression models in business, is valuable and today’s data volumes allows you to make use of it in multiple forms: 

1. Predictive Analytics:

This type of analysis uses historical data, finds patterns, looks out for trends and uses that information to build predictions about future trends.

Regression analysis can go far beyond forecasting impact on immediate revenue. For example, you can forecast the number of customers who will purchase a service and use that data to estimate the amount of workforce needed to run that service. Insurance companies make use of regression analysis to estimate credit health of policy holders and a possible number of claims in a given time period.

Predictive analytics helps companies:

• Reduce Costs
• Reduce the amount of tools needed
• Provide faster results
• Improve operational efficiency
• Help in fraud detection
• Risk management
• Optimize marketing campaigns

2. Operational Efficiency:

Regression models can also help optimize business processes. A factory director, for example, can build a regression model to understand the impact of the premises temperature on the overall productivity of all employees. In an ER hospital, we can analyze the relationship between the wait times of patients and the outcomes. 

3. Decision making:

Because the loads of data gathered on finances, operations and purchases, companies are now learning how to make use of data analytics to make data-driven decisions and not intuitive decisions. Linear and logistic regression, provides a more accurate analysis which can then be used to test hypotheses of situations prior to sending it to production.

Regression analysis is not only valuable in providing insights for decision making, but also to identify errors in judgement. For example, executives managing a store may think that adding after hours shopping will increase profit. Regression analysis, however, analyzes all the variables revolving around this action and may conclude that to support the increase in operating expenses due to longer working hours (such as additional employee labor charges) will decrease profit significantly. Regression analysis provides quantitative support for decisions and prevents mistakes, product of intuitiveness.

5. New Insights:

Over time businesses have gathered a large volume of cluttered data that can provide invaluable amounts of new insights. Unfortunately, this data is of no use without the appropriate analysis. Regression analysis can find a relationship between several variables by uncovering patterns that were not taken into account. “For example, analysis of data from point of sales systems and purchase accounts may highlight market patterns like increase in demand on certain days of the week or at certain times of the year. You can maintain optimal stock and personnel before a spike in demand arises by acknowledging these insights.” -Anurag

Data-driven decision eliminates the need to guess, and shields companies from making gut decisions. This greatly improves business performance by focusing on the areas with the most impact on the operationally and in revenue.

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What is Regression Analysis? Definition, Types, and Examples

Kate Williams

Last Updated: 22 January 2024

14 min read

Table Of Contents

• Regression Analysis Definition
• Regression Analysis FAQs
• Regression Analysis: Importance
• Types of Regression Analysis
• Uses By Businesses
• Regression Analysis Use Cases

If you want to find data trends or predict sales based on certain variables, then regression analysis is the way to go.

In this article, we will learn about regression analysis, types of regression analysis, business applications, and its use cases. Feel free to jump to a section that’s relevant to you.

• What is the definition of regression analysis?
• Regression analysis: FAQs
• Why is regression analysis important?
• Types of regression analysis and when to use them
• How is regression analysis used by businesses
• Use cases of regression analysis

What is Regression Analysis?

Need a quick regression definition? In simple terms, regression analysis identifies the variables that have an impact on another variable .

The regression model is primarily used in finance, investing, and other areas to determine the strength and character of the relationship between one dependent variable and a series of other variables.

Regression Analysis: FAQs

Let us look at some of the most commonly asked questions about regression analysis before we head deep into understanding everything about the regression method.

1. What is multiple regression analysis meaning?

Multiple regression analysis is a statistical method that is used to predict the value of a dependent variable based on the values of two or more independent variables.

2. In regression analysis, what is the predictor variable called?

The predictor variable is the name given to an independent variable that we use in regression analysis.

The predictor variable provides information about an associated dependent variable regarding a certain outcome. At their core, predictor variables are those that are linked with particular outcomes.

3. What is a residual plot in a regression analysis?

A residual plot is a graph that shows the residuals on the vertical axis and the independent variable on the horizontal axis.

Moreover, the residual plot is a representation of how close each data point is (vertically) from the graph of the prediction equation of the regression model. If the data point is above or below the graph of the prediction equation of the model, then it is supposed to fit the data.

4. What is linear regression analysis?

Linear regression analysis is used to predict the value of a variable based on the value of another variable. The variable that you want to predict is referred to as the dependent variable. The variable that you are using to predict the other value is called the independent variable.

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Why is Regression Analysis Important?

There are many business applications of regression analysis.

• For any machine learning problem which involves continuous numbers , regression analysis is essential. Some of those instances could be:
• Testing automobiles
• Weather analysis, and prediction
• Sales and promotions forecasting
• Financial forecasting
• Time series forecasting
• Regression analysis data also helps you understand whether the relationship between two different variables can give way to potential business opportunities .
• For example, if you change one variable (say delivery speed), regression analysis will tell you the kind of effect that it has on other variables (such as customer satisfaction, small value orders, etc).
• One of the best ways to solve regression issues in machine learning using a data model is through regression analysis. Plotting points on a chart, and running the best fit line , helps predict the possibility of errors.
• The insights from these patterns help businesses to see the kind of difference that it makes to their bottom line .

5 Types of Regression Analysis and When to Use Them

1. linear regression analysis.

• This type of regression analysis is one of the most basic types of regression and is used extensively in machine learning .
• Linear regression has a predictor variable and a dependent variable which is related to each linearly.
• Moreover, linear regression is used in cases where the relationship between the variables is related in a linear fashion.

Let’s say you are looking to measure the impact of email marketing on your sales. The linear analysis can be wrong as there will be aberrations. So, you should not use big data sets ( big data services ) for linear regression.

2. Logistic Regression Analysis

• If your dependent variable has discrete values , that is, if they can have only one or two values, then logistic regression SPSS is the way to go.
• The two values could be either 0 or 1, black or white, true or false, proceed or not proceed, and so on.
• To show the relationship between the target and independent variables, logistic regression uses a sigmoid curve.

This type of regression is best used when there are large data sets that have a chance of equal occurrence of values in target variables. There should not be a huge correlation between the independent variables in the dataset.

3. Lasso Regression Analysis

• Lasso regression is a regularization technique that reduces the model’s complexity.
• How does it do that? By limiting the absolute size of the regression coefficient .
• When doing so, the coefficient value becomes closer to zero. This does not happen with ridge regression.

Lass regression is advantageous as it uses feature selection – where it lets you select a set of features from the database to build your model. Since it uses only the required features, lasso regression manages to avoid overfitting.

4. Ridge Regression Analysis

• If there is a high correlation between independent variables , ridge regression is the recommended tool.
• It is also a regularization technique that reduces the complexity of the model .

Ridge regression manages to make the model less prone to overfitting by introducing a small amount of bias known as the ridge regression penalty, with the help of a bias matrix.

5. Polynomial Regression Analysis

• Polynomial regression models a non-linear dataset with the help of a linear model .
• Its working is similar to that of multiple linear regression. But it uses a non-linear curve and is mainly employed when data points are available in a non-linear fashion.
• It transforms the data points into polynomial features of a given degree and manages to model them in the form of a linear model.

Polynomial regression involves fitting the data points using a polynomial line. Since this model is susceptible to overfitting, businesses are advised to analyze the curve during the end so that they get accurate results.

While there are many more regression analysis techniques, these are the most popular ones.

How is regression analysis used by businesses?

Regression stats help businesses understand what their data points represent and how to use them with the help of business analytics techniques.

Using this regression model, you will understand how the typical value of the dependent variable changes based on how the other independent variables are held fixed.

Data professionals use this incredibly powerful statistical tool to remove unwanted variables and select the ones that are more important for the business.

Here are some uses of regression analysis:

1. Business Optimization

• The whole objective of regression analysis is to make use of the collected data and turn it into actionable insights .
• With the help of regression analysis, there won’t be any guesswork or hunches based on which decisions need to be made.
• Data-driven decision-making improves the output that the organization provides.
• Also, regression charts help organizations experiment with inputs that might not have been earlier thought of, but now that it is backed with data, the chances of success are also incredibly high.
• When there is a lot of data available, the accuracy of the insights will also be high.

2. Predictive Analytics

• For businesses that want to stay ahead of the competition, they need to be able to predict future trends. Organizations use regression analysis to understand what the future holds for them.
• To forecast trends, the data analysts predict how the dependent variables change based on the specific values given to them.
• You can use multivariate linear regression for tasks such as charting growth plans, forecasting sales volumes, predicting inventory required, and so on.
• Find out more about the area so that you can gather data from different sources
• Collect the data required for the relevant variables
• Specify and measure your regression model
• If you have a model which fits the data, then use it to come up with predictions

3. Decision-making

• For businesses to run effectively, they need to make better decisions and be aware of how each of their decisions will affect them. If they do not understand the consequences of their decisions, it can be difficult for their smooth functioning.
• Businesses need to collect information about each of their departments – sales, operations, marketing, finance, HR, expenditures, budgetary allocation, and so on. Using relevant parameters and analyzing them helps businesses improve their outcomes.
• Regression analysis helps businesses understand their data and gain insights into their operations . Business analysts use regression analysis extensively to make strategic business decisions.

4. Understanding failures

• One of the most important things that most businesses miss doing is not reflecting on their failures.
• Without contemplating why they met with failure for a marketing campaign or why their churn rate increased in the last two years, they will never find ways to make it right.
• Regression analysis provides quantitative support to enable this kind decision-making.

5. Predicting Success

• You can use regression analysis to predict the probability of success of an organization in various aspects.
• Additionally, regression in stats analyses the data point of various sales data, including current sales data, to understand and predict the success rate in the future.

6. Risk Analysis

• When analyzing data, data analysts, sometimes, make the mistake of considering correlation and causation as the same. However, businesses should know that correlation is not causation.
• Financial organizations use regression data to assess their risk and guide them to make sound business decisions.

7. Provides New Insights

• Looking at a huge set of data will help you get new insights. But data, without analysis, is meaningless.
• With the help of regression analysis, you can find the relationship between a variety of variables to uncover patterns.
• For example, regression models might indicate that there are more returns from a particular seller. So the eCommerce company can get in touch with the seller to understand how they send their products.

Each of these issues has different solutions to them. Without regression analysis, it might have been difficult to understand exactly what was the issue in the first place.

8. Analyze marketing effectiveness

• When the company wants to know if the funds they have invested in marketing campaigns for a particular brand will give them enough ROI, then regression analysis is the way to go.
• It is possible to check the isolated impact of each of the campaigns by controlling the factors that will have an impact on the sales.
• Businesses invest in a number of marketing channels – email marketing , paid ads, Instagram influencers, etc. Regression statistics is capable of capturing the isolated ROI as well as the combined ROI of each of these companies.

7 Use Cases of Regression Analysis

1. credit card.

• Credit card companies use regression analysis to understand various user factors such as the consumer’s future behavior, prediction of credit balance, risk of customer’s credit default, etc.
• All of these data points help the company implement specific EMI options based on the results.
• This will help credit card companies take note of the risky customers.
• Simple linear regression (also called Ordinary Least Squares (OLS)) gives an overall rationale for the placing of the line of the best fit among the data points.
• One of the most common applications using the statistical model is the Capital Asset Pricing Model (CAPM) which describes the relationship between the returns and risks of investing in a security.

3. Pharmaceuticals

• Pharmaceutical companies use the process to analyze the quantitative stability data to estimate the shelf life of a product. This is because it finds the nature of the relationship between an attribute and time.
• Medical researchers use regression analysis to understand if changes in drug dosage will have an impact on the blood pressure of patients. Pharma companies leveraging best engagement platforms of HCP to increase brand awareness in the virtual space.

For example, researchers will administer different dosages of a certain drug to patients and observe changes in their blood pressure. They will fit a simple regression model where they use dosage as the predictor variable and blood pressure as the response variable.

4. Text Editing

• Logistic regression is a popular choice in a number of natural language processing (NLP) tasks s uch as text preprocessing.
• After this, you can use logistic regression to make claims about the text fragment.
• Email sorting, toxic speech detection, topic classification for questions, etc, are some of the areas where logistic regression shows great results.

5. Hospitality

• You can use regression analysis to predict the intention of users and recognize them. For example, like where do the customers want to go? What they are planning to do?
• It can even predict if the customer hasn’t typed anything in the search bar, based on how they started.
• It is not possible to build such a huge and complex system from scratch. There are already several machine learning algorithms that have accumulated data and have simple models that make such predictions possible.

6. Professional sports

• Data scientists working with professional sports teams use regression analysis to understand the effect that training regiments will have on the performance of players .
• They will find out how different types of exercises, like weightlifting sessions or Zumba sessions, affect the number of points that player scores for their team (let’s say basketball).
• Using Zumba and weightlifting as the predictor variables, and the total points scored as the response variable, they will fit the regression model.

Depending on the final values, the analysts will recommend that a player participates in more or less weightlifting or Zumba sessions to maximize their performance.

7. Agriculture

• Agricultural scientists use regression analysis t o understand the effect of different fertilizers and how it affects the yield of the crops.
• For example, the analysts might use different types of fertilizers and water on fields to understand if there is an impact on the crop’s yield.
• Based on the final results, the agriculture analysts will change the number of fertilizers and water to maximize the crop output.

Wrapping Up

Using regression analysis helps you separate the effects that involve complicated research questions. It will allow you to make informed decisions, guide you with resource allocation, and increase your bottom line by a huge margin if you use the statistical method effectively.

If you are looking for an online survey tool to gather data for your regression analysis, SurveySparrow is one of the best choices. SurveySparrow has a host of features that lets you do as much as possible with a survey tool. Get on a call with us to understand how we can help you.

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Statistics By Jim

Making statistics intuitive

When Should I Use Regression Analysis?

By Jim Frost 183 Comments

Use regression analysis to describe the relationships between a set of independent variables and the dependent variable. Regression analysis produces a regression equation where the coefficients represent the relationship between each independent variable and the dependent variable. You can also use the equation to make predictions.

As a statistician , I should probably tell you that I love all statistical analyses equally—like parents with their kids. But, shhh, I have secret! Regression analysis is my favorite because it provides tremendous flexibility, which makes it useful in so many different circumstances. In fact, I’ve described regression analysis as taking correlation to the next level !

In this blog post, I explain the capabilities of regression analysis, the types of relationships it can assess, how it controls the variables, and generally why I love it! You’ll learn when you should consider using regression analysis.

Related post : What are Independent and Dependent Variables?

Use Regression to Analyze a Wide Variety of Relationships

• Model multiple independent variables
• Include continuous and categorical variables
• Use polynomial terms to model curvature
• Assess interaction terms to determine whether the effect of one independent variable depends on the value of another variable

These capabilities are all cool, but they don’t include an almost magical ability. Regression analysis can unscramble very intricate problems where the variables are entangled like spaghetti. For example, imagine you’re a researcher studying any of the following:

• Do socio-economic status and race affect educational achievement?
• Do education and IQ affect earnings?
• Do exercise habits and diet effect weight?
• Are drinking coffee and smoking cigarettes related to mortality risk?
• Does a particular exercise intervention have an impact on bone density that is a distinct effect from other physical activities?

More on the last two examples later!

All these research questions have entwined independent variables that can influence the dependent variables. How do you untangle a web of related variables? Which variables are statistically significant and what role does each one play? Regression comes to the rescue because you can use it for all of these scenarios!

Use Regression Analysis to Control the Independent Variables

As I mentioned, regression analysis describes how the changes in each independent variable are related to changes in the dependent variable. Crucially, regression also statistically controls every variable in your model.

What does controlling for a variable mean?

When you perform regression analysis, you need to isolate the role of each variable. For example, I participated in an exercise intervention study where our goal was to determine whether the intervention increased the subjects’ bone mineral density. We needed to isolate the role of the exercise intervention from everything else that can impact bone mineral density, which ranges from diet to other physical activity.

To accomplish this goal, you must minimize the effect of confounding variables. Regression analysis does this by estimating the effect that changing one independent variable has on the dependent variable while holding all the other independent variables constant. This process allows you to learn the role of each independent variable without worrying about the other variables in the model. Again, you want to isolate the effect of each variable.

Regression models help you prevent spurious correlations from confusing your results by controlling for confounders.

How do you control the other variables in regression?

A beautiful aspect of regression analysis is that you hold the other independent variables constant by merely including them in your model! Let’s look at this in action with an example.

A recent study analyzed the effect of coffee consumption on mortality. The first results indicated that higher coffee intake is related to a higher risk of death. However, coffee drinkers frequently smoke, and the researchers did not include smoking in their initial model. After they included smoking in the model, the regression results indicated that coffee intake lowers the risk of mortality while smoking increases it. This model isolates the role of each variable while holding the other variable constant. You can assess the effect of coffee intake while controlling for smoking. Conveniently, you’re also controlling for coffee intake when looking at the effect of smoking.

Note that the study also illustrates how excluding a relevant variable can produce misleading results. Omitting an important variable causes it to be uncontrolled, and it can bias the results for the variables that you do include in the model. This warning is particularly applicable for observational studies where the effects of omitted variables might be unbalanced. On the other hand, the randomization process in a true experiment tends to distribute the effects of these variables equally, which lessens omitted variable bias.

Related post : Confounding Variables and Omitted Variable Bias

How to Interpret Regression Output

To answer questions using regression analysis, you first need to fit and verify that you have a good model. Then, you look through the regression coefficients and p-values. When you have a low p-value (typically < 0.05), the independent variable is statistically significant. The coefficients represent the average change in the dependent variable given a one-unit change in the independent variable (IV) while controlling the other IVs.

For instance, if your dependent variable is income and your IVs include IQ and education (among other relevant variables), you might see output like this:

The low p-values indicate that both education and IQ are statistically significant. The coefficient for IQ indicates that each additional IQ point increases your income by an average of approximately $4.80 while controlling everything else in the model. Furthermore, an additional unit of education increases average earnings by $24.22 while holding the other variables constant.

Regression analysis is a form of inferential statistics . The p-values help determine whether the relationships that you observe in your sample also exist in the larger population . I’ve written an entire blog post about how to interpret regression coefficients and their p-values , which I highly recommend.

Obtaining Trustworthy Regression Results

With the vast power of using regression comes great responsibility. Sorry, but that’s the way it must be. To obtain regression results that you can trust, you need to do the following:

• Specify the correct model . As we saw, if you fail to include all the important variables in your model, the results can be biased.
• Check your residual plots . Be sure that your model fits the data adequately.
• Correlation between the independent variables is called multicollinearity. As we saw, some multicollinearity is OK. However, excessive multicollinearity can be a problem .

Using regression analysis gives you the ability to separate the effects of complicated research questions. You can disentangle the spaghetti noodles by modeling and controlling all relevant variables, and then assess the role that each one plays.

There are many different regression analysis procedures. Read my post to determine which type of regression is correct for your data .

If you’re learning regression and like the approach I use in my blog, check out my eBook!

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Reader Interactions

July 12, 2023 at 1:42 pm

Jim, I am trying to predict a categorical variable (college major category, where there are 5 different categories).

I have 3 different continuous variables (CAREER INTERESTS, which has 6 different subscales), PERSONALITY (which is the Big Five) and MORAL PREFERENCES (which uses the MFQ30 questionnaire, that has 5 subscales).

I am confused about what type of regression (hierarchical, etc.) I could use in this study. What are your thoughts?

July 17, 2023 at 12:18 am

Because your dependent variable is a categorical variable consider using Nominal Logistic Regression, also known as Multinomial Logistic Regression or Polytomous Logistic Regression. These terms are used interchangeably to describe a statistical method used for predicting the outcome of a categorical dependent variable based on one or more predictor variables.

January 9, 2023 at 12:03 am

First of all, Many thanks for this fantastic website that makes statistics seem a little bit simpler and more clear. It’s a fantastic resource. I have dataset of an experiment. It have dependent variable Choice reaction time(CRT) and independent variable Visual task. (This visual task includes two types of task; cognitive involved questions and minimizes cognitive questions. These questions are of three types questions which include choices/options(2,4,8)/bits(1,2,3) only two options to choose one answer, 4options questions and 8 options in questions. First i used Linear regression to check the best fitting of model(Hicks law) in SPSS. But unfortunately the value of r-square was very very low. Now, my professor push me to make new model by using that dataset. Please suggest me some steps and hints so i will start working on it.

December 14, 2022 at 3:59 am

Following are my research objectives a. To identify youth’s competencies in entreprenuership in the area. b. To identify the factor of youth involvement in agricultural entreprenuership in the area.

I have used opinion based question designed on 5-point likert scale item except demographic question in the beginning of my survey. The questionnaire contain simp-le opinion based question there is no dependent and independent items in questionnaire. My question is which analysis is suitable for my research? Regresion analysis or descriptive analysis or both?

December 14, 2022 at 5:57 pm

The question of whether there is a dependent variable and one or more independent variables is separate from the question of whether you need to use inferential or descriptive statistics. And regression analysis can be either a descriptive or inferential procedure. Although, it is almost always an inferential procedure. Let’s go through these issues.

If you just want to describe a sample and you’re not generalizing from the sample to a population, you’re performing descriptive statistics. In this case, you don’t need to use hypothesis testing and confidence intervals.

However, if you have a representative sample and you want to infer the properties of an entire population, then you need to perform hypothesis testing and look at confidence intervals. Read my post about the Difference between Descriptive and Inferential Statistics for more information.

Regression analysis can apply to either of these cases. You perform the same analysis but if you’re only describing the sample, you can ignore the p-values and confidence intervals. Instead, you’ll focus on using the coefficients to describe the relationships between the variables within the sample. There’s less to worry about but you only know what is happening within that sample and can’t apply the results to a larger population. Conversely, if you do want to generalize to a population, then you must consider the p-values and confidence intervals and determine whether the coefficients are statistically significant. Most analysts performing regression analysis do want to generalize to the population, making it an inferential procedure.

However, regression analysis does specify independent and dependent variables. If you don’t need to specify those types of variables, then just use a correlation. Likert data is ordinal data. And for that data type, you need use Spearman’s correlation. And, like regression analysis, correlation can be either a descriptive or inferential procedure. You either pay attention to the p-values (inferential) or not (descriptive). In both cases, you are interested in the correlation coefficients. You’ll see the relationships between the variables without need to specify independent and dependent variables. You could calculate medians or modes for each item but not the mean because that’s not appropriate for ordinal data.

I hope that helps!

December 12, 2022 at 9:18 am

Hi Jim, Supposing I’m interested in establishing an explanatory relationship between two variables, profits and average age of employees using regression analysis and I have access to data from the entire population of interest e.g. all the 30 firms in a particular industry, do I still need to perform statistical inference? What would be the meaning of p-values , F tests etc, given that I am not intending to generalize the results for firms outside the industry? Do I still need to perform power analysis given that I have access to the entire population of 30 firms? Is the population of 30 firms too small for reliable statistical deductions? Thanks in advance Jim.

December 13, 2022 at 5:11 pm

Hi Patrick,

If you are truly interested in only those 30 companies and have access to data for all their employees, then you don’t need to perform inferential statistics. You’ve got the entire population. Hence, you know the population parameters. Hypothesis tests account for sampling error. But when you measure the entire population, there is zero sampling error and, hence, zero need to perform a hypothesis test.

However, if your average ages are based on only a sample of the employees in the 30 firms, then you’re still working with samples. To generalize from the sample to the population of all employees at the 30 firms, you’d need to use hypothesis testing in that case.

So, you just need to determine whether you really have access to the data for the entire population.

December 8, 2022 at 1:52 am

Hi, Following are my research objectives a. To investigate effectiveness of asynchronous and synchronous mode of online education. b. To identify challenges that both teachers and students encounter in synchronous and asynchronous mode of online education. I have used pearson correlation to find relationship of effectiveness of synchronous mode with asynchronous mode and challenges of online mode and vice versa. I have used opinion based question designed on 5-point likert scale item. The questionnaire contain simp-le opinion based question there is no dependent and independent items in questionnaire. My question is that correlation is sufficient or i have to run other test for proving my hypothesis.

December 10, 2022 at 8:28 pm

Because you have Likert scale data, you should use Pearson’s correlation because that is more appropriate for ordinal data.

Another possibility would be to use a nonparametric test and evaluate the median difference between the asynchronous and synchronous modes of education for each item.

November 21, 2022 at 3:45 am

A scientist determined the intensity of solar radiation and temperature of plantains every hour throughout the day. He used correlation to describe the association between the two variables. A friend said he would get more information using regression. What are your views?

November 22, 2022 at 4:15 pm

Yes, I’d agree the regression provides more information that correlation. But it’s also important to understand how correlation and regression presents effect sizes differently because in some cases you might want to use correlation even though it provides less information.

Correlation gives you a standardized effect size (i.e., the correlation coefficient). Standardized effect sizes don’t provide information using the natural units of the data. In other words, you can’t relate a correlation coefficient to what’s going on with the natural data units. However, it does allow you to compare correlations between dissimilar variables.

Conversely, regression gives you unstandardized effect sizes in the coefficients. They tell you exactly what’s going on between an independent variable and dependent variable using the DV’s natural data units. But it’s harder to compare results between regression models with dissimilar DV units. Although regression has its own standardize measure of the overall strength of the model in the R-squared–but not the individual variables. Additionally, in regression, you can standardize the regression coefficients, which facilitates comparisons within a regression model but not between them.

In some cases, while correlation gives you less information, you might want to use it to facilitate comparisons between studies.

Regression allows you to predict the mean outcome. It also gives you to the tools to understand the amount of error between the predicted and observed values. Additionally, you can model a variety of different types of relationships (curved and interactions) Correlation doesn’t provide those.

So, yes, in general, regression provides more information, but it also provides a different take on the nature of the relationships.

February 1, 2022 at 6:39 am

First, congrats and many thanks on this wonderful website, which makes statistics look a bit easier and understandable. Its a great resource, both for students and professionals. Thanks again.

A request for bit of help, if you’d be kind enough to comment. Doing some research on pharmaceutical industry, regulations and its effects. I am looking at a) probable effects (if any) of drug price increases on other consumption categories (like food and travel), and b) the effects of pricing regulations on drug shortages. In ‘a’, I’ve got inflation data and average consumption expense by quintiles. In ‘b’, I’ve got last 6 year data on drug shortages, mainly due to government administered pricing. However, I’d need to show statistical significance (additionally, if it could predict anything statistically significant about drug shortages in the future).

What kind of stat methodology would be appropriate in terms of ‘a’ and ‘b’? Would appreciate your help.

December 11, 2021 at 7:39 pm

Thank you so much Sir.

August 7, 2021 at 7:01 am

Hello Mr. Jim,

Thank you very much for your opinion. Much helpful.

I’ve another case with 2 DV and multiple IDV and the scope is to determine the validity of data. So for this case, can I run MANOVA as regression analysis and look for significant value and null hypothesis for validity test?

Hoping to hear from you soon.

Kind Regards, A.Kaur

August 6, 2021 at 12:17 pm

Thank you for your reply Mr. Jim. My goal is to predict which approach best predicts CRI measure.

CRI-I: Disaster Management Cycle (DMC) based approach (Variable: PP, RS, RC, MP-contain all indices according to its phases) CRI- II: Sustainability based approach (Physical, Economy, Social-contain all indices according to its phases) CRI-III: Overall indices of data (24 indices from all the listed variable)

I’ve chosen PP and MP as my DV, and RS and RC as my IDV, since my goal focus on DMC.

Hope I’m clear now. And hoping to hear from you soon Mr. Jim. Thank you.

August 7, 2021 at 12:04 am

One approach would be to fit a regression model for each approach and the DV. Then assess the goodness-of-fit measures. You’d be particularly interested in the standard error of the regression . This measure tells you how wrong the model is typically. You’d be looking for the model that produces the lowest value because it indicates it’s less wrong at predicting the outcome.

July 31, 2021 at 1:31 pm

Good day Mr. Jim,

I’ve decided to run regression analysis after correlation test. My research is about reliability and validity of dataset for 3 approaches of community resilience index(CRI) based DMC, sustainability and overall indices approach. So now, I’m literally confused on how can to interpret data with regression analysis? Can I used OLS and GLM to interpret data?

3 approaches: 1:PP,RS,RC,MP {DMC} 2: PY,EC,SC {Sustainability} 3: Overall indices {24 indices}

For your information all those approaches are proposed in 1 dataset that contains 24 indices. Add on, I’ve previously conducted Likert questionnaire(5 scale) to collect my data.

I hope my question is clear. Hoping to hear from you soon.

August 4, 2021 at 4:38 pm

I’m sorry but I don’t completely understand what your goal is for your analysis. Are you trying to determine which approach best predicts sustainability? What are your IVs and DV. It wasn’t totally clear from your description. Thanks!

July 16, 2021 at 1:56 am

Going through your blog gave me a good understand when to use regression analysis, honestly it’s an amazing blog

July 19, 2021 at 10:22 pm

Thanks so much, Robin!

May 18, 2021 at 7:02 pm

Hey Jim, thanks for all the information. I would like to ask: are there any limitations in the multiple regression method? Is there other method in mathematics that can be more accurate than a regression?

Sincerly, Mythili

May 20, 2021 at 1:46 am

Hi Mythili,

There are definitely limitations for regression! That’s a broad question that could be answered with a book. But, a good place to start is to consider the assumptions for least squares regression . Click the link to learn more. You can think of those as limitations because if you violate the assumptions, you can’t necessarily trust the results! In fact, when you violate an assumption, you might need to switch to a different analysis or perform it a different way.

Additionally, the Gauss-Markov theorem states that least squares regression is the most efficient regression, but only when you satisfy those assumptions!

May 15, 2021 at 4:19 pm

Hi Sir, In regression analysis specifically multiple linear regression, should all variables (dependent and independent variables) be normally distributed?

Thank you, Helena

May 15, 2021 at 11:08 pm

In least squares regression analysis, you don’t assess the normality of the variables. Instead, you assess the normality of the residuals. However, there is some correlation because if you have dependent variable that follows a very non-normal distribution, it can be harder to obtain normal residuals. But it’s really the residuals that you need to focus on. I discuss that in my article about the least squares (OLS) regression assumptions .

April 18, 2021 at 11:12 pm

Hi Sir, I’m currently a senior high school student and currently struggling on my quantitative research. As a statistician what would you recommend a statistical treatment to use in identifying an impact? To answer the question “What is the impact of the development of educational brochure in minimizing cyber bullying in terms of? 3.1 Mental health 3.2 Self-Esteem”.

Waiting for your reply, desperate for answers lol Jane

April 16, 2021 at 7:21 am

Hi Jim, thank you

So would you advise an ordinal regression or another? i have a survey identifying if they use the new social media- which will place them into 2 groups. Then compare the 2 groups (1- use the new social media, 2- don’t use it) with a control (FB use) to compare their happiness scores (obtained from a survey aswell- higher score=more happier). The conclusions i can draw- would it be causal? or more an indication that for example the new users have lower happiness.

-Also is there a graph that can be drawn after a regression?

On a side note- when would it be advisable to do correlations? for example have both groups complete happiness score and conduct correlations for this and a regression to control for covariates? or is this not statistically advisable

April 16, 2021 at 3:46 pm

I highly recommend you get my book about regression analysis because I think it would be really helpful with these nuts and bolts types of questions. You can find it in My Web Store .

As for the type of regression, as I mentioned, that depends largely on what you use for your dependent variable. If it’s a single Likert item, then you’d use ordinal logistic regression. If it’s the sum or average of multiple Likert items, you can often use the regular least squares regression. But, I don’t have a good handle on exactly how you’re defining your dependent variable.

There are graphs you can create afterwards to illustrate the results. I cover those in my book. I don’t have a good post to refer you to that shows them. Fitted line plots are good when you have simple regression (just one independent variable), but when you have more there are other types.

You can do correlations but be aware that they don’t control for other variables. If there are confounders, your correlations might exhibit omitted variable bias and differ from the relationships you’ll find in the regression model. Personally, I would just stick to the regression results because they control for confounders that you include in the model.

April 15, 2021 at 4:46 pm

hi Sorry- as you can tell im a little confused on what best to do. As is it advisable to do 2 groups- users of the new social media and non users of that new social media. Then do a T-test to compare their happiness scores. Then have participants answer facebook use questionnaire to control for this by conducting a hierarchical regression where i enter this in- to identify how much this variance is explained by Facebook use?

Many thanks

April 15, 2021 at 10:28 pm

Hi Sam, you wouldn’t be able to do all of that with t-tests. I think regression is a better bet. You can still include an indicator variable to identify the two groups you mention AND include the controlling variables in that model. That way you can determine whether the difference between those two groups is statistically significant while controlling for the other IVs. All in one regression model!

April 15, 2021 at 8:26 am

Hi I wanted to ask if regression is the best test for me- I am looking at happiness scores and time spent on a new social media site. As other social media sites have a relationship with happiness and that people don’t use one social media site- i was going to control for this ‘other social media’ use. My 1st group would be the new social media site and Facebook users and the 2nd group would be Facebook users. They would do a happiness questionnaire and questionnaire about their time/use. Any advice I really appreciate it

I have read around and found partial correlations- do you advice that? So instead participants would complete a questionnaire on their use on this new social media, then also do a questionnaire on their Facebook use and do a happiness questionnaire. I would do a partial correlation between the new social media app use and happiness score, while controlling for Facebook use.

April 15, 2021 at 10:22 pm

This case sounds like a good time to use regression analysis. The type of regression depends largely on the nature of the dependent variable. It’s for a survey. Perhaps it’s a Likert scale item? If it’s an item, that’s an ordinal scale and you’d need to use ordinal logistic regression. If you’re summing multiple items for the DV, you might be able to use regular linear regression. Ordinal independent variables are a bit problematic. You’d need to use them as either continuous or categorical variables. You’d include the questions about FB use to control for that.

April 13, 2021 at 5:10 am

Thank you very much for your answer,

I understand your point of view. However that data set consist of companies investing the largest sums to R&D and not companies with also the best results. Some of them even shows up with a loss of operating profit. Is that still a factor of biasing my results?

Have a nice day, Natasha

April 12, 2021 at 11:36 am

thank you it was very useful

April 12, 2021 at 11:24 am

I am working on my thesis which is about evaluating the motivation of firms to invest in R&D of new products. I am specifically interested in automotive sector. I have a data of R&D ranking of the world top 2500 companies (by industry) which consist of data about their R&D expenses, (also R&D one-year growth), net sales (also net sales one-year growth), R&D intensity, Capex, operational profit, (also one-year growth), profitability, employees (also one-year growth), market cap (also one-year growth).

My question is that which type of analysis would you recommend to fulfill the topic requirements?

April 13, 2021 at 12:29 am

Hi Natasha,

You could certainly use regression analysis to see which variables related to R&D spending.

However, be aware that by using that list of companies, you are potentially biasing your results. For one thing, it’s a list of top R&D companies and you’d certainly want more of a mix of companies across the full range of R&D. You can learn from those who weren’t so good at R&D too. Also, by using a list of the top R&D companies, you’ll introduce some survival bias into the results because these are companies that made it and made it big (presumably). Again, you’d want mix of companies that had varying degrees of success and even some failures! If you limit your data to top companies and particularly top companies in R&D, you’ll limit how much can learn. You might still be able to learn some, but just be aware that you’re potentially biasing your results.

April 8, 2021 at 8:05 pm

Hi Mr. Jim! Thank you so much for your response. Well appreciated!

April 8, 2021 at 11:07 pm

You’re very welcome, Violetta!

April 8, 2021 at 2:08 am

Hi! I’m currently doing my research paper, and i am confused whether i can use regression analysis since my title is “New Normal Workplace Setting towards Employee’s Engagement with their Workloads” as for the moment I have used correlational approach since it deals with the relationship of two variables. But still im confused on what would be the best in my research. Hope i can get a response soon. Thank you so much!

April 8, 2021 at 3:56 pm

Hi Violetta,

If you’re just working with just two variables, you have a choice. You can use either correlation or regression. You can even use both together! It depends on the goals of your research. Correlation coefficient are standardized measures of an effect size while regression coefficients are unstandardized effect sizes. I write about the difference between standardized and unstandardized effect sizes . Click the link to read about that. I discuss both correlation and coefficient in that context. It should help you decide what is best for your research goals.

March 3, 2021 at 9:34 am

Hi Jim, I am undertaking a Msc dissertation and would like to ask questions on analysis please. The research is health related and I am looking at determinants of outcome. I have 5 continuous data independent variables and I would like to know if they have an association with the outcome of a treatment. They involve age, temperature and blood test values. The dependent variable is binary that is the treatment was yes successful or not. I am looking to do a logistic regression analysis. Questions I have: 1. Do I first need to do tests to find out if there is statistical significance of each variable before I do the regression analysis or can I go straight in? 2. If so will I need to carry out tests to find out if I have skewed data in order to know whether I need to do parametric or non parametric tests? Thank you.

March 3, 2021 at 6:01 pm

You should go in with a bunch theory and background knowledge about the independent variables you should include. Look to other research studies for guides. When you have a set of IVs identified, it’s usually ok to include them all and see what’s significant. An important caveat is if you have a small number of observations you don’t want to overfit your model . However, statistical significance shouldn’t be your only guide for which variables to include and exclude.

To read learn more about model specification, ready my post about specifying your regression model . I write about it in the context of linear regression rather than binary logistic regression, but the ideas are the same.

In terms of the distribution of your data, typically, you assess the residuals rather than the data itself. Usually, you can assess the residual plots .

January 4, 2021 at 12:01 pm

Looks like treating both ordinal variables as continuous seems to solve my problems with non-mutually exclusive levels of the variables if I enter the variables as categorical. My main concern is to look at the variable as a whole not by its levels so it might be what I need; the measurement ranges were based on a an established rating system and does not have any weight for my analysis. Tho, I’ll have to looks more into it as well as the residual plot etc before deciding. Thank you for highlighting this option!

Is it correct if I assign the numerical value to the levels like this? 1 to 5, from lowest to highest.

Spacing 1: less than 60mm 2: 60-200mm 3: 200-600mm 4:0.6-2m 5: more than 2m

length 1: less than 1m 2: 1-3m 3: 3-10m 4: 10-20m 4: more than 20m

As for the data repetition, what I mean was say data for Site A is:

Set 1 (quantity: 25) SP3 PER5 Set 2 (quantity: 30) SP4 PER6 set 3 (quantity: 56) SP2 PER3

so in the data input I’d entered set 1 data 25 times, set 2 data 30 times and set 3 data 56 times. From what I have gathered from fellow student and my lecturer, it is correct but I’d like a confirmation from a statistician. Thanks again!

December 31, 2020 at 5:44 am

I’m sorry, again the levels disappeared. maybe bc I used (>) and (<) so it's messing up the coding of the comment.

spacing levels:

SP1: less than 60mm SP2: 60-200mm SP3: 200-600mm SP4:0.6-2m SP5: more than 2m

length level:

PER1: more than 20m PER2: 10-20m PER3: 3-10m PER4: 1-3m PER4: less than 1m

Spacing and Length were recoded as ranges since they were estimate and not measured individually as it'd take too much time to measure each one (1 set of cracks may have at least 10 cracks, some can reach 50 or more and the measurement are not exactly the same between cracks belonging to the same set).

I've input the dummy like in my previous reply when running the model, tho the resulting equation I've provided does not include the length. Can ordinal variable be converted/treated into continuous variables?

Also, since each set has their own quantities, so I repeated the data in the input according to their quantity. Is that the right way of doing it?

January 2, 2021 at 7:10 pm

Technically those are ordinal variables. I write about this in more detail in my book about regression analysis , but you can enter these variables as either continuous variables (if you assign a numeric value to the groups) or as categorical variables. If you go the categorical route, you’ll need to use the indicator variable scheme and leave out a reference level approach as we discussed. The approach you should use depends on a combination of your analysis goals, the nature of your data, and the ability to adequately fit the model (i.e., properties of the residual plots).

I don’t exactly know what you mean by “repeated the data in the input.” However, you have levels for each categorical variable. Let’s use the lowest level for each variable as the reference level. Here’s how you’d use indicator variables to include both categorical variables in your model (some statistical software will do that for you behind the scenes).

Spacing variable: Leave out SP1. It’s the reference. Include and indicator variable for: SP2 SP3 SP4 SP5

Length Variable: Leave PER5 out as reference. Include indicator variables for: PER1 PER2 PER3 PER4

And just code each indicator variable appropriately based on the presence or absences of the corresponding characteristic. All zeros in a set of indictor variables for a categorical variable represents the reference level for that categorical variable.

As you can see, you’ll need to include many indicator variables (8), which is a drawback of entering them as categorical variables. You can quickly get into overfitting your model .

December 30, 2020 at 12:32 am

I’m sorry I had just noticed that the levels are missing

December 28, 2020 at 11:48 am

For my case, I’m studying the cracks set on a rock face and I have two independent categorical variables (spacing and length) that have 5 levels of measurement ranges each. Dependant variable is the blasted rock size i.e I want to know how the spacing and length of the existing cracks on a rock face would effect the size of blasted rocks.

E.g: For Spacing: SP1 = 2m

I’ve coded the levels to run the regression model into:

SP1 SP2 SP3 SP4 SP1 1 0 0 0 SP2 0 1 0 0 SP3 0 0 1 0 SP4 0 0 0 1 SP5 0 0 0 0

From the coding (leaving SP5 out as the reference level) above, after running the model, I have obtained the equation:

Blasted rock size (mm) = 1849.146 + 332.224SP1 + 137.624SP2 – 115.268SP3 – 103.604SP4

1 rock slope could consist of 2 or more crack sets hence the situation where more than 1 levels of spacing and length can be observed. As an example, rock face A consist of 3 crack sets with set #1 having SP1, set #2 with SP3 and set #3 have SP4. To predict blasted rock size for rock face A using the equation, I’ll have to insert “1” for SP1, SP3 and SP4. Which is actually the wrong way of doing it since they are not mutually exclusive? Or can I calculate each crack set separately using the same equation then average the of blasted rock size for these 3 crack sets?

From the method in your explanation, does this mean that I’ll have to separate each level into 10 different variables and code them as 1=yes and 0=no? If so, for spacing, will the coding be

SP1 SP2 SP3 SP4 SP5 SP1 1 0 0 0 0 SP2 0 1 0 0 0 SP3 0 0 1 0 0 SP4 0 0 0 1 0 SP5 0 0 0 0 1

in the input table which would be similar to the initial one except with SP5 included? But if I were to include all levels when running the model, SPSS would automatically excluded 1 level since I ran several rock faces (belonging to a single location) in a model so all levels of spacing and length are present in the data set.

The other way that I can think of is to create interaction for all possible combinations and dummy code them but wouldn’t that end up with a super long equation?

I’m sorry for imposing like this but I couldn’t grasp this problem on my own. Your help is very much appreciated.

December 31, 2020 at 12:51 am

Ah, ok, it sounds like you have two separate categorical variables. In that case, for each observation, you can have one level for each variable. Additionally, for each categorical variable, you’ll leave out one level for its own reference level.

I do have a question. spacing and length sound like continuous measurements. Why are you including them as categorical variables? There might be a good reason why but it almost seems like you can include them as continuous predictors. Perhaps you don’t have the raw measurements but instead they’re in groups? In which case, they might actually be ordinal variables. You can include ordinal variables as categorical variables. But sometimes they’ll still work as continuous variables.

December 26, 2020 at 12:12 am

I see, sorry I couldn’t fully understand your previous reply before this, thanks for the clarification. However, I am dealing with a situation where 2 or more levels of a variable could be observed simultaneously, is it theoretically right to use dummy or is there other method around it?

December 27, 2020 at 2:30 am

That sounds like you’re dealing with more than one variable rather than one categorical variable. Within an individual categorical variable, the levels of the variable are mutually exclusive. In your case, you need to sort out which categorical variables you have and be sure that the levels are mutually exclusive. If you looking at the presence and absence of certain characteristics, you can use a series of indicator variables. If these characteristics are not the mutually exclusive levels of a single categorical variable, you don’t use the rule about leaving one out.

For example, in a medical setting, you might include characteristics of a patient using a series of indicator variables: gender (1 = female 0 = male), high blood pressure (1 = Yes, 0 = No), On medication, etc. These are separate characteristics (not part of one larger categorical variable) and you can just include an indicator variable to indicate the presence or absence of that characteristic.

Perhaps that it what you need? But be aware that what you describe with multiple levels possible does not work for a single categorical variable. But the method I describe might be what you need if you’re talking about separate characteristics.

December 24, 2020 at 2:03 am

Thank you , sir

December 18, 2020 at 12:54 am

Thanks for the answer Jim,

does that mean predicted value for when both L4 and L1 are observed and when only L1 is observed without L4 is the same? (Y = 133)

thanks again!

December 18, 2020 at 1:03 am

The groups must be mutually exclusive. Hence, an observation could not be in both L1 and L4.

December 16, 2020 at 4:58 am

I have a question regarding categorical variables dummy coding, I can’t seem to find any post about this topic. Hope you don’t mind me asking here.

I ran a regression model with categorical variable containing 4 level: using the 4th level as the reference group. Meaning in the equation there will only be level 1 to 3 since level 4 is the reference. Say, the equation is Y = 120 + 13L1 – 6L2 + 15L3, to predict the Y with L4 then I’ll have Y = 120, right?

My question is what if I want to predict Y when there is L1 but no L4? if I calculate Y = 120 + 13L that would mean I am including L4 in the equation, or am I wrong about this?

Thank you in advance.

December 17, 2020 at 11:28 pm

I cover how this works in my book about regression analysis . If you’re using regression for a project, you might consider it.

It sounds like you’re approach is correct. You always leave one level out for the reference group. And, yes, given your equation, the predicted value for level 4 is 120.

For observations where the subject/item belongs to group 1, your equation stays the same, but you enter a 1 for L1 and 0s for L2 and L3. Hence, the predicted value is 133. In other words, you don’t change the equation given the level, you change the X values in the equation. When an observation belongs to group 4, you’ll enter 0s for L1, l2, and L3, which is why the predicted Y is 120. For a given categorical variable, you’ll only enter a single 1 for observations that belong to a non-reference group, and all 0s for observations belonging to the reference group. But the equation stays the same in all cases. I hope that makes sense!

December 14, 2020 at 5:35 am

May I just ask if there is a difference between a true and simple linear regression model? I can only think that their difference is the presence of a random error. Thanks a lot!

December 14, 2020 at 8:48 pm

Hi Anthony,

I’ve never heard the dichotomy state as being true vs. simple linear regression. I take true models to refer to the model that is correctly specified for the population. A simple regression model is just one that has a single predictor whereas multiple regression has more than one predictor. The true model has as many terms as are required, which includes predictors and other terms that fit curvature and interaction as needed.

December 13, 2020 at 3:04 pm

Hi Jim, I find your explanation to questions very good and so important. Thanks for that. Please I need your help in my thesis work. My question is if for example I want to measure say level of resilience capacity in a company’s safety management system. What tool would you advise. Regression or which other one ? Thanks Kwame

December 14, 2020 at 9:01 pm

The type of analysis you use depends the data you collect as well as a variety of other factors. The answer is entirely specific to your research question, field of study, data, etc. After you make those determinations, you can begin to figure out which type of analysis to use. I recommend researching your study area to answer all of those questions, including which type of analysis to use. If you need help after you start developing the answers to the preliminary question, I’d be able to provide more input.

Also, I really recommend reading my post about designing a study that includes statistical analyses . That’ll help you understand what type of information you need to collect and questions you need to answer.

November 12, 2020 at 11:12 pm

Thank you so much for your answer, Jim!

November 12, 2020 at 11:53 am

hello Jim, I have a question. I have one independent variable, and two dependent variables, I will explain the case before asking you a question. So, I obtain the data for independent variable using a questionnaire, and one of my dependent variable is also using a questionnaire. But, another dependent variable, which is my second variable, the data is from official website which is secondary data, different from the another variables. Then, I have a question, Is it okay if I use regression analysis to analyze these three variables? Or I have to use another statistical analysis that suit the best to analyze these variables? Thanks in advance.

November 12, 2020 at 4:37 pm

Most forms of regression analysis allow you to use one dependent variable and multiple independent variables. Because you have two dependent variables, you’ll need to fit two regression models, one for each dependent variable.

In regression, you need to be able to tie together all corresponding values of an observation for the dependent variable and the independent variables. We’ll use an example with people. To fit a regression model, for each person, you’ll need to know their values for the dependent variable and all the independent variables in the model. In your case, it sounds like you’re mixing data from an official website and a survey. If those data sources contain the same people and you can link their values as describes, that can work. However, if those data sources have different people, or you can’t link their scores, you won’t be able to perform regression analysis.

November 6, 2020 at 9:55 am

Hi Jim, if you’ve got three predictors and one dependent variable, is it ever worth doing linear regression on each individual predictor beforehand or should you just dive into the multiple regression? Thanks a lot!

November 6, 2020 at 8:48 pm

Hi Kristian,

You should probably just dive right into multiple regression. There’s a risk of being misled by starting out with regressions with individual predictors. It’s possible that omitted variable bias can increase or decrease the observed effect. By leaving out the other predictors, the model can’t control for them, which can cause that bias.

However, that said, it’s often a good idea to graph the relationship between pairs of variables using scatterplots to get an idea of the nature of each relationship. That’s a great place to start. Those plots not only reveal the direction of the relationship but also whether you need to model curvature.

I’d start with graphs and then try modeling with all the variables. You can always remove insignificant variables.

October 2, 2020 at 1:00 pm

Hi Jim, do you think it is correct to estimate a regression model based on historical data as Y=aX+b and then use the model for the forecast as Y=aX? Would this be biased?

if the variables involved are growth rates, would it be preferable to directly estimate the model without the intercept?

Thank you in advance Stefania

October 4, 2020 at 12:56 am

Hi Stefania,

The answer to that question depends on a very close understanding of the subject area. However, there are very few cases where fitting a model without a constant is advisable. Bias would be very likely. Read my article about the y-intercept , where I discuss this issue specifically.

September 30, 2020 at 3:22 am

Nice article. Thank you for sharing.

August 19, 2020 at 12:13 pm

If your outcome variable is a pass or fail, then it is binomial logistic. My undergrad thesis was on this topic. May be I can offer some help as this topic is of interest to me. Azad ( [email protected] )

August 6, 2020 at 2:36 am

Sir , what is cox regression analysis ?

August 6, 2020 at 12:52 am

A friend recommended your help with a stats question for my dissertation. I am currently looking at data regarding pass rate and student characteristics. I have collected multiple data points. One example is student pass rate (pass or rate) and observation hours (continuous variable (0-1000). Would this be a binomial logistic regression? Can that be performed in Excel?

Additionally I am looking at pass rate in relation to faculty characteristics. Another example is pass rate (percentage of 100% maybe continuous data 0-100) and categorical data (Level of degree – bachelor, masters, doctorate)? Additionally, pass rate (percentage of 100) and ratio of faculty to student within classroom (continuous Data) which test would be appropriate for this type of data comparison? Linear regression?

Thanks for your guidance!

July 24, 2020 at 7:14 am

Hi Jim. Concepts were well explained. Thank you so much for making this content available.

I have the data of Mortgage loan customers who are currently in default. There are various parameters why default would have happened. But predominantly there are two factors where we would have gone wrong while sanctioning the loan one is underwriting the loan( Credit Risk) and/or Property Valuation (Technical Risk). I have data of sub parameters coming under credit and technical risk at the point of sanction.

Now I want to arrive at an output where predominantly where did I go wrong. Either Technical/Credit risk or both. Which model of regression analysis can help in solving this.

July 3, 2020 at 3:40 am

dear sir, i ‘m currently final year undergradute of Bsc.Radiography degree, so i choosed risk estimation of cardiovascular diseses using several risk factors from regression analysis as my undergraduate research. i want to predict a percentage value for my cardiovascular risk estimation as a dependent variable using regression analysis. how can i do that sir,i’m very pleased to have your answer sir ? Thank you very much.

July 3, 2020 at 3:41 pm

Hi, It sounds like you might need to use binary logistic regression. If your dependent variable indicates the presence or absence (i.e., binary outcome measure) of a cardiovascular condition, binary logistic regression will predict the probability of having that condition given the values of your dependent variables.

June 26, 2020 at 8:35 pm

Thank you for all the information on your page , I am currently beginning to get into statistics and wanted to ask your advice about something

I am an business analyst with MI skills building dashboard etc and using sales data and kpi s

I am wondering for regression would a good independent variable be the significance of a salespersons sales performance over the teams total sales performance or am I on the wrong track with that ?

June 11, 2020 at 2:18 pm

Dear Jim… I am a first year ‘MBA’ student having least exposure to the research kind of things. Please have patience and explain me whether I can use regression to determine the impact of a variable on a ‘construct’?

June 7, 2020 at 6:49 pm

which criteria does an independent variable need to meet in order to use it in a regression analysis? How do you deal with data that does not meet these requirements?

June 8, 2020 at 3:13 pm

I recommend you read my post about specifying the correct regression model . That deals directly with which variables to include in the model. If you have further questions on the specifics, please post them in the comments section there.

June 5, 2020 at 7:15 am

How should we interpret the factor A that becomes not significant when fitting with factor B in a model? Can I conclude that factor B incorporates factor A and just ignore the effect of factor A?

May 28, 2020 at 2:17 am

Hello Mr.Jim and friends,

I have one dependent variable Y and six independent variables X1….X6. I have to find the effect of of all independent variables on Y , Specifically X6. to check wither it is effective or not 1) Can I use OLS regression 2) which other test i need to do before or after regression analysis

May 29, 2020 at 4:16 pm

If your dependent variable is continuous, then OLS is a good place to start. You’ll need to check the OLS assumptions for your model.

April 29, 2020 at 8:06 am

good,very explicit processes.

April 10, 2020 at 4:53 pm

I hope this comment reaches you in good health as we are living in some pretty tough times right now. Also, thank you for building this website as it is an excellent resource for novice statisticians such as myself. My question has to do with the first paragraph of this post. In it you state,

“Use regression analysis to describe the relationships between a set of independent variables and the dependent variable. Regression analysis produces a regression equation where the coefficients represent the relationship between each independent variable and the dependent variable. You can also use the equation to make predictions.”

Is it possible to use regression analysis to produce a regression equation when you have two independent variables and two dependent variables? Also, while I hopefully have you attention, would I need to do regression analysis twice(one for each dependent variable versus the independent variables)?

April 10, 2020 at 7:07 pm

Typically, you would separate regression models for each dependent variable. There are a few exception. For example, if you use multivariate ANOVA (MANOVA), you can include multiple dependent variables. If those DVs are correlated, using MANOVA provides some benefits. You can include covariates in the MANOVA model. For more informaton, read my post about MANOVA .

April 1, 2020 at 7:00 pm

n my study, I intervened with an instructional practice. My intervention has 4 independent variables (A, B, C, and D). In literature each subskill can be graded alone and we can get one whole score. In literature, the effect of the intervention is holistic (A, B, C, together predict the performance on D).

So, I conducted a multiple regression (enter method) before and after the intervention where individual scores of A, B, C were added as predictors on D.

I added Group (Experimental Vs Control ) to delete any difference at baseline between experimental and control. No significant effect was noticed except for individual score of A and C on D. Model had a weak fit.

However, after the intervention, I repeated the same regression. the group (experimental Vs Control) was the best predictor. No significant effect of A was noticed but significant effect of B and C was noticed — How do you think I can interpret the change in the significance value of A? It is relevant in literature but after the intervention it was not significant. Does the significance have to do with the increase of the significance of the Group?

January 26, 2020 at 2:51 pm

I’d like to ask a question that builds on your example of income regressed on IQ and education. In the dataset I am sure there would be a range of incomes. Let’s say you want to find ways to bring up the low income earners based on the data from this regression.

Can I use the coefficients from the regression to guide ideas on how to improve the lower income earners as an estimate of how much improvement would be expected? For example, if I take the lowest earner and find that he is also below average in IQ and education, could I suggest that he gets another degree and try to improve IQ test results to potentially gain $X (n*IQ + m*Edu) in income?

This example may not be strictly usable because I imagine there are many other factors for income. Assuming that we are confident that we’ve captured most of the variables that affect income, can the numbers be used in this way?

If this is not an appropriate application, how would one go about this? Thanks.

October 22, 2019 at 7:45 am

Hello I am completing a reflection paper for Math 221 I work in a call center can I use a regression analysis for this type of work?

October 20, 2019 at 4:48 am

I am a total novice when it comes to Statistics. My challenge is, I am working on the relationship between population growth of a town and class size of secondary schools in that same town (about 10 schools) over a period of years (2008-2018). Having gathered my data, I don’t know what to use in analyzing my data to show this relationship.

October 16, 2019 at 8:48 pm

Hi Jim! Im just a student whos trying to finish her science investigation 🙂 but i have a question. What is linear regression and how do we know if this method is appropriate for our data?

October 18, 2019 at 1:23 pm

Hi Marlene,

I think this blog post describes pretty well when to use regression analysis generally. Linear regression analysis is a specific form of regression. Linear refers to the form of the model–not whether it can fit curvature. I talk about this in my post about the differences between linear and nonlinear regression . I always suggest that you start with linear regression because it’s an easier to use analysis. However, sometimes linear regression can’t fit your data. It can fit curvature in your data but it can fit all types of curves. Nonlinear regression is more flexible in the types of curves.

As for determining whether linear regression is appropriate for your data, you need to see if it can provide an adequate fit to your data. To make that determination, please read my posts about residual plots because that’s how you can tell.

Best of luck with your research!! 🙂

August 27, 2019 at 4:50 pm

Hello Jim, thank you for this wonderful page. It has enlightened me when to use regression analysis. However, I am a complete beginner to using SPSS (and statistics at that) so I am hoping you can help me with my specific problem.

I intend to use a linear regression analysis. My dependent variable is continuous and I would think it’s ordinal (data was obtained through a 5-point Likert scale). I have two independent variables (also obtained through 5-point Likert scales). However, I also intend to use 7 control variables and this is where my problem lies. My control variables are all (I think) nominal (or is that called categorical in statistics?). They are as follows:

Age – 4 categories Gender – 2 categories Marital Status – 4 categories Education level – 11 categories Household income – 4 categories Nationality – 4 categories Country of origin – 9 categories

Do I input these control variables as it is? Or do I have to do something beforehand? I have heard about creating dummy variables. However, if I try creating dummy variables for each control variable, won’t I end up with many variables?

Please give me some advise regarding this. I am really stuck in this process for a while now. I look forward to hearing from you, thanks.

August 27, 2019 at 11:43 pm

There are several issues to address in your questions. I’ll provide some information. However, my regression ebook goes it into the details much further. So, I highly recommend you get that.

In terms of the dependent variable, the answer is clear. Likert scale data, if it’s the actual values of 1, 2, 3, 4, and 5, these are actually ordinal data and are not considered continuous. You’ll need to use ordinal logistic regression. If the DV is an average of multiple Likert score items for each individual, so an individual might have a 3.4, that is continuous data and you can try using linear least squares regression.

Categorical data and nominal data are the same. There are different naming conventions, but those synonyms.

For categorical data, it’s true that you need to recode them as indicator variables. However, most software should do that automatically behind the scenes. However, as you noticed, the recoding (even if your software does it for you) can involve creating many indicator variables (dummy variables), particularly when you have many categorical variables and/or many levels within a categorical variable. That can use up your degrees of freedom! My ebook covers this in more detail.

For Likert IV variables. Again, if it’s an average of multiple Likert items, you can probably include it as a continuous variable. However, if it’s the actual Likert values of 1, 2, 3, 4, and 5, then you’ll need to decide whether to include it as a continuous or categorical variable. There are pros and cons for both approaches. The best answer depends on both your data and your goals. My ebook describes this in more detail.

Yes, as a general rule, you want to include your control variables and IVs that you are specifically testing. Control variables are just more IVs, but they’re usually not your main focus of study. You include them so that you can account for them while testing your main variables of interest. Excluding relevant IVs that are significant can bias the estimates for the variables you’re interested in. However, if you include control variables and find they’re not significant, you can consider removing them from the model.

So, those are some pointers to start with!

June 22, 2019 at 1:02 am

Hi Jim and everyone! I’m starting some some statistical analysis and is been really useful. I have a question regarding variables and samples. I need to see if there is any relationship between days of the week and number of robberies. I already have the data but I wonder, if my variables (# of robberies in each day of the week (independent) and # of total roberies (dependent)) come from the same data sample, can it be a problem?

June 7, 2019 at 2:56 am

Thank you Jim this was really helpful

I have a question How do you interpret an independent variable lets say AGE with categories that are insignificant for example i run the regression analysis for the variable age with categories age as a whole was found to be significant but there appear insignificance within categories , it was as follows Age =0.002 <30 years =0.201 30-44 years=0.161 45+ ( ref cat)

I had another scenario occupation = 0.000 peasant farmers =0.061 petty businessmen=0.003 other occupation ( ref cat)

my research question was " what are effect of socio- demographic characteristics on men's attendance to education classes

I failed to interpret them , kindly help

June 7, 2019 at 10:07 am

For categorical variables, the linear regression procedure uses two tests of significance. It uses an F-test to determine the overall significance of the categorical variable across all its levels jointly. And, it uses separate t-tests to determine whether each individual level is different from the reference level. If you change the reference level, it can change the significance of t-tests because that changes the levels that the procedure directly compares. However, changing the reference level won’t change the F-test for the variable as a whole.

In your case, I’m guessing that the mean for <30 is on one side (high or low) compared to the reference category of 45+ while the mean of 30-44 is on the other side of 45+. These two categories are not far enough from 45+ to be significant. However, given the very low p-value for age, I'd guess that if you change the reference level from 45+ to one of the other two groups, you'll see significant p-values for at least one of the t-tests. The very low p-value for Age indicates that the means for the different levels are not all equal. However, given the reference level, you can't tell which means are different. Using a different reference level might provide more meaningful information.

For occupation, the low p-value for the F-test indicates that not all the means for the different types of occupations are equal. The t-test results indicate that the difference in means between petty businessmen and other (reference level) is statistically significant. The difference between peasant farmers and the reference category is not quite significant.

You don't include the coefficients, but those would indicate how those means differ.

Because you're using regression analysis, you should consider getting by regression ebook. I cover this topic, and others, in more detail in the book.

Best of luck with your analysis!

May 11, 2019 at 12:51 pm

Hi Jim, I have followed your discussion and I want to know if I can apply this analysis in case study

April 26, 2019 at 4:01 pm

Hi Jim really appreciate your excellency in regression analysis. please would help the steps to draw a single fitted line for several, say five IVs, against a sing DV with regard

April 26, 2019 at 4:18 pm

It sounds like you’re dealing with multiple regression because you have more than one IV. Each IV requires an axis (or dimension) on a graph. So, for a two-dimensional graph, you can use the X-axis (horizontal) for IV and the Y-axis for the DV. If you have two IVs, you could theoretically show them as hologram in three dimensions. Two dimensions for the IVs and one for the DV. However, when you get to three or more IVs, there’s just no way to graph them! You’d need four or more dimensions. So, what can you do?

You can view residual plots to see how the model with all 5 IVs fits the data. And, you can predict specific values by plugging numbers into the equation. But you can’t graph all 5 IVs against the DV at the same time.

You could graph them individually. Each IV by itself against the DV. However, that approach doesn’t control for the other variables in the model and can produce biased results.

The best thing you can do that shows the relationship between an individual IV and a DV while controlling for all the variables in a model is to use main effects plots and interaction plots. You can see interaction plots here . Unfortunately I don’t have a blog post about main effects plots, but I do write about them in my ebook, which I highly recommend you get to understand regression! Learn more about my ebook!

I hope this helps!

March 16, 2019 at 1:31 pm

Many thanks. I appreciate it.

March 15, 2019 at 10:47 am

I stumbled across your website in hopes of finding an answer to a couple of questions regarding the methodology of my political science paper. If you could help, I would be very grateful.

My research question is “Why do North-South regional trade agreements tend to generate economic convergence while South-South agreements sooner cause economic divergence?”. North = OECD developed countries and South = non-OECD developing countries.

This is my lineup of variables and hypotheses: DV: Economic convergence between country members in a regional trade agreement IV1: Complementarity (differentness) of relative factor abundance IV2: Market size of region IV3: Economic policy coordination (Harmonization of Foreign Direct Investment (FDI) policy)

H1: The higher the factor endowment difference between countries, the greater the convergence H2: The larger the market size, the greater the convergence H3: The greater the harmonization of FDI policies, the greater the convergence

I am not sure what the best methodological approach is. I will have to take North-South and South-South groups of countries and assign values for the groupings. I want to show the relationship between the IVs and DV, so I thought to use a regression. But there are at least two issues:

1. I feel the variables are not appropriate for a time series, which is usually used to show relationships. This is because e.g. the market size of a region will not be changing with time. Can I not do a time series and still have meaningful results?

2. The IVs are not completely independent of one another. How can I work with that?

Also, what kind of regression would be most appropriate in your view?

Many sincere thanks in advance. Irina

March 15, 2019 at 5:23 pm

I’m not an expert in that specific field, so I can’t give you concrete advice, but here are somethings to consider.

The question about whether you need to include time related information in the model depends on the nature of your data and whether you expect temporal effects to exist. If your data are essentially collected at the same time and refer to the same time period, you probably don’t need to account for time effects. If theory suggests that the outcome does not change over time, you probably don’t need to include variables for time effects.

However, if your data are collected at or otherwise describe different points in time, and you suspect that the relationships between the IVs and DV changes overtime, or there is an overall shift over time, yes, you’d need to account for the time effects in your model. In that case, failure to account for the effects of time can bias your other coefficients–basically there’s the potential for omitted variable bias .

I don’t know the subject area well enough to be able to answer those questions, but that’s what I’d think about.

You mention that the IVs are potentially correlated (multicollinearity). That might or might not be a problem. It depends on the degree of the correlation. Some correlation is OK and might not be a problem. I’d perform the analysis and check the VIFs, which measure multicollinearity. Read my post about multicollinearity , which discusses how to detect it, determine whether it’s a problem and some corrective measures.

I’d start with linear regression. Move away from that only if you have specific reason to do so.

March 10, 2019 at 3:59 am

I was wondering if you could help. I’m currently doing a lab report on Numerical cognition in Human and non human primates. Where we are looking at whether size , quantity and visibility of food effects choice. We have tested Humans so far and then are going to test chimps in the future. My Iv is Condition : visible and opague containers and my Dv is number of correct responses. So far I have compared the means of number of correct responses for both conditions using a one way repeated measures ANOVA but I don’t think this is correct. After having a look at your website, should I look to run a regression analysis instead ? Sorry for the confusion I’m really a rookie at this. Hope you can help !

March 11, 2019 at 11:26 am

Linear regression analysis and ANOVA are really the same type of analysis-linear models. They both use the same math “underneath the hood.” They each have their own historical traditions and terminology, but they’re really the same thing. In general, ANOVA tends to focus on categorical (nominal) independent variables while regression tends to focus on continuous IVs. However, you can add continuous variables into an ANOVA model and categorical variables into a regression model. If you fit the same model in ANOVA as regression, you’ll get the same results.

So, for your study, you can use either ANOVA or regression. However, because you have only one categorical IV, I’d normally suggest using one-way ANOVA. In fact, if you have only those two groups (visible vs opaque), you can use a 2-sample t-test.

Although, you mention repeated measures, you can use that if you in fact do have a pre-test and post-test conditions. You could even use a paired t-test if you have only the two groups and you have a pre- and post-tests.

There is one potential complication. You mention that the DV is a count of correct responses. Counts often do not follow the normal distribution but can follow other distributions such as the Poisson and Negative Binomial distributions. Although, counts can approximate the normal distribution when the mean is high enough (>~20). However, if you have two groups and each group has more than 15 observations, the analyses are robust to departures from the normal distribution.

I hope this helps! Best of luck with your analysis!

February 9, 2019 at 8:20 am

Thankyou so much for the reply . Appreciate it and I finally worked it out and got good mark on lab report, which was good :). Appreciate your time replying you explain things very clear so thankyou

January 17, 2019 at 9:49 am

Hi there. I am currently doing a lab report and have not done stats in years so hoping someone can help as due tommorow. When I do correlation bivariate test it shows the correlations not significant between a personaility trait and a particular cognitive task. Yet when I conduct a simple t test it shows a significant p value and gives the 95 % conf interval. If I was to compare that higher scores on one trait tends to mean higher scores on a particular cognitive task then should I be doing a regression then. We were told basic correlations so I did the bivariate option and just stated that the pearson’s r is not significant r=.. n= p =.84 for example. Yet if do a regression analysis for each it is significant. Why could this be?

January 18, 2019 at 9:45 am

There not quite enough details to know for sure what is happening–but here are some ideas.

Be aware that a series of pairwise correlations is not equivalent to performing regression analysis with multiple predictors. Suppose you have your outcome variable and two predictors (Y X1 X2). When you peform the pairwise correlations (X1 and Y, X2 and Y), each correlation does not account for the other X. However, when you include both X1 and X2 in a regression model, it estimates the relationship between each X and Y while accounting for the other X.

If the correlation and regression model results differ as you describe, you might well have a confounding variable, which biases your correlation results. I write about this in my post about omitted variable bias . You’d favor the regression results in this situation.

As for the difference between the 2-sample t-test and correlation, that’s not surprising because they are doing two entirely different things. The 2-sample t-test requires a continuous outcome variable and a categorical grouping variable and it tests the mean difference between the two groups. Correlations measure the linear association between two continuous variables. It’s not surprising the results can differ.

It sounds like you should probably use regression analysis and include your multiple continuous variables in the model along with your categorical grouping variables as independent variables to model your outcome variable.

January 17, 2019 at 12:39 am

This is Kathlene, and I am a Grade 12 student. I am currently doing my research. It’s a quantitative research. I am having a little trouble on how will i approach my statistical treatment. My research is entitled ” Emotional Quotient and Academic Performance Among Senior High School Students in Tarlac National High School: Basis to a Guidance Program. I was battling what to use to determine the relationship between the variables in my study. I’m thinking to use chi-square method but a friend said it would be more accurate to use the regression analysis method. Math is not really my field of study so i badly need your opinion regarding this.

I’m hoping you could lend me a helping hand.

January 17, 2019 at 9:27 am

Hi Kathlene,

It sounds like you’re in a great program! I wish more 12th grade students were conducting studies and analyzing their results! 🙂

To determine how to model the relationships between your variables, it depends on the type of variables you have. It sounds like your outcome variable is academic performance. If that’s a continuous variable, like GPA, then I’d agree with your friend that regression analysis would be a good place to start!

Chi-square assesses the relationship between categorical variables.

December 13, 2018 at 1:57 am

Hi Mr Jim, I am using orthogonal design having 7 factors with three levels. I have done regression analysis on Minitab software but i don’t know how to explain them or interpret them. I need your help in this regard.

December 13, 2018 at 9:13 am

I have a lot of content throughout my blog that will help you, including how to interpret the results. For a complete list for regression analysis, check out my regression tutorial .

Also, early next year I’ll be publishing a book about regression analysis as well that contains even more information.

If you have a more specific question after reading my other posts, you can ask them in the comments for the appropriate blog post.

Best of luck!

December 9, 2018 at 12:08 pm

By the way my gun laws vs VCR, is part of a regression model. Any help you can give, I’d greatly appreciate.

December 9, 2018 at 12:07 pm

Mr. Jim, I have a problem. I’m working on a research design on gun laws vs homicides with my dependent variable being violent crime rate. My sig is .308 The constant’s (VCR) standard error is 24.712 my n for violent crime rate is 430.44. I really need help ASAP. I don’t know how to interpret this well. Please help!!!

December 11, 2018 at 10:03 am

There’s not enough information for me to know how to interpret the results. How are you measuring gun laws? Also, VCR is your dependent variable, not the constant as you state. You don’t usually interpret the constant . All I can really say is that based on your p-value, it appears your independent variable is not statistically significant. You have insufficient evidence to conclude that there is a relationship between gun laws and homicides (or is it VCR?).

December 4, 2018 at 12:49 am

Your blog has been very useful. I have a query.. if I am conducting a multiple regression is it okay to have an outcome variable which is normally distributed ( i winsorized an outlier to achieve this) and have two other predictor variables which are not normally distributed? ( the normality tests scores were significant).

I have read in many places that you have to transform your data to achieve normality for the entire data set to conduct a multiple regression – but doing so has not helped me at all. Please advice.

December 4, 2018 at 10:42 am

I’m dubious about the Winsorizing process in general. Winsorizing reduces the effect of outliers. However, this process is fairly indiscriminate in terms of identifying outliers. It simply defines outliers as being more extreme than an upper and lower percentile and changes those extreme values to equal the specified percentiles. Identifying outliers should be a point by point investigation. Simply changing unusual values is not a good process. It might improve the fit of your data but it is an artificial improvement that overstates the true precision of the study area. If that point is truly an outlier, it might be better to remove it altogether, but make sure you a good explanation for why it’s an outlier.

For regression analysis, the distributions of your predictors and response don’t necessarily need to be normally distributed. However, it’s helpful, and generally sought, to have residuals that are normally distributed. So, check your residual plots! For more information, read my post about OLS assumptions so you know what you need to check!

If your residuals are nonnormally distributed, sometimes transforming the response can help. There are many transformations you can try. It’s a bit trial by error. I suggest you look into the Box-Cox and Johnson transformations. Both methods assess families of transformations and pick one that works bets for your data. However, it sounds like your outcome is already normally distributed so you might not need to do that.

Also, see what other researchers in your field have done with similar data. There’s little general advice I can offer other than to check the residuals and make sure they look good. If there are patterns in the residuals, make sure you’re fitting curvature that might be present. You can graph the various predictors by the residuals to find where the problem lies. You can also try transforming the variables as I describe earlier. While the variables don’t need to follow the normal distribution, if they’re very nonnormally distributed, it can cause problems in the residuals.

December 3, 2018 at 10:05 pm

Hi, I am confused about the assumption of independent observations in multiple linear regression. Here’s the case. I have heart rate data per five-minute for a day of 14 people. The dependent variable is the heart rate. During the day, the workers worked for 8 hours (8 am to 5 pm), so basically, I have 90 data points per worker for a day. So that makes it 1260 data points (90 times 14) to be included in the model. Is it valid to use multiple linear regression for this type of data?

December 4, 2018 at 10:47 am

It sounds like your model is more of a time series model. You can model those using regression analysis as well, but there are special concerns that you need to address. Your data are not independent. If someone has a height heart rate during one measurement, it’s very likely it’ll also be heighted 5 minutes later. The residuals are likely to be serially correlated, which violates one of the OLS assumptions .

You’ll likely need to include other variables in your model that capture this time dependent information, such as lagged variables. There are various considerations you’ll need to address that go beyond the scope of these comments. You’ll need to do some additional research into use regression analysis for time series data.

November 8, 2018 at 10:38 am

Ok.Thank you so much.

November 8, 2018 at 10:21 am

Thank you so much for your time! Actually i don’t have authentic data about property values (dependent variable) nor the concerning institutions have this data. Can i ask the property value directly to the property owner thorough walk interview?

November 8, 2018 at 10:31 am

You really need to have valid data. Using a self-reported valuation might be better than no data. However, be aware there might be differences between what the property owner says and the true market value. Your model would describe self-valuation rather than market valuation. Typically, I’ve seen studies like yours use actual sales prices.

November 8, 2018 at 12:20 am

Hello Sir! is it necessary fir dependent variable in multiple regression model to have values. i have number of independent variable( age of property, stories in building, location close to park)and single dependent variable (Property values). Some independent variable decrease the value of dependent variable, while some independent variables increase the value of the dependent variable? Can i put the value if my single dependent variable as ( a.<200000, b.<300000,c. d. 500000)?

November 8, 2018 at 9:39 am

Why would can’t you enter the actual property values? Ideally, that’s what you would do. If you are missing a value for a particular observation, you typically need to exclude the entire observation from the analysis. However, there are some ways to estimate missing values. For example, SPSS has advanced methods for imputing missing values. But, you should use those only to estimate a few missing values. Your plan should be to obtain the property values. If you can’t do that, it will be difficult to perform regression analysis.

There are some cases where you can’t record the exact values and it’s usually related to the observation time. This is known as censored data. A common example is in reliability analysis where you record failure times for a product. You run the experiment for a certain amount of time and you obtain some failures and know their failure times. However, some products don’t fail and you only know that their failure time is greater than the test time. There are censored regression models you can use in situations like that. However, I don’t think that applies to your subject-area, at least as far as I can tell.

November 5, 2018 at 5:52 pm

thank you so much Jim! this is really helpful 🙂

November 5, 2018 at 10:03 pm

You’re very welcome! Best of luck with your analysis!

November 5, 2018 at 5:16 pm

The variances (SD) for the 3 groups are 0.45, 0.7 and 1. Would you say that they vary by a lot? Another follow up question: does a narrower CI equals a better estimate?

November 5, 2018 at 5:26 pm

Yes, that’s definitely it!

I would suggest using Welch’s one-way ANOVA to analyze it and potentially use that analysis to calculate the CI. You’re essentially performing a one-way ANOVA. And, in ANOVA, there is the assumption of equal variances between groups, which your data do not satisfy. In regression, we’d refer to it as heteroscedasticity. In Welch’s ANOVA, you don’t need to satisfy that assumption. That makes it a simple solution for your case.

In terms of CIs, yes, narrower CIs indicate that the estimate is more precise than if you had a wider CI. Think of the CI as a margin of error around the estimate and it’s good to have a smaller margin of error. With a narrower CI, you can expect the actual mean to fall closer to the fitted value.

November 5, 2018 at 4:21 pm

Thank you so much for the quick response! I checked the residual plots, it gives me a pretty trend line at y=0, and my R square = 0.87. However the CI it gives me by using all 15 points (regression inference) is a little wider (2.012 – 3.655) than if I just use that 5 points(2.245 – 3.355). In this case, would you still prefer using all 15 points?

November 5, 2018 at 4:38 pm

That’s tricky. I hate to throw out data, but it does seem warranted. At least you have a good rationale for not using the data!

CIs of the mean for the a point at the end of a data range in a regression model do tend to be wider than in the middle of the range. Still, I’m not sure why it would be wider. Are the variances of the groups roughly equal? If not, that might well be the reason.

November 5, 2018 at 2:36 pm

suppose I have total of 15 data points at x=0, x=40, and x=80 (5 data points at each x value), now I can use regression to estimate y when x=60. But what if I want to estimate the average when x=0? Should I just use that 5 data points when x=0, or use the intercept from the regression line? Which is the best estimate for a 95% CI for the average y value when x=0?

Thank you 🙂

November 5, 2018 at 3:52 pm

Assuming that model provides a good fit to the data (check the residual plots), I’d use all the data to come up with the CI for the fitted value that corresponds to X = 0. That approach uses more data to calculate the estimate. Your CI might even be more precise (narrower) using all the data.

October 26, 2018 at 5:27 am

Hi, What make us use the linear regression instead of other types of regression. In other words, the motivation for selecting a linear model?

October 26, 2018 at 10:48 am

Typically, try linear regression first. If your data contain curvature, you might still be able to use linear regression. Linear regression is generally easier to use and includes some useful statistics that nonlinear regression can’t provide, such as p-values for the coefficients and R-squared.

However, if you can’t adequately fit the curvature in your data, it might be time to try nonlinear regression. While both types allow you fit curvature, nonlinear regression is more flexible because it allows your model to fit more types of curvature.

I’ve written a post about how to choose between linear and nonlinear regression that you should read. Within that post are various related links that talk about how to fit curves using both types of regression, along with additional information about both types.

October 26, 2018 at 1:17 am

Thank u so much for your reply. I am really gorgeous to know much more of this . I shall keep sending mails seeking your reply which i hope you will not mind

October 25, 2018 at 5:02 am

I have been unfortunate to get your reply to my comment on 18/09/2018

October 25, 2018 at 9:29 am

Sorry about the delay. As you can no doubt imagine, my schedule gets busy and things can fall through the cracks.

I replied under your original comment.

October 23, 2018 at 2:28 pm

Your blog has been really helpful! 🙂 I am currently completing my Masters Thesis and my primary outcome is to assess the relationship between Diabetes Distress and blood glucose control. I am a newbie to SPSS and I am at a loss as to how best to analyse my small (not normally distributed pre and post data transformation) data set.

I have been advised that regression analysis may be appropriate and better than correlations? However my data does not appear to be linear. My diabetes distress variables consist of a score of 1-6 based on a likert scale and also are categorical (low, moderate, high distress) and my blood glucose consists of continuous data and also a categorical variable of poorly controlled blood glucose and well controlled blood glucose.

At the moment I am struggling to complete this analysis. Any help would be greatly appreciated 🙂

October 21, 2018 at 5:06 pm

Dear Jim, thatk you very much for this post! Could you, please, explain the following.

You are writing: “you first need to fit and verify that you have a good model. Then, you look through the regression coefficients and p-values”

What if I have small r-squired, but the coefficiants are statistically significant with the small values?

October 15, 2018 at 5:37 am

Hi Jim Thanks for your enlightened explanations. However I want to engage you a bit. under how to interpret regression results, you indicated that a small p-Value indicates that the ” independent variable is statistically significant”. i tend not to agree> Note that since the null hypothesis is that the coefficient of the independent variable is equal to Zero, it’s rejection as evidenced by low p-Value should imply that it is the coefficient which is significantly different from zero and not the variable. almadi

October 15, 2018 at 9:56 am

Yes, you’re correct that the p-value tests whether the coefficient estimate is significantly different from zero. If it is, you can say that the coefficient is statistically significant. Alternatively, statisticians often say that the independent variable is statistically significant. In this context, these are two different ways of saying the same thing because the coefficient is a property of the variable itself.

September 18, 2018 at 5:44 am

As u must be well aware, govt releases price indices and these are broadly used to determine the effect of base prices during a given period of time.

Construction industry, normally uses these price indices running over a period of time to redetermine the prices based on the movement between the base date and current date, which is called as price adjustment

Govt after a few years of time releases a new series of price indices where we may not have the data of indices with old series which will necessitate us to use these new indices with a conversion factor to arrive at the equivalent value of the base price.

Where do you feel that Regression Analysis could be of help where we have to determine the current value of the base price using the new indices.

It is a bit amusing that someone was suggesting to me.

V.G.Subramanian

October 25, 2018 at 9:27 am

I agree that switching price indices can be a problem. If the indices overlap, you can perform regression analysis where the old index is the independent variable and the new index is the dependent variable. However, that is problematic if you don’t have both indices. If you had both indices, I suppose it wouldn’t be a problem to begin with!

Ideally, you’d understand the differences behind how the government calculates both indices, and you could use that to estimate the value of the other index.

I’m not particularly familiar with this practice, so I don’t have a whole lot of insight into it. I hope this helps somewhat!

September 2, 2018 at 8:15 am

Thank you for this, Jim. I’ve always felt a common sense explanation minus all the impressive math formulas is what is needed in statistics for data science. This is a big part of the basics I’ve been missing. I’m looking forward to your Logistic Regression Tutorial. How is that coming along for you?

September 2, 2018 at 2:59 pm

Hi Antonio,

Thanks so much for your kind words! They mean a lot to me! Yes, I totally agree, explanations should focus on being intuitive and helping people grasp the concepts.

I have written a post on binary logistic regression . Unfortunately, it’ll be awhile before I have a chance to write a more in-depth article–just too many subject to write about!

July 19, 2018 at 2:55 am

Dear sir, I have a few question about when to use ANOVA and when to use regression analysis. In my study i have conducted a experiment by considering temperature , pH, weight of a compound as a independent variables and extraction as a dependent variable ( i mentioned very generally but i have some specific independent and dependent variables along with these variables). I did statistical analysis by using one way ANOVA-Tukey’s test and i have used grouping method ( using alphabets a,b,c….) to show the significance based on the p value . My question is, for these type of data can i use regression analysis? and what is the main difference between Tukey’s test and regression analysis?

July 19, 2018 at 11:14 am

Both regression analysis and ANOVA are linear models. As linear models, both types of analyses have the same math “under the hood.” You can even use them interchangeably and get the same results. Traditionally, you use ANOVA when you have only, or mainly, categorical factors–although you can add in covariates (continuous variables). On the other hand, you tend to use regression when you have only, or mainly, continuous variables–although you can add in categorical variables.

Because ANOVA focuses on categorical factors and comparing multiple group means, statisticians have developed additional post hoc analyses to work with ANOVA, such as Tukey’s test. Typically, you’ll perform the ANOVA first and then the post hoc test. Suppose you perform a one-way ANOVA and obtain significant results. This significance tells you that not all of the group means are equal. However, it does not tell you which differences are statistically significant.

That point is where post hoc tests come in. These tests do two things. They’ll tell you which differences are statistically significant. They also control the family error rate for the group of comparisons. When you compare multiple differences like that, you increase the risk of a Type I error–which is when you say there is a difference but there really isn’t. When you compare multiple means, the Type I error rate will be higher than your significance level (alpha). These post hoc tests (other than Fishers) maintain the type I error rate so it continues to equal alpha, which is what you would expect.

So, use an ANOVA first. If you obtain significant results for a categorical factor, you can use post hoc tests like Tukey’s to explore the differences between the various factor levels.

I really need to write a blog post about this! I will soon!

In the meantime, I hope this helps!

May 28, 2018 at 5:28 am

Is it necessary to conduct correlation analysis before regression analysis?

May 30, 2018 at 11:02 am

Hi Kaushal,

No it’s not absolutely required. I actually prefer producing a series of scatterplots (or a matrix plot) so I can see the nature of the different relationships. That helps give me a better feel for the data along with the types of relationships. However, if you have a good theory and a solid background knowledge on which variables should be included in the model, you can go straight to modeling. I think it depends a lot on your existing level of knowledge.

That all said, I personally like knowing the correlation structure between all of the variables. It gives me a better feel for the data.

May 18, 2018 at 5:56 am

April 28, 2018 at 11:45 pm

Thank you Jim!

I really appreciate it!

April 28, 2018 at 7:38 am

Hi Jim, I hope you are having good time!

I would like to ask you a question, please!

I have 24 observations to perform a regression analysis (let’s say Zones), and I have many independent variables (IV). I would like to know what is the minimum number of observations I should have to perform a reasonable linear regression model. I would like to hear something from you about how to test many regression model with different IV, since I can not use many IV in a model where a have few observations (24).

Thank you in advance!

April 28, 2018 at 2:26 pm

Hi Patrik, great to hear from you again!

Those are great questions. For 24 observations, I’d say that you usually wouldn’t want more than 2 IVs. I write an entire post about how many variables you can include in a regression model . Including too many IVs (and other terms such as interactions and polynomials) is known as overfitting the model. Check that post out because it’ll provide guidance and show you the dangers of including too many.

There’s another issue a play too because you want to compare a number of different regression models to each other. If you compare many models, it’s a form of data mining. The risk here is that if you compare enough models, you will uncover chance correlations. These chance correlations look like the real thing but only appear in your sample and not the population. I’ve written a post about how using this type of data mining to choose a regression model causes problems . This concern is particularly problematic with a small sample size like yours. It can find “patterns” in randomly generated data.

So, there’s really two issues for you to watch out for–overfitting and chance correlations found through data mining!

Hope this helps!

April 5, 2018 at 5:51 am

Many Thanks Jim!!! You have no idea about how much you helped me.

Very well clarified!!!

God bless you always!!!

April 4, 2018 at 1:33 am

Hi Jim, I am everywhere in your post!

I am starting loving statistic, that’s why I am not quiet,

I have some questions for you:

To use OLS regression, one of the assumptions is that the dependent variable is normally distributed. To achieve this requirement what I should do with my data? Should I check the normality of my dependent variable, for example using Shapiro test (etc)? If I conclude that my dependent variable is not following the normal distribution I should start to see data transformation, right? Another way that I have used to see people analyzing the normality is by plotting the dependent variable with the independent variable and if the relationship doesn’t follow linear trend then they go to data transformation (which one you recommend me?) Should I perform the regression using my data (original) and then the residuals will show me non-normality if do exists?

When should I transform my independent variables, and what is the consequence of transforming them?

Sorry, I use to ask many questions in a single comment, but I think this is the way to understand the full picture of my doubt,

You are being so useful to me,

Thank you again!

April 4, 2018 at 11:11 am

Hi Patrik, I’m so happy to hear that you’re starting to love statistics! It’s a great field that is exciting. The thrill of discovery combined with getting the most value out of your data. I’m not sure if you’ve read my post about The Importance of Statistics , but if you haven’t, I recommend it. It explains why the field of statistics is more important than ever!

In OLS regression, the dependent variable does not have to be normally distributed. Instead, you need to assess the distribution of the residuals using residual plots . If your residuals are not normally distributed, there are a variety of possible reasons and different ways to resolve that issue. I always recommend that transforming your data is the last resort. For example, the residuals might be nonnormal because the model is specified incorrectly. Maybe there is curvature in the data that you aren’t modeling correctly? If so, transforming the data might mask the problem. You really want to specify the best possible model. However, if all else fails, you might need to transform the data. When you transform the data, you’ll need to back transform the results to make sense of the results because everything applies to the transformed data. Most statistical software should do this for you.

Be aware that you can’t trust R-squared and the standard error of the regression when you transform your dependent variable because they apply to the transformed data rather than the raw data (backtransformation won’t help there).

In terms of testing the normality of the residuals, I recommend using normal probability plots. You can usually tell at a glance whether they are normally distributed. If you need a test, I generally use the Anderson-Darling test–which you can see in action in my post about identifying the distribution of your data . By the way, as a case in point, the data in that post are not normal, but I use it as the dependent variable in OLS regression in this post about using regression to make predictions . The residuals are normally distributed even though the dependent variable is not.

March 29, 2018 at 2:27 am

In the coffee intake and smoking example, the first result showed that higher coffee intake leads to higher mortality, but after including smoking, coffee intake leads to lower or no mortality? Smoking was revealed to cause the mortality, but how did coffee intake now result in the opposite? Was a separate test taken for this result? Please let me know. S. CHATTERJEE

March 29, 2018 at 10:36 am

Hi, that’s a great question. It turns out that coffee and smoking are correlated. The negative effects of smoking on mortality are well documented. However, for some reason, the researchers did not originally include smoking in their model. Because drinking coffee and smoking are correlated, the variable for coffee consumption took on some of smoking’s effect on mortality.

Put another way, because smoking was not included in the model, it was not being controlled (held constant). So, as you increased coffee consumption, smoking also tended to increase because it is both positively correlated with coffee consumption and not in the model. Therefore, it appeared as though increased coffee consumption is correlated with higher mortality rates but only because smoking was not included in the model.

Presumably, the researchers had already collected data about smoking. So, all they had to do was include the smoking variable in their regression model. Voila, the model now controls for smoking and the new output displays the new estimate of the effect that coffee has on mortality.

This point illustrates a potential problem. If the researchers had not collected the smoking information, they would have really been stuck. Before conducting any study researchers need to do a lot of background research to be sure that they are collecting the correct data!

March 20, 2018 at 12:25 pm

Hi Jim Hope all thing is well,

I have faced problem with plotting, which is included the relationship between dependent variable (response) and the independent variables . when i do the main effect plots, i have the straight line increasing. y= x, this linear trending to change it i need to make y= square root for time

Im stuck with this thing i couldn’t find solution for it

March 5, 2018 at 9:37 am

I was wondering if you can help me? I am doing my dissertation and I have 1 within-subjects IV, and 3 between-subjects IVs.. most of my variables are categorical, but one is not categorical, it is a questionnaire which I am using to determine sleep quality, with both Likert scales and own answers to amount of sleep (hours), amount of times woke in the night etc. Can I use a regression when making use of both categorical data and other? I also have multiple DVs (angry/sad Likert ratings).. but I *could* combine those into one overall ’emotion’ DV. Any help would be much appreciated!

March 5, 2018 at 10:11 am

Hi Cara, because your DV use the Likert scale, you really should be using Ordinal Logistic Regression. This type of regression is designed for ordinal dependent variables like yours. As for the IVs, it can be tricky using ordinal variables. They’re not quite either continuous or categorical. My suggestion is to give them a try as continuous variable and check the residual plots to see how they look. If they look good, then it’s probably ok. However, if they don’t look good, you can try refitting the model using them as categorical variables and then rechecking the residual plots. If the residuals still don’t look good, you can then try using the chi-square test of independence for ordinal data.

As for combining the data, that would seem to be a subject-area specific decision, and I don’t know that area well enough to make an informed recommendation.

February 25, 2018 at 5:36 pm

Yes. But it may be that you miss my point. Because I argue that a proper and sound experiment will allow you to test for causality, regardless of if you deploy e.g. Pearsons r or regression. With no experimental design, neither Pearsons r nor a regression will test for an effect relationship between the variables. Randomisation makes a better case for controlling for variables that you are unaware of than picking a few, and then proclaim that your study found that x will cause an incrrase in y or that x has an effect on y. You may as well argue that you dont need to control for any variables and argue that any correlational study test for Effect relationships.

February 25, 2018 at 8:14 pm

Hi Martin, yes, that is exactly what I’m saying. Whether you can draw causal conclusion depends on whether you used a randomized experiment to collect your data. If it’s an observational study, you can’t assume it’s anything other than correlation. What you write in your comment agrees with what I’m saying.

The controlling for other variables that I mention in this post is a different matter. Yes, if you include a variable in a regression model, it is held constant while estimating the effects of the other variables. That doesn’t mean you can assume causality though.

February 25, 2018 at 5:04 pm

No statistical tool or method turns a survey or corrolation study into an experiment, i.e. regression does not test or imply cause effect relationship. A positive relationship between smoking and cancer in a regression analysis does not mean that smoking cause cancer. You have not controlled for what you are unaware of.

February 25, 2018 at 5:22 pm

Hi Martin, you are 100% correct about the fact that correlation doesn’t imply causation. This issue is one that I plan to cover in future posts.

There are two issues at play here. The type of study under which the data were collected and the statistical findings.

Being able to determine causation comes down to the difference between an observational study versus a randomized experiment. You actually use the same analyses to assess both types of designs. In an observational study, you can only establish correlation and not causality. However, in a randomized experiment, the same patterns and correlations in the data can suggest causality. So, regression analysis can help establish causality, but only when it’s performed on data that were collected through a randomized experiment.

February 6, 2018 at 7:11 am

Very nicely explanined. thank you

February 6, 2018 at 10:04 am

Thanks you, Hari!

December 1, 2017 at 2:40 am

Thanks for your reply and for the guidance.

I read your posts which are very helpful. After reading them, I concluded that only the independent variables which have a well-established association with the dependent variable should be included. Hence, in my case, variable Z should not be included given that the association of Z with dependent variable is not well-established.

Furthermore, suppose there is another variable (A) and literature suggests that it, in general, has an association with dependent variable. However, assume that A does not affect any independent variables so there is no omitted variable bias. In this case, if there is no data available for A (due to the study being conducted in different environment/context) then what statistical techniques can be deployed to address any problems caused due to the exclusion of A?

I look forward to your reply and I will be grateful for your reply.

Kind regards.

November 30, 2017 at 9:10 am

Thanks for the reply. I apologise if I am taking a considerable time out of your schedule.

Based on the literature, there isn’t any conclusive evidence that z is a determinant of y. So, that is why I intend to remove z. Some studies include it while some do not and some find significant association (between y and z) while some find the association insignificant. Hence, I think I can safely remove it.

Moreover, I will be grateful if you can answer another query. From an statistical viewpoint, is it fine if I use Generalized method of moments (GMM) for binary dependent variable?

November 30, 2017 at 2:24 pm

While I can’t offer you a concrete statement about whether you should include or exclude the variable (clearly there is disagreement in your own field), I do suggest that you read my article about specifying the correct regression model . I include a number of tips and considerations.

Unfortunately, I don’t know enough about GMM to make a recommendation. All of the examples I have seen personally are for continuous data, but I don’t know about binary data.

November 29, 2017 at 11:12 am

Thanks for your reply. I really appreciate it. Could you please also provide an answer to my query mentioned below for further clarification?

November 29, 2017 at 11:01 am

Further clarification on my above post. From internet I found that if a variable (z) that is related to y but unrelated to x then inclusion of z will reduce standard errors of x. So, if z is excluded, but f-stat and adjusted r-square are fine then does high standard errors create problems? I look forward to your reply.

November 29, 2017 at 11:50 am

Yes, what you read is correct. Typically, if Z is statistically significant, you should include it in your model. If you exclude it, the precision of your coefficient estimates will be lower (higher standard errors). You also risk a biased model because you are not including important information in the model–check the residual plots. The F-test of overall significance and adjusted R-squared depend on the other IVs in your model. If Z is by far the best variable, it’s possible that removing it will cause the F-test to not be significant and adjusted R-square might drop noticeably. Again, that depends on how the explanatory power of Z compares to the other IVs. Why do you want to remove a significant variable?

November 29, 2017 at 10:29 am

Thanks for the reply. Jim.

I am unable to understand “Your model won’t fit the data as well as before depending on the strength of the relationship between the dropped independent variable and the dependent variable”. Are you stating that other independent variables will be fine but r-square will become low? I will be grateful if you can explain this.

Kind regards

November 29, 2017 at 11:04 am

Hi, you indicated that the removed independent variable is related to the dependent variable, but it is not correlated with the other independent variables. Consequently, removing that independent variable should reduce R-squared. For one thing, that’s the typical result of removing variables, even when they’re not statistically significant. In this case, because it is not correlated to the other independent variables, you know that the removed variable is supplying unique information. Taking that variable out means that information is no longer included in the model. R-squared will definitely go down, possibly dramatically.

R-squared measures the strength of the relationship between the entire set of IVs and the DP. Read my post about R-squared for more information.

November 29, 2017 at 9:15 am

Hello, Jim.

What is the impact* on the independent variables in the model if I omit a variable that is a determinant of dependent variable but is not related to any of the independent variables?

*Here impact relates to the independent variables’ p-values and the coefficients.

November 29, 2017 at 10:13 am

If the independent variable is not correlated with the other independent variables, it’s likely that there would be a minimal effect on the other independent variables. Your model won’t fit the data as well as before depending on the strength of the relationship between the dropped independent variable and the dependent variable. You should also check the residual plots to be sure that by removing the variable you’re not introducing bias.

October 26, 2017 at 12:33 am

why do we use 5% level of significance usually for comparing instead of 1% or other

October 26, 2017 at 12:49 am

Hi, I actually write about this topic in a post about hypothesis testing . It’s basically a tradeoff between several different error rates–and a dash of tradition. Read that post and see if it answers your questions.

October 24, 2017 at 11:30 pm

Sir usually we take 5% level of significance for comparing why 0

October 24, 2017 at 11:35 pm

Hi Ghulam, yes, the significance level is usually 0.05. I’m not sure what you’re asking about in regards to zero? The p-values in the example output are all listed as 0.000, which is less than the significance level of 0.05, so they are statistically significant.

October 23, 2017 at 9:08 am

In my model, I use different independent variables. Now my question is before useing regression, do I need to check the distribution of data? if yes then please write the name tests. My title is Education and Productivity Nexus, : evidence from pharmaceutical sector in Bangladesh.

October 23, 2017 at 11:22 am

Hi Shamsun, typically you test the distribution of the residuals after you fit a model. I’ve written a blog post about checking your residual plots that should read.

I hope this helps! Jim

October 22, 2017 at 4:24 am

Thank you Mr. Jim

October 22, 2017 at 11:15 am

You’re very welcome!

October 22, 2017 at 2:31 am

In linear regression, can we use categorical variables as Independent variables? If yes, what should be the minimum or maximum categories in an Independent variable?

October 22, 2017 at 10:44 pm

Hi, yes you can use categorical variables as independent variables! The number of groups really depends on what makes sense for your study area. Of course, the minimum is two. There really is no maximum in theory. It depends on what makes sense for your study. However, in practice, having more groups requires a larger total sample size, which can become expensive. If you have 2-9 groups, you should have at least 15 in each group. For 10-12 groups, you should have 20. These numbers are based on simulation studies for ANOVA, but they also apply to categorical variables in regression. In a nutshell, figure out what makes sense for your study and then be sure to collect enough data!

I hope this help! Jim

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What is Regression Analysis and Why Should I Use It?

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Alchemer is an incredibly robust online survey software platform. It’s continually voted one of the best survey tools available on G2, FinancesOnline, and others. To make it even easier, we’ve created a series of blogs to help you better understand how to get the most from your Alchemer account.

Regression analysis is a powerful statistical method that allows you to examine the relationship between two or more variables of interest. 

While there are many types of regression analysis, at their core they all examine the influence of one or more independent variables on a dependent variable. 

Regression analysis provides detailed insight that can be applied to further improve products and services.

Here at Alchemer, we offer hands-on application training events during which customers  learn how to become super users of our software. 

In order to understand the value being delivered at these training events, we distribute follow-up surveys to attendees with the goals of learning what they enjoyed, what they didn’t, and what we can improve on for future sessions. 

The data collected from these feedback surveys allows us to measure the levels of satisfaction that our attendees associate with our events, and what variables influence those levels of satisfaction. 

Could it be the topics covered in the individual sessions of the event? The length of the sessions? The food or catering services provided? The cost to attend? Any of these variables have the potential to impact an attendee’s level of satisfaction.

By performing a regression analysis on this survey data, we can determine whether or not these variables have impacted overall attendee satisfaction, and if so, to what extent. 

This information then informs us about which elements of the sessions are being well received, and where we need to focus attention so that attendees are more satisfied in the future.

What is regression analysis and what does it mean to perform a regression?

Regression analysis is a reliable method of identifying which variables have impact on a topic of interest. The process of performing a regression allows you to confidently determine which factors matter most, which factors can be ignored, and how these factors influence each other.

In order to understand regression analysis fully, it’s essential to comprehend the following terms:

• Dependent Variable: This is the main factor that you’re trying to understand or predict. 
• Independent Variables: These are the factors that you hypothesize have an impact on your dependent variable.

In our application training example above, attendees’ satisfaction with the event is our dependent variable. The topics covered, length of sessions, food provided, and the cost of a ticket are our independent variables.

How does regression analysis work?

In order to conduct a regression analysis, you’ll need to define a dependent variable that you hypothesize is being influenced by one or several independent variables.

You’ll then need to establish a comprehensive dataset to work with. Administering surveys to your audiences of interest is a terrific way to establish this dataset. Your survey should include questions addressing all of the independent variables that you are interested in.

Let’s continue using our application training example. In this case, we’d want to measure the historical levels of satisfaction with the events from the past three years or so (or however long you deem statistically significant), as well as any information possible in regards to the independent variables. 

Perhaps we’re particularly curious about how the price of a ticket to the event has impacted levels of satisfaction. 

To begin investigating whether or not there is a relationship between these two variables, we would begin by plotting these data points on a chart, which would look like the following theoretical example.

(Plotting your data is the first step in figuring out if there is a relationship between your independent and dependent variables)

Our dependent variable (in this case, the level of event satisfaction) should be plotted on the y-axis, while our independent variable (the price of the event ticket) should be plotted on the x-axis.

Once your data is plotted, you may begin to see correlations. If the theoretical chart above did indeed represent the impact of ticket prices on event satisfaction, then we’d be able to confidently say that the higher the ticket price, the higher the levels of event satisfaction. 

But how can we tell the degree to which ticket price affects event satisfaction?

To begin answering this question, draw a line through the middle of all of the data points on the chart. This line is referred to as your regression line, and it can be precisely calculated using a standard statistics program like Excel.

We’ll use a theoretical chart once more to depict what a regression line should look like.

The regression line represents the relationship between your independent variable and your dependent variable. 

Excel will even provide a formula for the slope of the line, which adds further context to the relationship between your independent and dependent variables. 

The formula for a regression line might look something like Y = 100 + 7X + error term .

This tells you that if there is no “X”, then Y = 100. If X is our increase in ticket price, this informs us that if there is no increase in ticket price, event satisfaction will still increase by 100 points. 

You’ll notice that the slope formula calculated by Excel includes an error term. Regression lines always consider an error term because in reality, independent variables are never precisely perfect predictors of dependent variables. This makes sense while looking at the impact of  ticket prices on event satisfaction — there are clearly other variables that are contributing to event satisfaction outside of price.

Your regression line is simply an estimate based on the data available to you. So, the larger your error term, the less definitively certain your regression line is.

Why should your organization use regression analysis?

Regression analysis is helpful statistical method that can be leveraged across an organization to determine the degree to which particular independent variables are influencing dependent variables. 

The possible scenarios for conducting regression analysis to yield valuable, actionable business insights are endless.

The next time someone in your business is proposing a hypothesis that states that one factor, whether you can control that factor or not, is impacting a portion of the business, suggest performing a regression analysis to determine just how confident you should be in that hypothesis! This will allow you to make more informed business decisions, allocate resources more efficiently, and ultimately boost your bottom line.

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Regression Analysis

• Home > What We Do > Research Methods > Pricing and Value Research Techniques > Regression Analysis

From overall customer satisfaction to satisfaction with your product quality and price, regression analysis measures the strength of a relationship between different variables.

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To find out more about measuring customer satisfaction to help your business

How regression analysis works

While correlation analysis provides a single numeric summary of a relation (“the correlation coefficient”), regression analysis results in a prediction equation, describing the relationship between the variables. If the relationship is strong – expressed by the Rsquare value – it can be used to predict values of one variable given the other variables have known values. For example, how will the overall satisfaction score change if satisfaction with product quality goes up from 6 to 7?

Measuring customer satisfaction

Regression analysis can be used in customer satisfaction and employee satisfaction studies to answer questions such as: “Which product dimensions contribute most to someone’s overall satisfaction or loyalty to the brand?” This is often referred to as Key Drivers Analysis.

It can also be used to simulate the outcome when actions are taken. For example: “What will happen to the satisfaction score when product availability is improved?”

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4   Linear Regression

A quick review of regression, expectation, variance, and parameter estimation.

Input vector: \(X = (X_1, X_2, ... , X_p)\) .

Output Y is real-valued.

Predict Y from X by f ( X ) so that the expected loss function \(E(L(Y, f(X)))\) is minimized.

Review: Expectation

Intuitively, the expectation of a random variable is its “average” value under its distribution.

Formally, the expectation of a random variable X, denoted E[X], is its Lebesgue integral with respect to its distribution.

If X takes values in some countable numeric set \(\chi\) , then

\[E(X) =\sum_{x \in \chi}xP(X=x)\]

If \(X \in \mathbb{r eval=FALSE}^m\) has a density p , then

\[E(X) =\int_{\mathbb{r eval=FALSE}^m}xp(x)dx\]

Expectation is linear: \(E(aX +b)=aE(X) + b\)

Also, \(E(X+Y) = E(X) +E(Y)\)

The expectation is monotone: if X ≥ Y , then E ( X ) ≥ E ( Y )

Review: Variance

The variance of a random variable X is defined as:

\(Var(X) = E[(X-E[X])^2]=E[X^2]-(E[X])^2\)

and the variance obeys the following \(a, b \in \mathbb{r eval=FALSE}\) :

\[Var(aX + b) =a^2Var(X)\]

Review: Frequentist Basics

The data x 1 , … , x n is generally assumed to be independent and identically distributed (i.i.d.).

We would like to estimate some unknown value θ associated with the distribution from which the data was generated.

In general, our estimate will be a function of the data (i.e., a statistic) \[\hat{\theta} =f(x_1, x_2, ... , x_n)\]

Example: Given the results of n independent flips of a coin, determine the probability p with which it lands on heads.

Review: Parameter Estimation

In practice, we often seek to select a distribution (model) corresponding to our data.

If our model is parameterized by some set of values, then this problem is that of parameter estimation.

How can we obtain estimates in general? One Answer: Maximize the likelihood and the estimate is called the maximum likelihood estimate, MLE.

\[ \begin {align} \hat{\theta} & = argmax_{\theta} \prod_{i=1}^{n}p_{\theta}(x_i) \\ & =argmax_{\theta} \sum_{i=1}^{n}log (p_{\theta}(x_i)) \\ \end {align} \]

Let’s look at the setup for linear regression. We have an input vector: \(X = \left( X _ { 1 } , X _ { 2 } , \dots , X _ { p }\right)\) . This vector is p dimensional.

The output Y is a real value and is ordered.

We want to predict Y from X .

Before we actually do the prediction we have to train the function f ( X ). By the end of the training, I would have a function f ( X ) to map every X into an estimated Y . Then, we need some way to measure how good this predictor function is. This is measured by the expectation of a loss.

Why do we have a loss in the estimation?

Y is actually a random variable given X . For instance, consider predicting someone’s weight based on the person’s height. People can have different weights given the same height. If you think of the weight as Y and the height as X , Y is random given X . We, therefore, cannot have a perfect prediction for every subject because f ( X ) is a fixed function, impossible to be correct all the time. The loss measures how different the true Y is from your prediction.

Why do we have the overall loss expressed as an expectation?

The loss may be different for different subjects. In statistics, a common thing to do is to average the losses over the entire population.

Squared loss:

\[L ( Y , f ( X ) ) = ( Y - f ( X ) ) ^ { 2 }\]

We simply measure the difference between the two variables and square them so that we can handle negative and positive difference symmetrically.

Suppose the distribution of Y given X is known , the optimal predictor is:

\[\begin{array} { l } { f ^ {*} ( X ) = \operatorname { argmin } _ { f ( x ) } E ( Y - f ( x ) ) ^ { 2 } } \\ { = E ( Y | X ) } \end{array}\]

This is the conditional expectation of Y given X . The function E ( Y | X ) is called the regression function .

Example 3-1

We want to predict the number of physicians in a metropolitan area.

Problem : The number of active physicians in a Standard Metropolitan Statistical Area (SMSA), denoted by Y , is expected to be related to total population ( X 1 , measured in thousands), land area ( X 2 , measured in square miles), and total personal income ( X 3 , measured in millions of dollars). Data are collected for 141 SMSAs, as shown in the following table.

Our Goal: To predict Y from \(X _ { 1 } , X _ { 2 } , \text { and } X _ { 3 }\) .

This is a typical regression problem.

Upon successful completion of this lesson, you should be able to:

• Review of linear regression model focusing on prediction.
• Use least square estimation for linear regression.
• Apply model developed in training data to an independent test data.
• Context setting for more complex supervised prediction methods.

4.1 Linear Methods

The linear regression model:

\[ f(X)=\beta_{0} + \sum_{j=1}^{p}X_{j}\beta_{j}\]

This is just a linear combination of the measurements that are used to make predictions, plus a constant, (the intercept term). This is a simple approach. However, It might be the case that the regression function might be pretty close to a linear function, and hence the model is a good approximation.

What if the model is not true?

• It still might be a good approximation - the best we can do.
• Sometimes because of the lack of training data or smarter algorithms, this is the most we can estimate robustly from the data.

Comments on \(X_j\) :

• We assume that these are quantitative inputs [or dummy indicator variables representing levels of a qualitative input]
• We can also perform transformations of the quantitative inputs, e.g., log(•), √(•). In this case, this linear regression model is still a linear function in terms of the coefficients to be estimated. However, instead of using the original \(X_{j}\) , we have replaced them or augmented them with the transformed values. Regardless of the transformations performed on \(X _ { j } f ( x )\) is still a linear function of the unknown parameters.
• Some basic expansions: \(X _ { 2 } = X _ { 1 } ^ { 2 } , X _ { 3 } = X _ { 1 } ^ { 3 } , X _ { 4 } = X _ { 1 } \cdot X _ { 2 }\) .

Below is a geometric interpretation of a linear regression.

For instance, if we have two variables, \(X_{1}\) and \(X_{2}\) , and we predict Y by a linear combination of \(X_{1}\) and \(X_{2}\) , the predictor function corresponds to a plane (hyperplane) in the three-dimensional space of \(X_{1}\) , \(X_{2}\) , Y . Given a pair of \(X_{1}\) and \(X_{2}\) we could find the corresponding point on the plane to decide Y by drawing a perpendicular line to the hyperplane, starting from the point in the plane spanned by the two predictor variables.

For accurate prediction, hopefully, the data will lie close to this hyperplane, but they won’t lie exactly in the hyperplane (unless perfect prediction is achieved). In the plot above, the red points are the actual data points. They do not lie on the plane but are close to it.

How should we choose this hyperplane?

We choose a plane such that the total squared distance from the red points (real data points) to the corresponding predicted points in the plane is minimized. Graphically, if we add up the squares of the lengths of the line segments drawn from the red points to the hyperplane, the optimal hyperplane should yield the minimum sum of squared lengths.

The issue of finding the regression function \(E ( Y | X )\) is converted to estimating \(\beta _ { j } , j = 0,1 , \dots , p\) .

Remember in earlier discussions we talked about the trade-off between model complexity and accurate prediction on training data. In this case, we start with a linear model, which is relatively simple. The model complexity issue is taken care of by using a simple linear function. In basic linear regression, there is no explicit action taken to restrict model complexity. Although variable selection, which we cover in Lesson 5: Variable Selection, can be considered a way to control model complexity.

With the model complexity under check, the next thing we want to do is to have a predictor that fits the training data well.

Let the training data be:

\[\left\{ \left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \dots , \left( x _ { N } , y _ { N } \right) \right\} , \text { where } x _ { i } = \left( x _ { i 1 } , x _ { i 2 } , \ldots , x _ { i p } \right)\]

Denote \(\beta = \left( \beta _ { 0 } , \beta _ { 1 } , \ldots , \beta _ { p } \right) ^ { T }\) .

Without knowing the true distribution for X and Y , we cannot directly minimize the expected loss.

Instead, the expected loss \(E ( Y - f ( X ) ) ^ { 2 }\) is approximated by the empirical loss \(R S S ( \beta ) / N\) :

\[ \begin {align}RSS(\beta)&=\sum_{i=1}^{N}\left(y_i - f(x_i)\right)^2 &=\sum_{i=1}^{N}\left(y_i - \beta_0 -\sum_{j=1}^{p}x_{ij}\beta_{j}\right)^2 \\ \end {align} \]

This empirical loss is basically the accuracy you computed based on the training data. This is called the residual sum of squares, RSS .

The x ’s are known numbers from the training data.

Here is the input matrix X of dimension N × ( p +1):

\[\begin{pmatrix} 1 & x_{1,1} &x_{1,2} & ... &x_{1,p} \\ 1 & x_{2,1} & x_{2,2} & ... &x_{2,p} \\ ... & ... & ... & ... & ... \\ 1 & x_{N,1} &x_{N,2} &... & x_{N,p} \end{pmatrix}\]

Earlier we mentioned that our training data had N number of points. So, in the example where we were predicting the number of doctors, there were 101 metropolitan areas that were investigated. Therefore, N =101. Dimension p = 3 in this example. The input matrix is augmented with a column of 1’s (for the intercept term). So, above you see the first column contains all 1’s. Then if you look at every row, every row corresponds to one sample point and the dimensions go from one to p . Hence, the input matrix X is of dimension N × ( p +1).

Output vector y :

\[ y= \begin{pmatrix} y_{1}\\ y_{2}\\ ...\\ y_{N} \end{pmatrix} \]

Again, this is taken from the training data set.

The estimated \(\beta\) is \(\hat{\beta}\) and this is also put in a column vector, \(\left( \beta _ { 0 } , \beta _ { 1 } , \dots , \beta _ { p } \right)\) .

The fitted values (not the same as the true values) at the training inputs are

\[\hat{y}_{i}=\hat{\beta}_{0}+\sum_{j=1}^{p}x_{ij}\hat{\beta}_{j}\]

\[ \hat{y}= \begin{pmatrix} \hat{y}_{1}\\ \hat{y}_{2}\\ ...\\ \hat{y}_{N} \end{pmatrix} \]

For instance, if you are talking about sample i , the fitted value for sample i would be to take all the values of the x ’s for sample i , (denoted by \(x_{ij}\) ) and do a linear summation for all of these \(x_{ij}\) ’s with weights \(\hat{\beta}_{j}\) and the intercept term \(\hat{\beta}_{0}\) .

4.2 Point Estimate

• The least square estimation of \(\hat{\beta}\) is:

\[\hat{\beta} =(X^{T}X)^{-1}X^{T}y \]

• The fitted value vector is:

\[\hat{y} =X\hat{\beta}=X(X^{T}X)^{-1}X^{T}y \]

• Hat matrix:

\[H=X(X^{T}X)^{-1}X^{T} \]

Geometric Interpretation

Each column of X is a vector in an N -dimensional space (not the \(p + 1\) * dimensional feature vector space). Here, we take out columns in matrix X , and this is why they live in N*-dimensional space. Values for the same variable across all of the samples are put in a vector. I represent this input matrix as the matrix formed by the column vectors:

\[X = \left( X _ { 0 } , x _ { 1 } , \ldots , x _ { p } \right)\]

Here \(x_0\) is the column of 1’s for the intercept term. It turns out that the fitted output vector \(\hat{y}\) is a linear combination of the column vectors \(x _ { j } , j = 0,1 , \dots , p\) . Go back and look at the matrix and you will see this.

This means that \(\hat{y}\) lies in the subspace spanned by \(x _ { j } , j = 0,1 , \dots , p\) .

The dimension of the column vectors is N , the number of samples. Usually, the number of samples is much bigger than the dimension p . The true y can be any point in this N -dimensional space. What we want to find is an approximation constraint in the \(p+1\) dimensional space such that the distance between the true y and the approximation is minimized. It turns out that the residual sum of squares is equal to the square of the Euclidean distance between y and \(\hat{y}\) .

\[RSS(\hat{\beta})=\parallel y - \hat{y}\parallel^2 \]

For the optimal solution, \(y-\hat{y}\) has to be perpendicular to the subspace, i.e., \(\hat{y}\) is the projection of y on the subspace spanned by \(x _ { j } , j = 0,1 , \dots , p\) .

Geometrically speaking let’s look at a really simple example. Take a look at the diagram below. What we want to find is a \(\hat{y}\) that lies in the hyperplane defined or spanned by \(x _ {1}\) and \(x _ {2}\) . You would draw a perpendicular line from y to the plane to find \(\hat{y}\) . This comes from a basic geometric fact. In general, if you want to find some point in a subspace to represent some point in a higher dimensional space, the best you can do is to project that point to your subspace.

The difference between your approximation and the true vector has to be perpendicular to the subspace.

The geometric interpretation is very helpful for understanding coefficient shrinkage and subset selection (covered in Lesson 5 and 6).

4.3 Example Results

Let’s take a look at some results for our earlier example about the number of active physicians in a Standard Metropolitan Statistical Area (SMSA - data). If I do the optimization using the equations, I obtain these values below:

\[\hat{Y}_{i}= –143.89+0.341X_{i1}–0.019X_{i2}+0.254X_{i3} \]

\[RSS(\hat{\beta})=52,942,438 \]

Let’s take a look at some scatter plots. We plot one variable versus another. For instance, in the upper left-hand plot, we plot the pairs of \(x_{1}\) and y . These are two-dimensional plots, each variable plotted individually against any other variable.

STAT 501 on Linear Regression goes deeper into which scatter plots are more helpful than others. These can be indicative of potential problems that exist in your data. For instance, in the plots above you can see that \(x_{3}\) is almost a perfectly linear function of \(x_{1}\) . This might indicate that there might be some problems when you do the optimization. What happens is that if \(x_{3}\) is a perfectly linear function of \(x_{1}\) , then when you solve the linear equation to determine the \(β\) ’s, there is no unique solution. The scatter plots help to discover such potential problems.

In practice, because there is always measurement error, you rarely get a perfect linear relationship. However, you might get something very close. In this case, the matrix, \(X ^ { T } X\) , will be close to singular, causing large numerical errors in computation. Therefore, we would like to have predictor variables that are not so strongly correlated.

4.4 Theoretical Justification

If the linear model is true.

Here is some theoretical justification for why we do parameter estimation using least squares.

If the linear model is true, i.e., if the conditional expectation of Y given X indeed is a linear function of the X j ’s, and Y is the sum of that linear function and an independent Gaussian noise, we have the following properties for least squares estimation.

\[ E(Y|X)=\beta_0+\sum_{j=1}^{p}X_{j}\beta{j} \]

The least squares estimation of \(\beta\) is unbiased,

\[E(\hat{\beta}_{j}) =\beta_j, j=0,1, ... , p \]

To draw inferences about \(\beta\) , further assume: \(Y = E(Y | X) + \epsilon\) where \(\epsilon \sim N(0,\sigma^2)\) and is independent of X .

\(X_{ij}\) are regarded as fixed, \(Y_i\) are random due to \(\epsilon\) .

The estimation accuracy of \(\hat{\beta}\) , the variance of \(\hat{\beta}\) is given here:

\[Var(\hat{\beta})=(X^{T}X)^{-1}\sigma^2\]

You should see that the higher \(\sigma^2\) is, the variance of \(\hat{\beta}\) will be higher. This is very natural. Basically, if the noise level is high, you’re bound to have a large variance in your estimation. But then, of course, it also depends on \(X^T X\) . This is why in experimental design, methods are developed to choose X so that the variance tends to be small.

Note that \(\hat{\beta}\) is a vector and hence its variance is a covariance matrix of size ( p + 1) × ( p + 1). The covariance matrix not only tells the variance for every individual \(\beta_j\) , but also the covariance for any pair of \(\beta_j\) and \(\beta_k\) , \(j \ne k\) .

Gauss-Markov Theorem

This theorem says that the least squares estimator is the best linear unbiased estimator.

Assume that the linear model is true. For any linear combination of the parameters \(\beta_0 , \cdots ,beta_p\) you get a new parameter denoted by \(\theta = a^{T}\beta\) . Then \(a^{T}\hat{\beta}\) is just a weighted sum of \(\hat{\beta}_0, ..., \hat{\beta}_p\) and is an unbiased estimator since \(\hat{\beta}\) is unbiased.

We want to estimate \(θ\) and the least squares estimate of \(θ\) is:

\[ \begin {align} \hat{\theta} & = a^T\hat{\beta}\\ & = a^T(X^{T}X)^{-1}Xy \\ & \doteq \tilde{a}^{T}y, \\ \end{align} \]

which is linear in y . The Gauss-Markov theorem states that for any other linear unbiased estimator, \(c^Ty\) , the linear estimator obtained from the least squares estimation on \(\theta\) is guaranteed to have a smaller variance than \(c^Ty\) :

\[Var(\tilde{a}^{T}y) \le Var(c^{T}y).\]

Keep in mind that you’re only comparing with linear unbiased estimators. If the estimator is not linear, or is not unbiased, then it is possible to do better in terms of squared loss.

\(\beta_j\) , j = 0, 1, …, p are special cases of \(a^T\beta\) , where \(a^T\) only has one non-zero element that equals 1.

4.5 R Scripts

1. acquire data.

Diabetes data

The diabetes data set is taken from the UCI machine learning database on Kaggle: Pima Indians Diabetes Database

• 768 samples in the dataset
• 8 quantitative variables
• 2 classes; with or without signs of diabetes

Save the data into your working directory for this course as “diabetes.data.” Then load data into R as follows:

In RawData , the response variable is its last column; and the remaining columns are the predictor variables.

2. Fitting a Linear Model

In order to fit linear regression models in R, lm can be used for linear models, which are specified symbolically. A typical model takes the form of response~predictors where response is the (numeric) response vector and predictors is a series of predictor variables.

Take the full model and the base model (no predictors used) as examples:

For the full model, coefficients shows the least square estimation for \(\hat{\beta}\) and fitted.values are the fitted values for the response variable.

The results for the coefficients should be as follows:

The fitted values should start with 0.6517572852.

Source Code

Master Regression Analysis for Machine Learning Models

Regression analysis provides the core statistical techniques for developing predictive machine learning models. By understanding relationships between independent input variables and a target output, regression allows us to forecast future observations.

This comprehensive technical guide explores essential regression concepts for machine learning experts, data scientists, and statisticians. Readers will gain an in-depth understanding of:

• Linear regression and extensions like polynomial models
• Using splines and regularization to prevent overfitting
• Best practices for feature engineering and assumption checking
• Model evaluation methods and real-world case studies

By the end, you will have the advanced capabilities to confidently build, assess, and deploy regression models. Let‘s get started.

Introduction to Regression Analysis

Regression analysis refers to an umbrella of statistical methods for modeling the relationship between two or more variables. It enables us to predict a continuous numerical target variable based on changes in other input features. Regression algorithms quantify the "influence" of each variable on the target to uncover predictive patterns from the data.

While many regression models exist, linear regression using ordinary least squares is one of the most fundamental and interpretable. By fitting a linear equation to minimize residuals, it identifies interpretable slopes and intercepts to forecast future observations. However, linear regression makes several key assumptions:

Key Linear Regression Assumptions

• Linear relationship between dependent and independent variables
• Statistical independence of error terms
• Homoscedasticity – constant error term variance
• Lack of perfect multicollinearity among variables

Later sections will explore both testing for violations and mitigation strategies in depth. First, let‘s solidify foundational linear regression concepts.

Simple Linear Regression

In simple linear regression, a single explanatory variable x is used to predict the quantitative target variable y. The model takes the form:

$$y = \beta_0 + \beta_1 x $$

Where $\beta_0$ represents the intercept and $\beta_1$ encodes the slope for variable x. Given training data, we estimate optimal values for the intercept and slope by minimizing the residual sum of squares between predicted and actual y-values.

Once fit, this simple model enables explaining relationships and making predictions. It forms the building block for more complex regression techniques.

Using Linear Regression in Practice

Due to its ubiquity, interpretability, and ease of use, simple linear regression can provide surprisingly powerful predictions in many business settings. Common applications include:

• Forecasting – Predicting future sales, demand, or production capacity based on historical trends
• Finance – Modeling how macroeconomic indicators impact firm revenues or valuation
• Healthcare – Quantifying how clinical factors influence patient outcomes or length of stay

Later sections will demonstrate the flexibility of more complex regression approaches. First, let‘s solidify foundational linear regression concepts.

Going In-Depth on Assumptions

While the linear regression equation is straightforward, properly applying the method requires testing and validating key assumptions:

• Linear relationship – The true relationship between input x and output y is approximately linear.
• Statistical independence – The error terms are uncorrelated.
• Homoscedasticity – The errors exhibit constant variance.
• No perfect multicollinearity – The predictors are not overly correlated.

Verifying these assumptions is necessary to ensure reliable coefficient estimates and predictions.

Graphical and Statistical Tests

We can check assumptions using both data visualizations and statistical tests:

• Scatter plots – Assess linearity and homoscedasticity
• Residual plots – Check independence and equal variance
• Correlation matrices – Identify multicollinearity candidates
• Durbin-Watson – Autocorrelation of residuals
• Variance Inflation Factors – Quantify multicollinearity

For example, non-random patterning in a residuals plot indicates violations of independence or homoscedasticity. We‘ll explore deeper residual analysis methods later on.

Strategies for Meeting Assumptions

When assumptions do not hold, several mitigation strategies exist:

• Transforming variables – Taking logs or standardized scores
• Removing outliers – Trimming or winsorization
• Robust regression – Downweighting high-influence points
• Correcting heteroscedasticity – Weighted least squares

In practice, some degree of violation is often acceptable if key model outputs remain reliable. Perfectly meeting all assumptions is secondary to predictive accuracy and usefulness.

Real-World Considerations

Healthcare spending data provides an insightful example for checking assumptions in practice. Predicting average spend based on illness severity likely violates homoscedasticity. The error variance increases for more severe (and expensive) cases. Strategies like weighted regression directly account for heterogeneous variation.

In any industry, thoughtfully exploring assumptions provides crucial confidence in applying models. Next, we‘ll cover extensions to linear regression for more flexibility.

Introducing Multiple Regression

While simple linear regression only utilizes a single predictor, multiple regression expands this to two or more explanatory variables. The model takes the form:

$$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + … + \beta_px_p$$

Where $x_1$ through $x_p$ refer to p distinct input variables. By increasing parameters, multiple regression flexibility captures interactions between variables to improve predictive accuracy. It also enables directly quantifying the marginal effect of each variable on the outcome $y$, accounting for the other predictors.

Comparing Simple and Multiple Regression

Consider predicting healthcare costs. A simple model uses only patient illness severity. The R-squared is 0.60, indicating severity explains 60% of spend variance. However, including additional inputs like age, gender, and previous conditions increases R-squared to 0.85. Conceptually, these supplementary variables explain nuanced differences in cost not captured by severity alone.

Multiple regression offers greater depth, but still assumes linear variable relationships. Next, we explore nonlinear modeling options.

Modeling Nonlinear Trends with Polynomial Regression

Thus far, our regression examples assumed the input variables demonstrate a linear correlation with the output y. However, real-world data frequently exhibits nonlinear trends – especially in social sciences, healthcare, finance, and other complex system domains.

Polynomial regression introduces nonlinearity by adding polynomial terms of predictors $x$ as additional model terms. For example, a quadratic equation takes the form:

$$y = \beta_0 + \beta_1x + \beta_2x^2$$

This model includes both the linear term $x$ and the squared term $x^2$. By adding parameters, we increase flexibility to capture curvature and more complex variable relationships.

We can evaluate even higher-order polynomial terms for cubic, quartic, or high-degree patterns if appropriate. However, these highly complex fits come with downsides. Risks include:

• Overfitting on training data noise
• Multicollinearity between polynomial terms
• Odd behavior near endpoint extrapolation

Smoothing with Regression Splines

Splines bring together linear segments, with "knots" specifying the cutpoints between regions. They provide a powerful alternative to high-degree polynomials, enabling flexible fitting without sacrificing smoothness or overfitting as easily.

Popular spline options include:

• Cubic splines – Cubic curves between knots
• B-splines – Basis function compositions
• Natural cubic splines – With continuity constraints

In the next section, we‘ll explore regularization techniques to directly prevent overfitting as model flexibility increases.

Preventing Overfitting with Regularization

Regularization methods augment model training to avoid overfitting on noise within training data. This focuses on learning generalizable patterns less likely to cause issues predicting future samples. Two standard regularization techniques are ridge regression and lasso regression.

Ridge Regression

Ridge regression works by penalizing model coefficients as part of the training process. Specifically, it adds an $L2$ term equal to the sum of squared magnitudes:

$$Penalty = \lambda\sum\beta_i^2$$

Where the $\lambda$ hyperparameter controls the regularization strength. By shrinking coefficients, ridge regression smooths the learned regression function to avoid highly complex solutions prone to overfitting. An example 5-fold cross-validated grid search helps tune optimal regularization strength.

The Lasso Method

The lasso technique uses an $L1$ penalty equal to the absolute sum of coefficients instead. Mathematically, this induces sparsity in the solution – shrinking the less important coefficients exactly to zero.

$$Penalty = \lambda\sum|\beta_i|$$

The lasso often increases model interpretability by performing embedded feature selection. It eliminates noise variables altogether. Computationally, adding L1 regularization terms also enables efficient cyclic coordinate descent optimization approaches.

Through smart regularization, we can heavily restrict model flexibility while retaining strong out-of-sample prediction accuracy.

Evaluating Model Performance

To choose between regression approaches and quantify real-world usefulness, numerical performance metrics are crucial. We evaluate on an unseen holdout test set after data splitting and model tuning on development validation data.

Quantitative Metrics

Key regression scoring metrics include:

• Residual analysis – Residual plots and summary statistics
• R-squared – Variance explained, adjusted for model size
• Mean Absolute Error (MAE) – Magnitude of average residuals
• Root Mean Squared Error (RMSE) – Penalizes larger errors

Additional statistical tests like F-tests and t-tests help assess the overall fit and individual variable contributions. We also analyze metrics across samples to check for biases.

Detecting and Amending Issues

Beyond overall performance, visualizing residuals against both actual values and input variables is invaluable for detecting issues like heteroscedasticity. We can then use weighted least squares or transforms to correct problems.

Industry expectations also guide acceptable accuracy. For example, marketing response models may tolerate 10%+ absolute percentage errors, while program trading models demand precise reliability. We adjust complexity accordingly.

In production systems, regression models require ongoing statistical monitoring to detect concept drift and retrain if necessary.

Business and Decision Making

For business use cases, consider upside and economics alongside raw numeric accuracy:

• Does the model enable beneficial new decisions? What operational levers or strategies become available?
• How much business value does this create? Estimate total cost savings or profit upside.

Quantifying practical impact and value helps justify investments into further improving modeling and data infrastructure.

Feature Engineering for Regression

Beyond algorithm selection, transforming raw input data into informative features is critical for success. We must encode categorical variables appropriately, handle missing data fields, remove unnecessary noise, and avoid data leaks.

Encoding Categorical Variables

For categorical inputs like product types, ordinal encoding assigns integers corresponding to some inherent order. Binary dummy variables are useful for nominal categories lacking a logical ranking. Discretizing continuous variables also enables nonlinear modeling.

Imputing Missing Values

Rather than casewise deletion which reduces sample size, we estimate missing continuous values via mean, mode or regression imputation. Categorical missing data get imputed with the mode or a new category identifier. Ensuring sufficient training samples with an imputation strategy prevents data leakage.

Variable Transformation

Applying mathematical transformations before model fitting changes how variables interact and contribute to predictions. Common examples include scaling with z-scores and using logarithmic values to analyze percentage changes.subject-matter context guides appropriate transformations.

Creating Interaction Features

Introducing explicit multiplication terms between variables allows estimating interactive effects beyond coefficients of individual inputs. For example, the combination of age and previous conditions likely impacts healthcare costs nonlinearly.

Conclusion and Recap

This guide explored essential linear regression techniques, nonlinear extensions, regularization methods, and best practices for applying predictive modeling. Key takeaways include:

• Simple linear regression effectively models numerous real-world problems but requires meeting assumptions about the data distribution and relationships
• Multiple regression increases flexibility and predictive accuracy by adding coefficients for additional explanatory variables
• Polynomials add nonlinearity but risk overfitting without regularization; splines balance flexibility and smoothness
• Regularization penalties constrain model complexity to focus on generalized patterns and prevent overfitting noise
• Feature engineering through thoughtful imputation, encodings, transformations and interaction modeling further improves predictive modeling

For hands-on practice building regression models in Python, check out the full Master Machine Learning video course. You’ll master regression analysis and predictive modeling through intuitive coding labs and real-world case studies. The comprehensive curriculum progresses from linear regression through regularized nonlinear neural network-based approaches. Start your machine learning journey today!

Dr. Alex Mitchell is a dedicated coding instructor with a deep passion for teaching and a wealth of experience in computer science education. As a university professor, Dr. Mitchell has played a pivotal role in shaping the coding skills of countless students, helping them navigate the intricate world of programming languages and software development.

Beyond the classroom, Dr. Mitchell is an active contributor to the freeCodeCamp community, where he regularly shares his expertise through tutorials, code examples, and practical insights. His teaching repertoire includes a wide range of languages and frameworks, such as Python, JavaScript, Next.js, and React, which he presents in an accessible and engaging manner.

Dr. Mitchell’s approach to teaching blends academic rigor with real-world applications, ensuring that his students not only understand the theory but also how to apply it effectively. His commitment to education and his ability to simplify complex topics have made him a respected figure in both the university and online learning communities.

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