simple linear regression null hypothesis

Understanding the Null Hypothesis for Linear Regression

Linear regression is a technique we can use to understand the relationship between one or more predictor variables and a response variable .

If we only have one predictor variable and one response variable, we can use simple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x

• ŷ: The estimated response value.
• β 0 : The average value of y when x is zero.
• β 1 : The average change in y associated with a one unit increase in x.
• x: The value of the predictor variable.

Simple linear regression uses the following null and alternative hypotheses:

• H 0 : β 1 = 0
• H A : β 1 ≠ 0

The null hypothesis states that the coefficient β 1 is equal to zero. In other words, there is no statistically significant relationship between the predictor variable, x, and the response variable, y.

The alternative hypothesis states that β 1 is not equal to zero. In other words, there is a statistically significant relationship between x and y.

If we have multiple predictor variables and one response variable, we can use multiple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x 1 + β 2 x 2 + … + β k x k

• β 0 : The average value of y when all predictor variables are equal to zero.
• β i : The average change in y associated with a one unit increase in x i .
• x i : The value of the predictor variable x i .

Multiple linear regression uses the following null and alternative hypotheses:

• H 0 : β 1 = β 2 = … = β k = 0
• H A : β 1 = β 2 = … = β k ≠ 0

The null hypothesis states that all coefficients in the model are equal to zero. In other words, none of the predictor variables have a statistically significant relationship with the response variable, y.

The alternative hypothesis states that not every coefficient is simultaneously equal to zero.

The following examples show how to decide to reject or fail to reject the null hypothesis in both simple linear regression and multiple linear regression models.

Example 1: Simple Linear Regression

Suppose a professor would like to use the number of hours studied to predict the exam score that students will receive in his class. He collects data for 20 students and fits a simple linear regression model.

The following screenshot shows the output of the regression model:

The fitted simple linear regression model is:

Exam Score = 67.1617 + 5.2503*(hours studied)

To determine if there is a statistically significant relationship between hours studied and exam score, we need to analyze the overall F value of the model and the corresponding p-value:

• Overall F-Value:  47.9952
• P-value:  0.000

Since this p-value is less than .05, we can reject the null hypothesis. In other words, there is a statistically significant relationship between hours studied and exam score received.

Example 2: Multiple Linear Regression

Suppose a professor would like to use the number of hours studied and the number of prep exams taken to predict the exam score that students will receive in his class. He collects data for 20 students and fits a multiple linear regression model.

The fitted multiple linear regression model is:

Exam Score = 67.67 + 5.56*(hours studied) – 0.60*(prep exams taken)

To determine if there is a jointly statistically significant relationship between the two predictor variables and the response variable, we need to analyze the overall F value of the model and the corresponding p-value:

• Overall F-Value:  23.46
• P-value:  0.00

Since this p-value is less than .05, we can reject the null hypothesis. In other words, hours studied and prep exams taken have a jointly statistically significant relationship with exam score.

Note: Although the p-value for prep exams taken (p = 0.52) is not significant, prep exams combined with hours studied has a significant relationship with exam score.

Additional Resources

Understanding the F-Test of Overall Significance in Regression How to Read and Interpret a Regression Table How to Report Regression Results How to Perform Simple Linear Regression in Excel How to Perform Multiple Linear Regression in Excel

The Complete Guide: How to Report Regression Results

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Simple Linear Regression Assumptions

Simple linear regression assumptions #.

RStudio: RMarkdown , Quarto

Goodness of fit of regression: analysis of variance.

\(F\) -statistics.

Diagnostic plots.

Figure depicts the statistical model for regression: #

First we start with \(X\) , then compute the mean \(\beta_0 + \beta_1 X\) then add error \(\epsilon\) yielding

Geometry of least squares #

Full model #.

This is the model

Its fitted values are

Reduced model #

Regression sum of squares #.

The closer \(\hat{Y}\) is to the \({1}\) axis, the less “variation” there is along the \(X\) axis.

This closeness can be measured by the length of the vector \(\hat{Y}-\bar{Y} \cdot 1\) .

Its length is

An important right triangle #

Sides of the triangle: SSR, SSE

Hypotenuse: SST

Degrees of freedom in the right triangle #

Sides of the triangle: SSR has 1 d.f., SSE has n-2 d.f.

Hypotenuse: SST has n-1 d.f.

Mean squares #

Each sum of squares gets an extra bit of information associated to them, called their degrees of freedom .

Roughly speaking, the degrees of freedom can be determined by dimension counting.

Computing degrees of freedom #

The \(SSE\) has \(n-2\) degrees of freedom because it is the squared length of a vector that lies in \(n-2\) dimensions. To see this, note that it is perpendicular to the 2-dimensional plane formed by the \(X\) axis and the \(1\) axis.

The \(SST\) has \(n-1\) degrees of freedom because it is the squared length of a vector that lies in \(n-1\) dimensions. In this case, this vector is perpendicular to the \(1\) axis.

The \(SSR\) has 1 degree of freedom because it is the squared length of a vector that lies in the 2-dimensional plane but is perpendicular to the \(1\) axis.

A different visualization #

These sums of squares can be visualized by other means as well. We will illustrate with a synthetic dataset.

SST: total sum of squares #

Description #

This figure depicts the **total sum of squares, \(SST\) ** – the sum of the squared differences between the Y values and the sample mean of the Y values.

SSE: error sum of squares #

This figure depicts the **error sum of squares, \(SSE\) ** - the sum of the squared differences between the \(Y\) values and the \(\hat{Y}\) values, i.e. the fitted values of the regression model.

SSR: regression sum of squares #

This figure depicts the **regression sum of squares, \(SSR\) ** - the sum of the squared differences between the \(\hat{Y}\) values and the sample mean of the \(Y\) values.

Definition of \(R^2\) #

As noted above, if the regression model fits very well, then \(SSR\) will be large relative to \(SST\) . The \(R^2\) score is just the ratio of these sums of squares.

Let’s verify this on the big_bang data.

Let’s verify our claim \(SST=SSE+SSR\) :

\(R^2\) and correlation #

Finally, \(R=\sqrt{R^2}\) is called the (absolute) correlation coefficient because it is equal to the absolute value of sample correlation coefficient of \(X\) and \(Y\) .

\(F\) -statistics #

After a \(t\) -statistic, the next most commonly encountered statistic is a \(\chi^2\) statistic, or its closely related cousin, the \(F\) statistic.

A \(\chi^2_k\) random variable is the distribution of the squared length of a centered normal vector in \(k\) dimensions ( proper definition needs slightly more detail ).

Sums of squares are squared lengths !

\(F\) statistic for simple linear regression #

Can be thought of as a ratio of a difference in sums of squares normalized by our “best estimate” of variance .

\(F\) statistics and \(R^2\) #

The \(R^2\) is also closely related to the \(F\) statistic reported as the goodness of fit in summary of lm .

Simple manipulations yield

\(F\) test under \(H_0\) #

Under \(H_0:\beta_1=0\) ,

and from our “right triangle”, these vectors are orthogonal.

The null hypothesis \(H_0:\beta_1=0\) implies that \(SSR \sim \chi^2_1 \cdot \sigma^2\) .

\(F\) -statistics and mean squares #

An \(F\) -statistic is a ratio of mean squares : it has a numerator, \(N\) , and a denominator, \(D\) that are independent.

We say \(F\) has an \(F\) distribution with parameters \(df_{{\rm num}}, df_{{\rm den}}\) and write \(F \sim F_{df_{{\rm num}}, df_{{\rm den}}}\)

\(F\) statistics are computed to test some \(H_0\) .

When that \(H_0\) is true, the \(F\) statistic has this \(F\) distribution (with appropriate degrees of freedom).

Relation between \(F\) and \(t\) statistics. #

If \(T \sim t_{\nu}\) , then

In other words, the square of a \(t\) -statistic is an \(F\) -statistic. Because it is always positive, an \(F\) -statistic has no direction associated with it.

Verifying \(F\) -statistic calculation #

Let’s check this in our example.

The \(t\) statistic for education is the \(t\) -statistic for the parameter \(\beta_1\) under \(H_0:\beta_1=0\) . Its value is 6.024 above. If we square it, we should get about the same as the F-statistic .

Interpretation of an \(F\) -statistic #

In regression, the numerator is usually a difference in goodness of fit of two (nested) models.

The denominator is \(\hat{\sigma}^2\) – an estimate of \(\sigma^2\) .

In our example today: the bigger model is the simple linear regression model, the smaller is the model with constant mean (one sample model).

If the \(F\) is large, it says that the bigger model explains a lot more variability in \(Y\) (relative to \(\sigma^2\) ) than the smaller one.

Analysis of variance #

The \(F\) -statistic has the form

Right triangle with full and reduced model: sum of squares #

Sides of the triangle: \(SSE_R-SSE_F\) , \(SSE_F\)

Hypotenuse: \(SSE_R\)

Right triangle with full and reduced model: degrees of freedom #

Sides of the triangle: \(df_R-df_F\) , \(df_F\)

Hypotenuse: \(df_R\)

The \(F\) -statistic for simple linear regression revisited #

The null hypothesis is

The usual \(\alpha\) rejection rule would be to reject \(H_0\) if the \(F_{\text{obs}}\) the observed \(F\) statistic is greater than \(F_{1,n-2,1-\alpha}\) .

In our case, the observed \(F\) was 36.3, \(n-2=22\) and the appropriate 5% threshold is computed below to be 4.30. Therefore, we strongly reject \(H_0\) .

Case study B: breakdown time for insulating fluid #

A designed experiment to estimate average breakdown time under different voltages.

Another model #

There were only 7 distinct values of Voltage : can be treated as a category (i.e. factor )

A different reduced & full model #

Our “right triangle” again (only degrees of freedom this time): #

Sides of the triangle: \(df_R-df_F=5\) , \(df_F=69\)

Hypotenuse: \(df_R=74\)

Diagnostics for simple linear regression #

What can go wrong #.

Using a linear regression function can be wrong: maybe regression function should be quadratic.

We assumed independent Gaussian errors with the same variance. This may be incorrect.

The errors may not be normally distributed.

The errors may not be independent.

The errors may not have the same variance.

Detecting problems is more art then science , i.e. we cannot test for all possible problems in a regression model.

Inspecting residuals #

The basic idea of most diagnostic measures is the following:

If the model is correct then residuals \(e_i = Y_i -\widehat{Y}_i, 1 \leq i \leq n\) should look like a sample of (not quite independent) \(N(0, \sigma^2)\) random variables.

A poorly fitting model #

Figure: \(Y\) vs. \(X\) and fitted regression line #

Residuals for anscombe’s data #.

Figure: residuals vs. \(X\) #

Quadratic model #.

Let’s add a quadratic term to our model (a multiple linear regression model ).

Figure: \(Y\) and fitted quadratic model vs. \(X\) #

Inspecting residuals of quadratic model #.

The residuals of the quadratic model have no apparent pattern in them, suggesting this is a better fit than the simple linear regression model.

Figure: residuals of quadratic model vs. \(X\) #

Assessing normality of errors: qq-plot for linear model #.

Figure: quantiles of poorly fitting model’s residuals vs. expected Gaussian quantiles #

Qq-plot for quadratic model #.

Figure: quantiles of quadratic model’s residuals vs. expected Gaussian quantiles #

qqnorm does not seem vastly different \(\implies\) several diagnostic tools can be useful in assessing a model.

Default plots in R #

Assessing constant variance assumption #

Removing an outlier #

Heteroscedastic errors #

When we plot the residuals against the fitted values for this model (even with the outlier removed) we see that the variance clearly depends on \(GSS\) . They also do not seem symmetric around 0 so perhaps the Gaussian model is not appropriate.

Plots from lm #

We can see some of these plots in R :

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.

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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

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

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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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 .

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6.4 - the hypothesis tests for the slopes.

At the beginning of this lesson, we translated three different research questions pertaining to heart attacks in rabbits ( Cool Hearts dataset ) into three sets of hypotheses we can test using the general linear F -statistic. The research questions and their corresponding hypotheses are:

Hypotheses 1

Is the regression model containing at least one predictor useful in predicting the size of the infarct?

• \(H_{0} \colon \beta_{1} = \beta_{2} = \beta_{3} = 0\)
• \(H_{A} \colon\) At least one \(\beta_{j} ≠ 0\) (for j = 1, 2, 3)

Hypotheses 2

Is the size of the infarct significantly (linearly) related to the area of the region at risk?

• \(H_{0} \colon \beta_{1} = 0 \)
• \(H_{A} \colon \beta_{1} \ne 0 \)

Hypotheses 3

(Primary research question) Is the size of the infarct area significantly (linearly) related to the type of treatment upon controlling for the size of the region at risk for infarction?

• \(H_{0} \colon \beta_{2} = \beta_{3} = 0\)
• \(H_{A} \colon \) At least one \(\beta_{j} ≠ 0\) (for j = 2, 3)

Let's test each of the hypotheses now using the general linear F -statistic:

\(F^*=\left(\dfrac{SSE(R)-SSE(F)}{df_R-df_F}\right) \div \left(\dfrac{SSE(F)}{df_F}\right)\)

To calculate the F -statistic for each test, we first determine the error sum of squares for the reduced and full models — SSE ( R ) and SSE ( F ), respectively. The number of error degrees of freedom associated with the reduced and full models — \(df_{R}\) and \(df_{F}\), respectively — is the number of observations, n , minus the number of parameters, p , in the model. That is, in general, the number of error degrees of freedom is n - p . We use statistical software, such as Minitab's F -distribution probability calculator, to determine the P -value for each test.

Testing all slope parameters equal 0 Section  

To answer the research question: "Is the regression model containing at least one predictor useful in predicting the size of the infarct?" To do so, we test the hypotheses:

• \(H_{0} \colon \beta_{1} = \beta_{2} = \beta_{3} = 0 \)
• \(H_{A} \colon\) At least one \(\beta_{j} \ne 0 \) (for j = 1, 2, 3)

The full model

The full model is the largest possible model — that is, the model containing all of the possible predictors. In this case, the full model is:

\(y_i=(\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i3})+\epsilon_i\)

The error sum of squares for the full model, SSE ( F ), is just the usual error sum of squares, SSE , that appears in the analysis of variance table. Because there are 4 parameters in the full model, the number of error degrees of freedom associated with the full model is \(df_{F} = n - 4\).

The reduced model

The reduced model is the model that the null hypothesis describes. Because the null hypothesis sets each of the slope parameters in the full model equal to 0, the reduced model is:

\(y_i=\beta_0+\epsilon_i\)

The reduced model suggests that none of the variations in the response y is explained by any of the predictors. Therefore, the error sum of squares for the reduced model, SSE ( R ), is just the total sum of squares, SSTO , that appears in the analysis of variance table. Because there is only one parameter in the reduced model, the number of error degrees of freedom associated with the reduced model is \(df_{R} = n - 1 \).

Upon plugging in the above quantities, the general linear F -statistic:

\(F^*=\dfrac{SSE(R)-SSE(F)}{df_R-df_F} \div \dfrac{SSE(F)}{df_F}\)

becomes the usual " overall F -test ":

\(F^*=\dfrac{SSR}{3} \div \dfrac{SSE}{n-4}=\dfrac{MSR}{MSE}\)

That is, to test \(H_{0}\) : \(\beta_{1} = \beta_{2} = \beta_{3} = 0 \), we just use the overall F -test and P -value reported in the analysis of variance table:

Analysis of Variance

Regression Equation

Inf = - 0.135 + 0.613 Area - 0.2435 X2 - 0.0657 X3

There is sufficient evidence ( F = 16.43, P < 0.001) to conclude that at least one of the slope parameters is not equal to 0.

In general, to test that all of the slope parameters in a multiple linear regression model are 0, we use the overall F -test reported in the analysis of variance table.

Testing one slope parameter is 0 Section  

Now let's answer the second research question: "Is the size of the infarct significantly (linearly) related to the area of the region at risk?" To do so, we test the hypotheses:

Again, the full model is the model containing all of the possible predictors:

The error sum of squares for the full model, SSE ( F ), is just the usual error sum of squares, SSE . Alternatively, because the three predictors in the model are \(x_{1}\), \(x_{2}\), and \(x_{3}\), we can denote the error sum of squares as SSE (\(x_{1}\), \(x_{2}\), \(x_{3}\)). Again, because there are 4 parameters in the model, the number of error degrees of freedom associated with the full model is \(df_{F} = n - 4 \).

Because the null hypothesis sets the first slope parameter, \(\beta_{1}\), equal to 0, the reduced model is:

\(y_i=(\beta_0+\beta_2x_{i2}+\beta_3x_{i3})+\epsilon_i\)

Because the two predictors in the model are \(x_{2}\) and \(x_{3}\), we denote the error sum of squares as SSE (\(x_{2}\), \(x_{3}\)). Because there are 3 parameters in the model, the number of error degrees of freedom associated with the reduced model is \(df_{R} = n - 3\).

The general linear statistic:

simplifies to:

\(F^*=\dfrac{SSR(x_1|x_2, x_3)}{1}\div \dfrac{SSE(x_1,x_2, x_3)}{n-4}=\dfrac{MSR(x_1|x_2, x_3)}{MSE(x_1,x_2, x_3)}\)

Getting the numbers from the Minitab output:

we determine that the value of the F -statistic is:

\(F^* = \dfrac{SSR(x_1 \vert x_2, x_3)}{1} \div \dfrac{SSE(x_1, x_2, x_3)}{28} = \dfrac{0.63742}{0.01946}=32.7554\)

The P -value is the probability — if the null hypothesis were true — that we would get an F -statistic larger than 32.7554. Comparing our F -statistic to an F -distribution with 1 numerator degree of freedom and 28 denominator degrees of freedom, Minitab tells us that the probability is close to 1 that we would observe an F -statistic smaller than 32.7554:

F distribution with 1 DF in Numerator and 28 DF in denominator

Therefore, the probability that we would get an F -statistic larger than 32.7554 is close to 0. That is, the P -value is < 0.001. There is sufficient evidence ( F = 32.8, P < 0.001) to conclude that the size of the infarct is significantly related to the size of the area at risk after the other predictors x2 and x3 have been taken into account.

But wait a second! Have you been wondering why we couldn't just use the slope's t -statistic to test that the slope parameter, \(\beta_{1}\), is 0? We can! Notice that the P -value ( P < 0.001) for the t -test ( t * = 5.72):

Coefficients

is the same as the P -value we obtained for the F -test. This will always be the case when we test that only one slope parameter is 0. That's because of the well-known relationship between a t -statistic and an F -statistic that has one numerator degree of freedom:

\(t_{(n-p)}^{2}=F_{(1, n-p)}\)

For our example, the square of the t -statistic, 5.72, equals our F -statistic (within rounding error). That is:

\(t^{*2}=5.72^2=32.72=F^*\)

So what have we learned in all of this discussion about the equivalence of the F -test and the t -test? In short:

Compare the output obtained when \(x_{1}\) = Area is entered into the model last :

Inf = - 0.135 - 0.2435 X2 - 0.0657 X3 + 0.613 Area

to the output obtained when \(x_{1}\) = Area is entered into the model first :

The t -statistic and P -value are the same regardless of the order in which \(x_{1}\) = Area is entered into the model. That's because — by its equivalence to the F -test — the t -test for one slope parameter adjusts for all of the other predictors included in the model.

• We can use either the F -test or the t -test to test that only one slope parameter is 0. Because the t -test results can be read right off of the Minitab output, it makes sense that it would be the test that we'll use most often.
• But, we have to be careful with our interpretations! The equivalence of the t -test to the F -test has taught us something new about the t -test. The t -test is a test for the marginal significance of the \(x_{1}\) predictor after the other predictors \(x_{2}\) and \(x_{3}\) have been taken into account. It does not test for the significance of the relationship between the response y and the predictor \(x_{1}\) alone.

Testing a subset of slope parameters is 0 Section  

Finally, let's answer the third — and primary — research question: "Is the size of the infarct area significantly (linearly) related to the type of treatment upon controlling for the size of the region at risk for infarction?" To do so, we test the hypotheses:

• \(H_{0} \colon \beta_{2} = \beta_{3} = 0 \)
• \(H_{A} \colon\) At least one \(\beta_{j} \ne 0 \) (for j = 2, 3)

Because the null hypothesis sets the second and third slope parameters, \(\beta_{2}\) and \(\beta_{3}\), equal to 0, the reduced model is:

\(y_i=(\beta_0+\beta_1x_{i1})+\epsilon_i\)

The ANOVA table for the reduced model is:

Because the only predictor in the model is \(x_{1}\), we denote the error sum of squares as SSE (\(x_{1}\)) = 0.8793. Because there are 2 parameters in the model, the number of error degrees of freedom associated with the reduced model is \(df_{R} = n - 2 = 32 – 2 = 30\).

\begin{align} F^*&=\dfrac{SSE(R)-SSE(F)}{df_R-df_F} \div\dfrac{SSE(F)}{df_F}\\&=\dfrac{0.8793-0.54491}{30-28} \div\dfrac{0.54491}{28}\\&= \dfrac{0.33439}{2} \div 0.01946\\&=8.59.\end{align}

Alternatively, we can calculate the F-statistic using a partial F-test :

\begin{align}F^*&=\dfrac{SSR(x_2, x_3|x_1)}{2}\div \dfrac{SSE(x_1,x_2, x_3)}{n-4}\\&=\dfrac{MSR(x_2, x_3|x_1)}{MSE(x_1,x_2, x_3)}.\end{align}

To conduct the test, we regress y = InfSize on \(x_{1}\) = Area and \(x_{2}\) and \(x_{3 }\)— in order (and with "Sequential sums of squares" selected under "Options"):

Inf = - 0.135 + 0.613 Area - 0.2435 X2 - 0.0657 X3

yielding SSR (\(x_{2}\) | \(x_{1}\)) = 0.31453, SSR (\(x_{3}\) | \(x_{1}\), \(x_{2}\)) = 0.01981, and MSE = 0.54491/28 = 0.01946. Therefore, the value of the partial F -statistic is:

\begin{align} F^*&=\dfrac{SSR(x_2, x_3|x_1)}{2}\div \dfrac{SSE(x_1,x_2, x_3)}{n-4}\\&=\dfrac{0.31453+0.01981}{2}\div\dfrac{0.54491}{28}\\&= \dfrac{0.33434}{2} \div 0.01946\\&=8.59,\end{align}

which is identical (within round-off error) to the general F-statistic above. The P -value is the probability — if the null hypothesis were true — that we would observe a partial F -statistic more extreme than 8.59. The following Minitab output:

F distribution with 2 DF in Numerator and 28 DF in denominator

tells us that the probability of observing such an F -statistic that is smaller than 8.59 is 0.9988. Therefore, the probability of observing such an F -statistic that is larger than 8.59 is 1 - 0.9988 = 0.0012. The P -value is very small. There is sufficient evidence ( F = 8.59, P = 0.0012) to conclude that the type of cooling is significantly related to the extent of damage that occurs — after taking into account the size of the region at risk.

Summary of MLR Testing Section  

For the simple linear regression model, there is only one slope parameter about which one can perform hypothesis tests. For the multiple linear regression model, there are three different hypothesis tests for slopes that one could conduct. They are:

• Hypothesis test for testing that all of the slope parameters are 0.
• Hypothesis test for testing that a subset — more than one, but not all — of the slope parameters are 0.
• Hypothesis test for testing that one slope parameter is 0.

We have learned how to perform each of the above three hypothesis tests. Along the way, we also took two detours — one to learn about the " general linear F-test " and one to learn about " sequential sums of squares. " As you now know, knowledge about both is necessary for performing the three hypothesis tests.

The F -statistic and associated p -value in the ANOVA table is used for testing whether all of the slope parameters are 0. In most applications, this p -value will be small enough to reject the null hypothesis and conclude that at least one predictor is useful in the model. For example, for the rabbit heart attacks study, the F -statistic is (0.95927/(4–1)) / (0.54491/(32–4)) = 16.43 with p -value 0.000.

To test whether a subset — more than one, but not all — of the slope parameters are 0, there are two equivalent ways to calculate the F-statistic:

• Use the general linear F-test formula by fitting the full model to find SSE(F) and fitting the reduced model to find SSE(R) . Then the numerator of the F-statistic is (SSE(R) – SSE(F)) / ( \(df_{R}\) – \(df_{F}\)) .
• Alternatively, use the partial F-test formula by fitting only the full model but making sure the relevant predictors are fitted last and "sequential sums of squares" have been selected. Then the numerator of the F-statistic is the sum of the relevant sequential sums of squares divided by the sum of the degrees of freedom for these sequential sums of squares. The denominator of the F -statistic is the mean squared error in the ANOVA table.

For example, for the rabbit heart attacks study, the general linear F-statistic is ((0.8793 – 0.54491) / (30 – 28)) / (0.54491 / 28) = 8.59 with p -value 0.0012. Alternatively, the partial F -statistic for testing the slope parameters for predictors \(x_{2}\) and \(x_{3}\) using sequential sums of squares is ((0.31453 + 0.01981) / 2) / (0.54491 / 28) = 8.59.

To test whether one slope parameter is 0, we can use an F -test as just described. Alternatively, we can use a t -test, which will have an identical p -value since in this case, the square of the t -statistic is equal to the F -statistic. For example, for the rabbit heart attacks study, the F -statistic for testing the slope parameter for the Area predictor is (0.63742/1) / (0.54491/(32–4)) = 32.75 with p -value 0.000. Alternatively, the t -statistic for testing the slope parameter for the Area predictor is 0.613 / 0.107 = 5.72 with p -value 0.000, and \(5.72^{2} = 32.72\).

Incidentally, you may be wondering why we can't just do a series of individual t-tests to test whether a subset of the slope parameters is 0. For example, for the rabbit heart attacks study, we could have done the following:

• Fit the model of y = InfSize on \(x_{1}\) = Area and \(x_{2}\) and \(x_{3}\) and use an individual t-test for \(x_{3}\).
• If the test results indicate that we can drop \(x_{3}\) then fit the model of y = InfSize on \(x_{1}\) = Area and \(x_{2}\) and use an individual t-test for \(x_{2}\).

The problem with this approach is we're using two individual t-tests instead of one F-test, which means our chance of drawing an incorrect conclusion in our testing procedure is higher. Every time we do a hypothesis test, we can draw an incorrect conclusion by:

• rejecting a true null hypothesis, i.e., make a type I error by concluding the tested predictor(s) should be retained in the model when in truth it/they should be dropped; or
• failing to reject a false null hypothesis, i.e., make a type II error by concluding the tested predictor(s) should be dropped from the model when in truth it/they should be retained.

Thus, in general, the fewer tests we perform the better. In this case, this means that wherever possible using one F-test in place of multiple individual t-tests is preferable.

Hypothesis tests for the slope parameters Section  

The problems in this section are designed to review the hypothesis tests for the slope parameters, as well as to give you some practice on models with a three-group qualitative variable (which we'll cover in more detail in Lesson 8). We consider tests for:

• whether one slope parameter is 0 (for example, \(H_{0} \colon \beta_{1} = 0 \))
• whether a subset (more than one but less than all) of the slope parameters are 0 (for example, \(H_{0} \colon \beta_{2} = \beta_{3} = 0 \) against the alternative \(H_{A} \colon \beta_{2} \ne 0 \) or \(\beta_{3} \ne 0 \) or both ≠ 0)
• whether all of the slope parameters are 0 (for example, \(H_{0} \colon \beta_{1} = \beta_{2} = \beta_{3}\) = 0 against the alternative \(H_{A} \colon \) at least one of the \(\beta_{i}\) is not 0)

(Note the correct specification of the alternative hypotheses for the last two situations.)

Sugar beets study

A group of researchers was interested in studying the effects of three different growth regulators ( treat , denoted 1, 2, and 3) on the yield of sugar beets (y = yield , in pounds). They planned to plant the beets in 30 different plots and then randomly treat 10 plots with the first growth regulator, 10 plots with the second growth regulator, and 10 plots with the third growth regulator. One problem, though, is that the amount of available nitrogen in the 30 different plots varies naturally, thereby giving a potentially unfair advantage to plots with higher levels of available nitrogen. Therefore, the researchers also measured and recorded the available nitrogen (\(x_{1}\) = nit , in pounds/acre) in each plot. They are interested in comparing the mean yields of sugar beets subjected to the different growth regulators after taking into account the available nitrogen. The Sugar Beets dataset contains the data from the researcher's experiment.

Preliminary Work

The plot shows a similar positive linear trend within each treatment category, which suggests that it is reasonable to formulate a multiple regression model that would place three parallel lines through the data.

Because the qualitative variable treat distinguishes between the three treatment groups (1, 2, and 3), we need to create two indicator variables, \(x_{2}\) and \(x_{3}\), say, to fit a linear regression model to these data. The new indicator variables should be defined as follows:

Use Minitab's Calc >> Make Indicator Variables command to create the new indicator variables in your worksheet

Minitab creates an indicator variable for each treatment group but we can only use two, for treatment groups 1 and 2 in this case (treatment group 3 is the reference level in this case).

Then, if we assume the trend in the data can be summarized by this regression model:

\(y_{i} = \beta_{0}\) + \(\beta_{1}\)\(x_{1}\) + \(\beta_{2}\)\(x_{2}\) + \(\beta_{3}\)\(x_{3}\) + \(\epsilon_{i}\)

where \(x_{1}\) = nit and \(x_{2}\) and \(x_{3}\) are defined as above, what is the mean response function for plots receiving treatment 3? for plots receiving treatment 1? for plots receiving treatment 2? Are the three regression lines that arise from our formulated model parallel? What does the parameter \(\beta_{2}\) quantify? And, what does the parameter \(\beta_{3}\) quantify?

The fitted equation from Minitab is Yield = 84.99 + 1.3088 Nit - 2.43 \(x_{2}\) - 2.35 \(x_{3}\), which means that the equations for each treatment group are:

• Group 1: Yield = 84.99 + 1.3088 Nit - 2.43(1) = 82.56 + 1.3088 Nit
• Group 2: Yield = 84.99 + 1.3088 Nit - 2.35(1) = 82.64 + 1.3088 Nit
• Group 3: Yield = 84.99 + 1.3088 Nit

The three estimated regression lines are parallel since they have the same slope, 1.3088.

The regression parameter for \(x_{2}\) represents the difference between the estimated intercept for treatment 1 and the estimated intercept for reference treatment 3.

The regression parameter for \(x_{3}\) represents the difference between the estimated intercept for treatment 2 and the estimated intercept for reference treatment 3.

Testing whether all of the slope parameters are 0

\(H_0 \colon \beta_1 = \beta_2 = \beta_3 = 0\) against the alternative \(H_A \colon \) at least one of the \(\beta_i\) is not 0.

\(F=\dfrac{SSR(X_1,X_2,X_3)\div3}{SSE(X_1,X_2,X_3)\div(n-4)}=\dfrac{MSR(X_1,X_2,X_3)}{MSE(X_1,X_2,X_3)}\)

\(F = \dfrac{\frac{16039.5}{3}}{\frac{1078.0}{30-4}} = \dfrac{5346.5}{41.46} = 128.95\)

Since the p -value for this F -statistic is reported as 0.000, we reject \(H_{0}\) in favor of \(H_{A}\) and conclude that at least one of the slope parameters is not zero, i.e., the regression model containing at least one predictor is useful in predicting the size of sugar beet yield.

Tests for whether one slope parameter is 0

\(H_0 \colon \beta_1= 0\) against the alternative \(H_A \colon \beta_1 \ne 0\)

t -statistic = 19.60, p -value = 0.000, so we reject \(H_{0}\) in favor of \(H_{A}\) and conclude that the slope parameter for \(x_{1}\) = nit is not zero, i.e., sugar beet yield is significantly linearly related to the available nitrogen (controlling for treatment).

\(F=\dfrac{SSR(X_1|X_2,X_3)\div1}{SSE(X_1,X_2,X_3)\div(n-4)}=\dfrac{MSR(X_1|X_2,X_3)}{MSE(X_1,X_2,X_3)}\)

Use the Minitab output to calculate the value of this F statistic. Does the value you obtain equal \(t^{2}\), the square of the t -statistic as we might expect?

\(F-statistic= \dfrac{\frac{15934.5}{1}}{\frac{1078.0}{30-4}} = \dfrac{15934.5}{41.46} = 384.32\), which is the same as \(19.60^{2}\).

Because \(t^{2}\) will equal the partial F -statistic whenever you test for whether one slope parameter is 0, it makes sense to just use the t -statistic and P -value that Minitab displays as a default. But, note that we've just learned something new about the meaning of the t -test in the multiple regression setting. It tests for the ("marginal") significance of the \(x_{1}\) predictor after \(x_{2}\) and \(x_{3}\) have already been taken into account.

Tests for whether a subset of the slope parameters is 0

\(H_0 \colon \beta_2=\beta_3= 0\) against the alternative \(H_A \colon \beta_2 \ne 0\) or \(\beta_3 \ne 0\) or both \(\ne 0\).

\(F=\dfrac{SSR(X_2,X_3|X_1)\div2}{SSE(X_1,X_2,X_3)\div(n-4)}=\dfrac{MSR(X_2,X_3|X_1)}{MSE(X_1,X_2,X_3)}\)

\(F = \dfrac{\frac{10.4+27.5}{2}}{\frac{1078.0}{30-4}} = \dfrac{18.95}{41.46} = 0.46\).

F distribution with 2 DF in Numerator and 26 DF in denominator

p-value \(= 1-0.363677 = 0.636\), so we fail to reject \(H_{0}\) in favor of \(H_{A}\) and conclude that we cannot rule out \(\beta_2 = \beta_3 = 0\), i.e., there is no significant difference in the mean yields of sugar beets subjected to the different growth regulators after taking into account the available nitrogen.

Note that the sequential mean square due to regression, MSR(\(X_{2}\),\(X_{3}\)|\(X_{1}\)), is obtained by dividing the sequential sum of square by its degrees of freedom (2, in this case, since two additional predictors \(X_{2}\) and \(X_{3}\) are considered). Use the Minitab output to calculate the value of this F statistic, and use Minitab to get the associated P -value. Answer the researcher's question at the \(\alpha= 0.05\) level.

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Null hypothesis for linear regression

I am confused about the null hypothesis for linear regression.

If a variable in a linear model has $p < 0.05$ (when R prints out stars), I would say the variable is a statistically significant part of the model.

What does that translate to in terms of null hypothesis?

Am I rejecting the null hypothesis that the coefficient for that variable is $0,$ or am I accepting a null hypothesis that the coefficient is $\ne 0$ ?

Is there a difference between those statements?

• hypothesis-testing
• interpretation

3 Answers 3

The issue applies to null hypotheses more broadly than regression

You should get used to stating nulls before you look at p-values.

Am I rejecting the null hypothesis that the coefficient for that variable is 0

Yes, as long as it's the population coefficient, ($\beta_i$) you're talking about (obviously - with continuous response - the estimate of the coefficient isn't 0).

or am I accepting a null hypothesis that the coefficient is != 0?

Null hypotheses would generally be null - either 'no effect' or some conventionally accepted value. In this case, the population coefficient being 0 is a classical 'no effect' null.

More prosaically, when testing a point hypothesis against a composite alternative (a two-sided alternative in this case), one takes the point hypothesis as the null, because that's the one under which we can compute the distribution of the test statistic (more generally, using an open set for a null presents certain problems, even when both are composite). With a pair of point hypotheses, one is (at least mechanically) free to make either one the null (and even then one still would generally want to make the one that's most clearly "null" the null -- if either of them is; that is to choose the 'no effect' or conventionally-accepted one the null).

• $\begingroup$ Thanks. Now I finally understand why it's called the 'null hypothesis.' Related question: is there a way to accept the null hypothesis - to say you have enough evidence to state there is no relationship between the vars? $\endgroup$ –  wrschneider Commented Jan 30, 2015 at 15:37
• $\begingroup$ Well, that could depend on who you ask and partly on the circumstances. At least with a point null vs a composite alternative (the most common situation), I say the answer is no, since failure to reject doesn't indicate the null is actually true, only that the effect is probably quite small (relative to what the sample size could detect). In some of the other cases, more of an argument might be made, but I'd still say 'no'. Of course, ordinary null-hypothesis significance testing isn't the only possible choice, and some of the other choices would give more of a basis to do that. $\endgroup$ –  Glen_b Commented Jan 30, 2015 at 15:56
• 1 $\begingroup$ @wrschneider99 Obligatory plug for tests for equivalence as an alternative way of thinking about "accepting the null hypothesis." $\endgroup$ –  Alexis Commented Feb 13, 2015 at 3:53
• $\begingroup$ @Alexis +1 definitely worth raising. $\endgroup$ –  Glen_b Commented Feb 13, 2015 at 4:24
• $\begingroup$ @Glen_b: I know this is a broad question, but if you could give me a quick answer or ref: How does one go about choosing the right statistic, or any statistic to test a hypothesis? In this case we test wether the slope is 0. How do we choose/find a statistic that would allow us to test this claim? How do we come up with, e.g., the F-test when determining equality of variance? How is the process in general? Appreciate a ref. if question is too broad. $\endgroup$ –  MSIS Commented Oct 30, 2020 at 12:53

The P-Value in regression output in R tests the null hypothesis that the coefficient equals 0.

Any regression equation is given by y = a + b*x + u, where 'a' and 'b' are the intercept and slope of the best fit line and 'u' is the disturbance term. Imagine b=0; the equation would then be y = a + 0*x + u = a + u. Notice that the 'x' has disappeared. It simply means that there is no relationship between y and x. Thus are test hypothesis would be, Ho: b=0; Ha: b != 0

Next step is to compare the critical values with the test statistic. If the test stat lies within the Rejection region then we knockout the Null hypothesis.

Alternatively, the p-value also relays this information by pointing out how much probability remains between the test stat and the end tails.

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