statistics linear regression hypothesis testing

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

12.2.1: Hypothesis Test for Linear Regression

• Last updated
• Save as PDF
• Page ID 34850

• Rachel Webb
• Portland State University

To test to see if the slope is significant we will be doing a two-tailed test with hypotheses. The population least squares regression line would be \(y = \beta_{0} + \beta_{1} + \varepsilon\) where \(\beta_{0}\) (pronounced “beta-naught”) is the population \(y\)-intercept, \(\beta_{1}\) (pronounced “beta-one”) is the population slope and \(\varepsilon\) is called the error term.

If the slope were horizontal (equal to zero), the regression line would give the same \(y\)-value for every input of \(x\) and would be of no use. If there is a statistically significant linear relationship then the slope needs to be different from zero. We will only do the two-tailed test, but the same rules for hypothesis testing apply for a one-tailed test.

We will only be using the two-tailed test for a population slope.

The hypotheses are:

\(H_{0}: \beta_{1} = 0\) \(H_{1}: \beta_{1} \neq 0\)

The null hypothesis of a two-tailed test states that there is not a linear relationship between \(x\) and \(y\). The alternative hypothesis of a two-tailed test states that there is a significant linear relationship between \(x\) and \(y\).

Either a t-test or an F-test may be used to see if the slope is significantly different from zero. The population of the variable \(y\) must be normally distributed.

F-Test for Regression

An F-test can be used instead of a t-test. Both tests will yield the same results, so it is a matter of preference and what technology is available. Figure 12-12 is a template for a regression ANOVA table,

where \(n\) is the number of pairs in the sample and \(p\) is the number of predictor (independent) variables; for now this is just \(p = 1\). Use the F-distribution with degrees of freedom for regression = \(df_{R} = p\), and degrees of freedom for error = \(df_{E} = n - p - 1\). This F-test is always a right-tailed test since ANOVA is testing the variation in the regression model is larger than the variation in the error.

Use an F-test to see if there is a significant relationship between hours studied and grade on the exam. Use \(\alpha\) = 0.05.

T-Test for Regression

If the regression equation has a slope of zero, then every \(x\) value will give the same \(y\) value and the regression equation would be useless for prediction. We should perform a t-test to see if the slope is significantly different from zero before using the regression equation for prediction. The numeric value of t will be the same as the t-test for a correlation. The two test statistic formulas are algebraically equal; however, the formulas are different and we use a different parameter in the hypotheses.

The formula for the t-test statistic is \(t = \frac{b_{1}}{\sqrt{ \left(\frac{MSE}{SS_{xx}}\right) }}\)

Use the t-distribution with degrees of freedom equal to \(n - p - 1\).

The t-test for slope has the same hypotheses as the F-test:

Use a t-test to see if there is a significant relationship between hours studied and grade on the exam, use \(\alpha\) = 0.05.

• Prompt Library
• DS/AI Trends
• Stats Tools
• Interview Questions
• Generative AI
• Machine Learning
• Deep Learning

Linear regression hypothesis testing: Concepts, Examples

In relation to machine learning , linear regression is defined as a predictive modeling technique that allows us to build a model which can help predict continuous response variables as a function of a linear combination of explanatory or predictor variables. While training linear regression models, we need to rely on hypothesis testing in relation to determining the relationship between the response and predictor variables. In the case of the linear regression model, two types of hypothesis testing are done. They are T-tests and F-tests . In other words, there are two types of statistics that are used to assess whether linear regression models exist representing response and predictor variables. They are t-statistics and f-statistics. As data scientists , it is of utmost importance to determine if linear regression is the correct choice of model for our particular problem and this can be done by performing hypothesis testing related to linear regression response and predictor variables. Many times, it is found that these concepts are not very clear with a lot many data scientists. In this blog post, we will discuss linear regression and hypothesis testing related to t-statistics and f-statistics . We will also provide an example to help illustrate how these concepts work.

Table of Contents

What are linear regression models?

A linear regression model can be defined as the function approximation that represents a continuous response variable as a function of one or more predictor variables. While building a linear regression model, the goal is to identify a linear equation that best predicts or models the relationship between the response or dependent variable and one or more predictor or independent variables.

There are two different kinds of linear regression models. They are as follows:

• Simple or Univariate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and one predictor or independent variable. The form of the equation that represents a simple linear regression model is Y=mX+b, where m is the coefficients of the predictor variable and b is bias. When considering the linear regression line, m represents the slope and b represents the intercept.
• Multiple or Multi-variate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and more than one predictor or independent variable. The form of the equation that represents a multiple linear regression model is Y=b0+b1X1+ b2X2 + … + bnXn, where bi represents the coefficients of the ith predictor variable. In this type of linear regression model, each predictor variable has its own coefficient that is used to calculate the predicted value of the response variable.

While training linear regression models, the requirement is to determine the coefficients which can result in the best-fitted linear regression line. The learning algorithm used to find the most appropriate coefficients is known as least squares regression . In the least-squares regression method, the coefficients are calculated using the least-squares error function. The main objective of this method is to minimize or reduce the sum of squared residuals between actual and predicted response values. The sum of squared residuals is also called the residual sum of squares (RSS). The outcome of executing the least-squares regression method is coefficients that minimize the linear regression cost function .

The residual e of the ith observation is represented as the following where [latex]Y_i[/latex] is the ith observation and [latex]\hat{Y_i}[/latex] is the prediction for ith observation or the value of response variable for ith observation.

[latex]e_i = Y_i – \hat{Y_i}[/latex]

The residual sum of squares can be represented as the following:

[latex]RSS = e_1^2 + e_2^2 + e_3^2 + … + e_n^2[/latex]

The least-squares method represents the algorithm that minimizes the above term, RSS.

Once the coefficients are determined, can it be claimed that these coefficients are the most appropriate ones for linear regression? The answer is no. After all, the coefficients are only the estimates and thus, there will be standard errors associated with each of the coefficients.  Recall that the standard error is used to calculate the confidence interval in which the mean value of the population parameter would exist. In other words, it represents the error of estimating a population parameter based on the sample data. The value of the standard error is calculated as the standard deviation of the sample divided by the square root of the sample size. The formula below represents the standard error of a mean.

[latex]SE(\mu) = \frac{\sigma}{\sqrt(N)}[/latex]

Thus, without analyzing aspects such as the standard error associated with the coefficients, it cannot be claimed that the linear regression coefficients are the most suitable ones without performing hypothesis testing. This is where hypothesis testing is needed . Before we get into why we need hypothesis testing with the linear regression model, let’s briefly learn about what is hypothesis testing?

Train a Multiple Linear Regression Model using R

Before getting into understanding the hypothesis testing concepts in relation to the linear regression model, let’s train a multi-variate or multiple linear regression model and print the summary output of the model which will be referred to, in the next section. 

The data used for creating a multi-linear regression model is BostonHousing which can be loaded in RStudioby installing mlbench package. The code is shown below:

install.packages(“mlbench”) library(mlbench) data(“BostonHousing”)

Once the data is loaded, the code shown below can be used to create the linear regression model.

attach(BostonHousing) BostonHousing.lm <- lm(log(medv) ~ crim + chas + rad + lstat) summary(BostonHousing.lm)

Executing the above command will result in the creation of a linear regression model with the response variable as medv and predictor variables as crim, chas, rad, and lstat. The following represents the details related to the response and predictor variables:

• log(medv) : Log of the median value of owner-occupied homes in USD 1000’s
• crim : Per capita crime rate by town
• chas : Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
• rad : Index of accessibility to radial highways
• lstat : Percentage of the lower status of the population

The following will be the output of the summary command that prints the details relating to the model including hypothesis testing details for coefficients (t-statistics) and the model as a whole (f-statistics) 

Hypothesis tests & Linear Regression Models

Hypothesis tests are the statistical procedure that is used to test a claim or assumption about the underlying distribution of a population based on the sample data. Here are key steps of doing hypothesis tests with linear regression models:

• Hypothesis formulation for T-tests: In the case of linear regression, the claim is made that there exists a relationship between response and predictor variables, and the claim is represented using the non-zero value of coefficients of predictor variables in the linear equation or regression model. This is formulated as an alternate hypothesis. Thus, the null hypothesis is set that there is no relationship between response and the predictor variables . Hence, the coefficients related to each of the predictor variables is equal to zero (0). So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis for each test states that a1 = 0, a2 = 0, a3 = 0 etc. For all the predictor variables, individual hypothesis testing is done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. Thus, if there are, say, 5 features, there will be five hypothesis tests and each will have an associated null and alternate hypothesis.
• Hypothesis formulation for F-test : In addition, there is a hypothesis test done around the claim that there is a linear regression model representing the response variable and all the predictor variables. The null hypothesis is that the linear regression model does not exist . This essentially means that the value of all the coefficients is equal to zero. So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis states that a1 = a2 = a3 = 0.
• F-statistics for testing hypothesis for linear regression model : F-test is used to test the null hypothesis that a linear regression model does not exist, representing the relationship between the response variable y and the predictor variables x1, x2, x3, x4 and x5. The null hypothesis can also be represented as x1 = x2 = x3 = x4 = x5 = 0. F-statistics is calculated as a function of sum of squares residuals for restricted regression (representing linear regression model with only intercept or bias and all the values of coefficients as zero) and sum of squares residuals for unrestricted regression (representing linear regression model). In the above diagram, note the value of f-statistics as 15.66 against the degrees of freedom as 5 and 194. 
• Evaluate t-statistics against the critical value/region : After calculating the value of t-statistics for each coefficient, it is now time to make a decision about whether to accept or reject the null hypothesis. In order for this decision to be made, one needs to set a significance level, which is also known as the alpha level. The significance level of 0.05 is usually set for rejecting the null hypothesis or otherwise. If the value of t-statistics fall in the critical region, the null hypothesis is rejected. Or, if the p-value comes out to be less than 0.05, the null hypothesis is rejected.
• Evaluate f-statistics against the critical value/region : The value of F-statistics and the p-value is evaluated for testing the null hypothesis that the linear regression model representing response and predictor variables does not exist. If the value of f-statistics is more than the critical value at the level of significance as 0.05, the null hypothesis is rejected. This means that the linear model exists with at least one valid coefficients. 
• Draw conclusions : The final step of hypothesis testing is to draw a conclusion by interpreting the results in terms of the original claim or hypothesis. If the null hypothesis of one or more predictor variables is rejected, it represents the fact that the relationship between the response and the predictor variable is not statistically significant based on the evidence or the sample data we used for training the model. Similarly, if the f-statistics value lies in the critical region and the value of the p-value is less than the alpha value usually set as 0.05, one can say that there exists a linear regression model.

Why hypothesis tests for linear regression models?

The reasons why we need to do hypothesis tests in case of a linear regression model are following:

• By creating the model, we are establishing a new truth (claims) about the relationship between response or dependent variable with one or more predictor or independent variables. In order to justify the truth, there are needed one or more tests. These tests can be termed as an act of testing the claim (or new truth) or in other words, hypothesis tests.
• One kind of test is required to test the relationship between response and each of the predictor variables (hence, T-tests)
• Another kind of test is required to test the linear regression model representation as a whole. This is called F-test.

While training linear regression models, hypothesis testing is done to determine whether the relationship between the response and each of the predictor variables is statistically significant or otherwise. The coefficients related to each of the predictor variables is determined. Then, individual hypothesis tests are done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. If at least one of the null hypotheses is rejected, it represents the fact that there exists no relationship between response and that particular predictor variable. T-statistics is used for performing the hypothesis testing because the standard deviation of the sampling distribution is unknown. The value of t-statistics is compared with the critical value from the t-distribution table in order to make a decision about whether to accept or reject the null hypothesis regarding the relationship between the response and predictor variables. If the value falls in the critical region, then the null hypothesis is rejected which means that there is no relationship between response and that predictor variable. In addition to T-tests, F-test is performed to test the null hypothesis that the linear regression model does not exist and that the value of all the coefficients is zero (0). Learn more about the linear regression and t-test in this blog – Linear regression t-test: formula, example .

Recent Posts

• Linear Regression T-test: Formula, Example - May 7, 2024
• Feature Engineering in Machine Learning: Python Examples - May 3, 2024
• Feature Selection vs Feature Extraction: Machine Learning - May 2, 2024

Ajitesh Kumar

One response.

Very informative

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

• Search for:
• Excellence Awaits: IITs, NITs & IIITs Journey

ChatGPT Prompts (250+)

• Generate Design Ideas for App
• Expand Feature Set of App
• Create a User Journey Map for App
• Generate Visual Design Ideas for App
• Generate a List of Competitors for App
• Linear Regression T-test: Formula, Example
• Feature Engineering in Machine Learning: Python Examples
• Feature Selection vs Feature Extraction: Machine Learning
• Model Selection by Evaluating Bias & Variance: Example
• Bias-Variance Trade-off in Machine Learning: Examples

Data Science / AI Trends

• • Prepend any arxiv.org link with talk2 to load the paper into a responsive chat application
• • Custom LLM and AI Agents (RAG) On Structured + Unstructured Data - AI Brain For Your Organization
• • Guides, papers, lecture, notebooks and resources for prompt engineering
• • Common tricks to make LLMs efficient and stable
• • Machine learning in finance

Free Online Tools

• Create Scatter Plots Online for your Excel Data
• Histogram / Frequency Distribution Creation Tool
• Online Pie Chart Maker Tool
• Z-test vs T-test Decision Tool
• Independent samples t-test calculator

Recent Comments

I found it very helpful. However the differences are not too understandable for me

Very Nice Explaination. Thankyiu very much,

in your case E respresent Member or Oraganization which include on e or more peers?

Such a informative post. Keep it up

Thank you....for your support. you given a good solution for me.

Linear regression - Hypothesis testing

by Marco Taboga , PhD

This lecture discusses how to perform tests of hypotheses about the coefficients of a linear regression model estimated by ordinary least squares (OLS).

Table of contents

Normal vs non-normal model

The linear regression model, matrix notation, tests of hypothesis in the normal linear regression model, test of a restriction on a single coefficient (t test), test of a set of linear restrictions (f test), tests based on maximum likelihood procedures (wald, lagrange multiplier, likelihood ratio), tests of hypothesis when the ols estimator is asymptotically normal, test of a restriction on a single coefficient (z test), test of a set of linear restrictions (chi-square test), learn more about regression analysis.

The lecture is divided in two parts:

in the first part, we discuss hypothesis testing in the normal linear regression model , in which the OLS estimator of the coefficients has a normal distribution conditional on the matrix of regressors;

in the second part, we show how to carry out hypothesis tests in linear regression analyses where the hypothesis of normality holds only in large samples (i.e., the OLS estimator can be proved to be asymptotically normal).

We also denote:

We now explain how to derive tests about the coefficients of the normal linear regression model.

It can be proved (see the lecture about the normal linear regression model ) that the assumption of conditional normality implies that:

How the acceptance region is determined depends not only on the desired size of the test , but also on whether the test is:

one-tailed (only one of the two things, i.e., either smaller or larger, is possible).

For more details on how to determine the acceptance region, see the glossary entry on critical values .

The F test is one-tailed .

A critical value in the right tail of the F distribution is chosen so as to achieve the desired size of the test.

Then, the null hypothesis is rejected if the F statistics is larger than the critical value.

In this section we explain how to perform hypothesis tests about the coefficients of a linear regression model when the OLS estimator is asymptotically normal.

As we have shown in the lecture on the properties of the OLS estimator , in several cases (i.e., under different sets of assumptions) it can be proved that:

These two properties are used to derive the asymptotic distribution of the test statistics used in hypothesis testing.

The test can be either one-tailed or two-tailed . The same comments made for the t-test apply here.

Like the F test, also the Chi-square test is usually one-tailed .

The desired size of the test is achieved by appropriately choosing a critical value in the right tail of the Chi-square distribution.

The null is rejected if the Chi-square statistics is larger than the critical value.

Want to learn more about regression analysis? Here are some suggestions:

R squared of a linear regression ;

Gauss-Markov theorem ;

Generalized Least Squares ;

Multicollinearity ;

Dummy variables ;

Selection of linear regression models

Partitioned regression ;

Ridge regression .

How to cite

Please cite as:

Taboga, Marco (2021). "Linear regression - Hypothesis testing", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/linear-regression-hypothesis-testing.

Most of the learning materials found on this website are now available in a traditional textbook format.

• F distribution
• Beta distribution
• Conditional probability
• Central Limit Theorem
• Binomial distribution
• Mean square convergence
• Delta method
• Almost sure convergence
• Mathematical tools
• Fundamentals of probability
• Probability distributions
• Asymptotic theory
• Fundamentals of statistics
• About Statlect
• Cookies, privacy and terms of use
• Loss function
• Almost sure
• Type I error
• Precision matrix
• Integrable variable
• To enhance your privacy,
• we removed the social buttons,
• but don't forget to share .

Teach yourself statistics

Hypothesis Test for Regression Slope

This lesson describes how to conduct a hypothesis test to determine whether there is a significant linear relationship between an independent variable X and a dependent variable Y .

The test focuses on the slope of the regression line

Y = Β 0 + Β 1 X

where Β 0 is a constant, Β 1 is the slope (also called the regression coefficient), X is the value of the independent variable, and Y is the value of the dependent variable.

If we find that the slope of the regression line is significantly different from zero, we will conclude that there is a significant relationship between the independent and dependent variables.

Test Requirements

The approach described in this lesson is valid whenever the standard requirements for simple linear regression are met.

• The dependent variable Y has a linear relationship to the independent variable X .
• For each value of X, the probability distribution of Y has the same standard deviation σ.
• The Y values are independent.
• The Y values are roughly normally distributed (i.e., symmetric and unimodal ). A little skewness is ok if the sample size is large.

The test procedure consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

If there is a significant linear relationship between the independent variable X and the dependent variable Y , the slope will not equal zero.

H o : Β 1 = 0

H a : Β 1 ≠ 0

The null hypothesis states that the slope is equal to zero, and the alternative hypothesis states that the slope is not equal to zero.

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the following elements.

• Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use a linear regression t-test (described in the next section) to determine whether the slope of the regression line differs significantly from zero.

Analyze Sample Data

Using sample data, find the standard error of the slope, the slope of the regression line, the degrees of freedom, the test statistic, and the P-value associated with the test statistic. The approach described in this section is illustrated in the sample problem at the end of this lesson.

SE = s b 1 = sqrt [ Σ(y i - ŷ i ) 2 / (n - 2) ] / sqrt [ Σ(x i - x ) 2 ]

• Slope. Like the standard error, the slope of the regression line will be provided by most statistics software packages. In the hypothetical output above, the slope is equal to 35.

t = b 1 / SE

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

The local utility company surveys 101 randomly selected customers. For each survey participant, the company collects the following: annual electric bill (in dollars) and home size (in square feet). Output from a regression analysis appears below.

Is there a significant linear relationship between annual bill and home size? Use a 0.05 level of significance.

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

H o : The slope of the regression line is equal to zero.

H a : The slope of the regression line is not equal to zero.

• Formulate an analysis plan . For this analysis, the significance level is 0.05. Using sample data, we will conduct a linear regression t-test to determine whether the slope of the regression line differs significantly from zero.

We get the slope (b 1 ) and the standard error (SE) from the regression output.

b 1 = 0.55       SE = 0.24

We compute the degrees of freedom and the t statistic, using the following equations.

DF = n - 2 = 101 - 2 = 99

t = b 1 /SE = 0.55/0.24 = 2.29

where DF is the degrees of freedom, n is the number of observations in the sample, b 1 is the slope of the regression line, and SE is the standard error of the slope.

• Interpret results . Since the P-value (0.0242) is less than the significance level (0.05), we cannot accept the null hypothesis.

Linear Regression Analysis using SPSS Statistics

Introduction.

Linear regression is the next step up after correlation. It is used when we want to predict the value of a variable based on the value of another variable. The variable we want to predict is called the dependent variable (or sometimes, the outcome variable). The variable we are using to predict the other variable's value is called the independent variable (or sometimes, the predictor variable). For example, you could use linear regression to understand whether exam performance can be predicted based on revision time; whether cigarette consumption can be predicted based on smoking duration; and so forth. If you have two or more independent variables, rather than just one, you need to use multiple regression .

This "quick start" guide shows you how to carry out linear regression using SPSS Statistics, as well as interpret and report the results from this test. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for linear regression to give you a valid result. We discuss these assumptions next.

SPSS Statistics

Assumptions.

When you choose to analyse your data using linear regression, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using linear regression. You need to do this because it is only appropriate to use linear regression if your data "passes" seven assumptions that are required for linear regression to give you a valid result. In practice, checking for these seven assumptions just adds a little bit more time to your analysis, requiring you to click a few more buttons in SPSS Statistics when performing your analysis, as well as think a little bit more about your data, but it is not a difficult task.

Before we introduce you to these seven assumptions, do not be surprised if, when analysing your own data using SPSS Statistics, one or more of these assumptions is violated (i.e., not met). This is not uncommon when working with real-world data rather than textbook examples, which often only show you how to carry out linear regression when everything goes well! However, don’t worry. Even when your data fails certain assumptions, there is often a solution to overcome this. First, let’s take a look at these seven assumptions:

• Assumption #1: Your dependent variable should be measured at the continuous level (i.e., it is either an interval or ratio variable). Examples of continuous variables include revision time (measured in hours), intelligence (measured using IQ score), exam performance (measured from 0 to 100), weight (measured in kg), and so forth. You can learn more about interval and ratio variables in our article: Types of Variable .
• Assumption #2: Your independent variable should also be measured at the continuous level (i.e., it is either an interval or ratio variable). See the bullet above for examples of continuous variables.

• Assumption #5: You should have independence of observations , which you can easily check using the Durbin-Watson statistic, which is a simple test to run using SPSS Statistics. We explain how to interpret the result of the Durbin-Watson statistic in our enhanced linear regression guide.

• Assumption #7: Finally, you need to check that the residuals (errors) of the regression line are approximately normally distributed (we explain these terms in our enhanced linear regression guide). Two common methods to check this assumption include using either a histogram (with a superimposed normal curve) or a Normal P-P Plot. Again, in our enhanced linear regression guide, we: (a) show you how to check this assumption using SPSS Statistics, whether you use a histogram (with superimposed normal curve) or Normal P-P Plot; (b) explain how to interpret these diagrams; and (c) provide a possible solution if your data fails to meet this assumption.

You can check assumptions #3, #4, #5, #6 and #7 using SPSS Statistics. Assumptions #3 should be checked first, before moving onto assumptions #4, #5, #6 and #7. We suggest testing the assumptions in this order because assumptions #3, #4, #5, #6 and #7 require you to run the linear regression procedure in SPSS Statistics first, so it is easier to deal with these after checking assumption #1 and #2. Just remember that if you do not run the statistical tests on these assumptions correctly, the results you get when running a linear regression might not be valid. This is why we dedicate a number of sections of our enhanced linear regression guide to help you get this right. You can find out more about our enhanced content as a whole on our Features: Overview page, or more specifically, learn how we help with testing assumptions on our Features: Assumptions page.

In the section, Procedure , we illustrate the SPSS Statistics procedure to perform a linear regression assuming that no assumptions have been violated. First, we introduce the example that is used in this guide.

A salesperson for a large car brand wants to determine whether there is a relationship between an individual's income and the price they pay for a car. As such, the individual's "income" is the independent variable and the "price" they pay for a car is the dependent variable. The salesperson wants to use this information to determine which cars to offer potential customers in new areas where average income is known.

Setup in SPSS Statistics

In SPSS Statistics, we created two variables so that we could enter our data: Income (the independent variable), and Price (the dependent variable). It can also be useful to create a third variable, caseno , to act as a chronological case number. This third variable is used to make it easy for you to eliminate cases (e.g., significant outliers) that you have identified when checking for assumptions. However, we do not include it in the SPSS Statistics procedure that follows because we assume that you have already checked these assumptions. In our enhanced linear regression guide, we show you how to correctly enter data in SPSS Statistics to run a linear regression when you are also checking for assumptions. You can learn about our enhanced data setup content on our Features: Data Setup page. Alternately, see our generic, "quick start" guide: Entering Data in SPSS Statistics .

Test Procedure in SPSS Statistics

The five steps below show you how to analyse your data using linear regression in SPSS Statistics when none of the seven assumptions in the previous section, Assumptions , have been violated. At the end of these four steps, we show you how to interpret the results from your linear regression. If you are looking for help to make sure your data meets assumptions #3, #4, #5, #6 and #7, which are required when using linear regression and can be tested using SPSS Statistics, you can learn more about our enhanced guides on our Features: Overview page.

Note: The procedure that follows is identical for SPSS Statistics versions 18 to 28 , as well as the subscription version of SPSS Statistics, with version 28 and the subscription version being the latest versions of SPSS Statistics. However, in version 27 and the subscription version , SPSS Statistics introduced a new look to their interface called " SPSS Light ", replacing the previous look for versions 26 and earlier versions , which was called " SPSS Standard ". Therefore, if you have SPSS Statistics versions 27 or 28 (or the subscription version of SPSS Statistics), the images that follow will be light grey rather than blue. However, the procedure is identical .

Published with written permission from SPSS Statistics, IBM Corporation.

You will be presented with the Linear Regression dialogue box:

Access all 96 SPSS Statistics guides in Laerd Statistics

Output of Linear Regression Analysis

SPSS Statistics will generate quite a few tables of output for a linear regression. In this section, we show you only the three main tables required to understand your results from the linear regression procedure, assuming that no assumptions have been violated. A complete explanation of the output you have to interpret when checking your data for the six assumptions required to carry out linear regression is provided in our enhanced guide. This includes relevant scatterplots, histogram (with superimposed normal curve), Normal P-P Plot, casewise diagnostics and the Durbin-Watson statistic. Below, we focus on the results for the linear regression analysis only.

The first table of interest is the Model Summary table, as shown below:

This table provides the R and R 2 values. The R value represents the simple correlation and is 0.873 (the " R " Column), which indicates a high degree of correlation. The R 2 value (the " R Square " column) indicates how much of the total variation in the dependent variable, Price , can be explained by the independent variable, Income . In this case, 76.2% can be explained, which is very large.

The next table is the ANOVA table, which reports how well the regression equation fits the data (i.e., predicts the dependent variable) and is shown below:

This table indicates that the regression model predicts the dependent variable significantly well. How do we know this? Look at the " Regression " row and go to the " Sig. " column. This indicates the statistical significance of the regression model that was run. Here, p < 0.0005, which is less than 0.05, and indicates that, overall, the regression model statistically significantly predicts the outcome variable (i.e., it is a good fit for the data).

The Coefficients table provides us with the necessary information to predict price from income, as well as determine whether income contributes statistically significantly to the model (by looking at the " Sig. " column). Furthermore, we can use the values in the " B " column under the " Unstandardized Coefficients " column, as shown below:

to present the regression equation as:

Price = 8287 + 0.564(Income)

If you are unsure how to interpret regression equations or how to use them to make predictions, we discuss this in our enhanced linear regression guide. We also show you how to write up the results from your assumptions tests and linear regression output if you need to report this in a dissertation/thesis, assignment or research report. We do this using the Harvard and APA styles. You can learn more about our enhanced content on our Features: Overview page.

We also have a "quick start" guide on how to perform a linear regression analysis in Stata .

Save 10% on All AnalystPrep 2024 Study Packages with Coupon Code BLOG10 .

• Payment Plans
• Product List
• Partnerships

• Try Free Trial
• Study Packages
• Levels I, II & III Lifetime Package
• Video Lessons
• Study Notes
• Practice Questions
• Levels II & III Lifetime Package
• About the Exam
• About your Instructor
• Part I Study Packages
• Part I & Part II Lifetime Package
• Part II Study Packages
• Exams P & FM Lifetime Package
• Quantitative Questions
• Verbal Questions
• Data Insight Questions
• Live Tutoring
• About your Instructors
• EA Practice Questions
• Data Sufficiency Questions
• Integrated Reasoning Questions

Hypothesis Testing in Regression Analysis

Hypothesis testing is used to confirm if the estimated regression coefficients bear any statistical significance.  Either the confidence interval approach or the t-test approach can be used in hypothesis testing. In this section, we will explore the t-test approach.

The t-test Approach

The following are the steps followed in the performance of the t-test:

• Set the significance level for the test.
• Formulate the null and the alternative hypotheses.

$$t=\frac{\widehat{b_1}-b_1}{s_{\widehat{b_1}}}$$

\(b_1\) = True slope coefficient.

\(\widehat{b_1}\) = Point estimate for \(b_1\)

\(b_1 s_{\widehat{b_1\ }}\) = Standard error of the regression coefficient.

• Compare the absolute value of the t-statistic to the critical t-value (t_c). Reject the null hypothesis if the absolute value of the t-statistic is greater than the critical t-value i.e., \(t\ >\ +\ t_{critical}\ or\ t\ <\ –t_{\text{critical}}\).

Example: Hypothesis Testing of the Significance of Regression Coefficients

An analyst generates the following output from the regression analysis of inflation on unemployment:

$$\small{\begin{array}{llll}\hline{}& \textbf{Regression Statistics} &{}&{}\\ \hline{}& \text{Multiple R} & 0.8766 &{} \\ {}& \text{R Square} & 0.7684 &{} \\ {}& \text{Adjusted R Square} & 0.7394 & {}\\ {}& \text{Standard Error} & 0.0063 &{}\\ {}& \text{Observations} & 10 &{}\\ \hline {}& & & \\ \hline{} & \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t-Stat}\\ \hline \text{Intercept} & 0.0710 & 0.0094 & 7.5160 \\\text{Forecast (Slope)} & -0.9041 & 0.1755 & -5.1516\\ \hline\end{array}}$$

At the 5% significant level, test the null hypothesis that the slope coefficient is significantly different from one, that is,

$$ H_{0}: b_{1} = 1\ vs. \ H_{a}: b_{1}≠1 $$

The calculated t-statistic, \(\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}\) is equal to:

$$\begin{align*}\text{t}& = \frac{-0.9041-1}{0.1755}\\& = -10.85\end{align*}$$

The critical two-tail t-values from the table with \(n-2=8\) degrees of freedom are:

$$\text{t}_{c}=±2.306$$

Notice that \(|t|>t_{c}\) i.e., (\(10.85>2.306\))

Therefore, we reject the null hypothesis and conclude that the estimated slope coefficient is statistically different from one.

Note that we used the confidence interval approach and arrived at the same conclusion.

Question Neeth Shinu, CFA, is forecasting price elasticity of supply for a certain product. Shinu uses the quantity of the product supplied for the past 5months as the dependent variable and the price per unit of the product as the independent variable. The regression results are shown below. $$\small{\begin{array}{lccccc}\hline \textbf{Regression Statistics} & & & & & \\ \hline \text{Multiple R} & 0.9971 & {}& {}&{}\\ \text{R Square} & 0.9941 & & & \\ \text{Adjusted R Square} & 0.9922 & & & & \\ \text{Standard Error} & 3.6515 & & & \\ \text{Observations} & 5 & & & \\ \hline {}& \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t Stat} & \textbf{P-value}\\ \hline\text{Intercept} & -159 & 10.520 & (15.114) & 0.001\\ \text{Slope} & 0.26 & 0.012 & 22.517 & 0.000\\ \hline\end{array}}$$ Which of the following most likely reports the correct value of the t-statistic for the slope and most accurately evaluates its statistical significance with 95% confidence?     A. \(t=21.67\); slope is significantly different from zero.     B. \(t= 3.18\); slope is significantly different from zero.     C. \(t=22.57\); slope is not significantly different from zero. Solution The correct answer is A . The t-statistic is calculated using the formula: $$\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}$$ Where: \(b_{1}\) = True slope coefficient \(\widehat{b_{1}}\) = Point estimator for \(b_{1}\) \(\widehat{S_{b_{1}}}\) = Standard error of the regression coefficient $$\begin{align*}\text{t}&=\frac{0.26-0}{0.012}\\&=21.67\end{align*}$$ The critical two-tail t-values from the t-table with \(n-2 = 3\) degrees of freedom are: $$t_{c}=±3.18$$ Notice that \(|t|>t_{c}\) (i.e \(21.67>3.18\)). Therefore, the null hypothesis can be rejected. Further, we can conclude that the estimated slope coefficient is statistically different from zero.

Offered by AnalystPrep

Analysis of Variance (ANOVA)

Predicted value of a dependent variable, hypothesis testing.

A hypothesis is an assumptive statement about a problem, idea, or some other... Read More

Calculating Unconditional Probability ...

We can use the total probability rule to determine the unconditional probability of... Read More

Measures of Central Tendency

Measures of central tendency are values that tend to occur at the center... Read More

The Central Limit Theorem

The central limit theorem asserts that “given a population described by any probability... Read More