null hypothesis test regression

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13.6 Testing the Regression Coefficients

Learning objectives.

• Conduct and interpret a hypothesis test on individual regression coefficients.

Previously, we learned that the population model for the multiple regression equation is

[latex]\begin{eqnarray*} y & = & \beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_kx_k +\epsilon \end{eqnarray*}[/latex]

where [latex]x_1,x_2,\ldots,x_k[/latex] are the independent variables, [latex]\beta_0,\beta_1,\ldots,\beta_k[/latex] are the population parameters of the regression coefficients, and [latex]\epsilon[/latex] is the error variable.  In multiple regression, we estimate each population regression coefficient [latex]\beta_i[/latex] with the sample regression coefficient [latex]b_i[/latex].

In the previous section, we learned how to conduct an overall model test to determine if the regression model is valid.  If the outcome of the overall model test is that the model is valid, then at least one of the independent variables is related to the dependent variable—in other words, at least one of the regression coefficients [latex]\beta_i[/latex] is not zero.  However, the overall model test does not tell us which independent variables are related to the dependent variable.  To determine which independent variables are related to the dependent variable, we must test each of the regression coefficients.

Testing the Regression Coefficients

For an individual regression coefficient, we want to test if there is a relationship between the dependent variable [latex]y[/latex] and the independent variable [latex]x_i[/latex].

• No Relationship .  There is no relationship between the dependent variable [latex]y[/latex] and the independent variable [latex]x_i[/latex].  In this case, the regression coefficient [latex]\beta_i[/latex] is zero.  This is the claim for the null hypothesis in an individual regression coefficient test:  [latex]H_0: \beta_i=0[/latex].
• Relationship.  There is a relationship between the dependent variable [latex]y[/latex] and the independent variable [latex]x_i[/latex].  In this case, the regression coefficients [latex]\beta_i[/latex] is not zero.  This is the claim for the alternative hypothesis in an individual regression coefficient test:  [latex]H_a: \beta_i \neq 0[/latex].  We are not interested if the regression coefficient [latex]\beta_i[/latex] is positive or negative, only that it is not zero.  We only need to find out if the regression coefficient is not zero to demonstrate that there is a relationship between the dependent variable and the independent variable. This makes the test on a regression coefficient a two-tailed test.

In order to conduct a hypothesis test on an individual regression coefficient [latex]\beta_i[/latex], we need to use the distribution of the sample regression coefficient [latex]b_i[/latex]:

• The mean of the distribution of the sample regression coefficient is the population regression coefficient [latex]\beta_i[/latex].
• The standard deviation of the distribution of the sample regression coefficient is [latex]\sigma_{b_i}[/latex].  Because we do not know the population standard deviation we must estimate [latex]\sigma_{b_i}[/latex] with the sample standard deviation [latex]s_{b_i}[/latex].
• The distribution of the sample regression coefficient follows a normal distribution.

Steps to Conduct a Hypothesis Test on a Regression Coefficient

[latex]\begin{eqnarray*} H_0: &  &  \beta_i=0 \\ \\ \end{eqnarray*}[/latex]

[latex]\begin{eqnarray*} H_a: &  & \beta_i \neq 0 \\ \\ \end{eqnarray*}[/latex]

• Collect the sample information for the test and identify the significance level [latex]\alpha[/latex].

[latex]\begin{eqnarray*}t & = & \frac{b_i-\beta_i}{s_{b_i}} \\ \\ df &  = & n-k-1 \\  \\ \end{eqnarray*}[/latex]

• The results of the sample data are significant.  There is sufficient evidence to conclude that the null hypothesis [latex]H_0[/latex] is an incorrect belief and that the alternative hypothesis [latex]H_a[/latex] is most likely correct.
• The results of the sample data are not significant.  There is not sufficient evidence to conclude that the alternative hypothesis [latex]H_a[/latex] may be correct.
• Write down a concluding sentence specific to the context of the question.

The required [latex]t[/latex]-score and p -value for the test can be found on the regression summary table, which we learned how to generate in Excel in a previous section.

The human resources department at a large company wants to develop a model to predict an employee’s job satisfaction from the number of hours of unpaid work per week the employee does, the employee’s age, and the employee’s income.  A sample of 25 employees at the company is taken and the data is recorded in the table below.  The employee’s income is recorded in $1000s and the job satisfaction score is out of 10, with higher values indicating greater job satisfaction.

Previously, we found the multiple regression equation to predict the job satisfaction score from the other variables:

[latex]\begin{eqnarray*} \hat{y} & = & 4.7993-0.3818x_1+0.0046x_2+0.0233x_3 \\ \\ \hat{y} & = & \mbox{predicted job satisfaction score} \\ x_1 & = & \mbox{hours of unpaid work per week} \\ x_2 & = & \mbox{age} \\ x_3 & = & \mbox{income (\$1000s)}\end{eqnarray*}[/latex]

At the 5% significance level, test the relationship between the dependent variable “job satisfaction” and the independent variable “hours of unpaid work per week”.

Hypotheses:

[latex]\begin{eqnarray*} H_0: & & \beta_1=0 \\   H_a: & & \beta_1 \neq 0 \end{eqnarray*}[/latex]

The regression summary table generated by Excel is shown below:

The  p -value for the test on the hours of unpaid work per week regression coefficient is in the bottom part of the table under the P-value column of the Hours of Unpaid Work per Week row .  So the  p -value=[latex]0.0082[/latex].

Conclusion:  

Because p -value[latex]=0.0082 \lt 0.05=\alpha[/latex], we reject the null hypothesis in favour of the alternative hypothesis.  At the 5% significance level there is enough evidence to suggest that there is a relationship between the dependent variable “job satisfaction” and the independent variable “hours of unpaid work per week.”

• The null hypothesis [latex]\beta_1=0[/latex] is the claim that the regression coefficient for the independent variable [latex]x_1[/latex] is zero.  That is, the null hypothesis is the claim that there is no relationship between the dependent variable and the independent variable “hours of unpaid work per week.”
• The alternative hypothesis is the claim that the regression coefficient for the independent variable [latex]x_1[/latex] is not zero.  The alternative hypothesis is the claim that there is a relationship between the dependent variable and the independent variable “hours of unpaid work per week.”
• When conducting a test on a regression coefficient, make sure to use the correct subscript on [latex]\beta[/latex] to correspond to how the independent variables were defined in the regression model and which independent variable is being tested.  Here the subscript on [latex]\beta[/latex] is 1 because the “hours of unpaid work per week” is defined as [latex]x_1[/latex] in the regression model.
• The p -value for the tests on the regression coefficients are located in the bottom part of the table under the P-value column heading in the corresponding independent variable row. 
• Because the alternative hypothesis is a [latex]\neq[/latex], the p -value is the sum of the area in the tails of the [latex]t[/latex]-distribution.  This is the value calculated out by Excel in the regression summary table.
• The p -value of 0.0082 is a small probability compared to the significance level, and so is unlikely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis.  In other words, the regression coefficient [latex]\beta_1[/latex] is not zero, and so there is a relationship between the dependent variable “job satisfaction” and the independent variable “hours of unpaid work per week.”  This means that the independent variable “hours of unpaid work per week” is useful in predicting the dependent variable.

At the 5% significance level, test the relationship between the dependent variable “job satisfaction” and the independent variable “age”.

[latex]\begin{eqnarray*} H_0: & & \beta_2=0 \\   H_a: & & \beta_2 \neq 0 \end{eqnarray*}[/latex]

The  p -value for the test on the age regression coefficient is in the bottom part of the table under the P-value column of the Age row .  So the  p -value=[latex]0.8439[/latex].

Because p -value[latex]=0.8439 \gt 0.05=\alpha[/latex], we do not reject the null hypothesis.  At the 5% significance level there is not enough evidence to suggest that there is a relationship between the dependent variable “job satisfaction” and the independent variable “age.”

• The null hypothesis [latex]\beta_2=0[/latex] is the claim that the regression coefficient for the independent variable [latex]x_2[/latex] is zero.  That is, the null hypothesis is the claim that there is no relationship between the dependent variable and the independent variable “age.”
• The alternative hypothesis is the claim that the regression coefficient for the independent variable [latex]x_2[/latex] is not zero.  The alternative hypothesis is the claim that there is a relationship between the dependent variable and the independent variable “age.”
• When conducting a test on a regression coefficient, make sure to use the correct subscript on [latex]\beta[/latex] to correspond to how the independent variables were defined in the regression model and which independent variable is being tested.  Here the subscript on [latex]\beta[/latex] is 2 because “age” is defined as [latex]x_2[/latex] in the regression model.
• The p -value of 0.8439 is a large probability compared to the significance level, and so is likely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely correct, and so the conclusion of the test is to not reject the null hypothesis.  In other words, the regression coefficient [latex]\beta_2[/latex] is zero, and so there is no relationship between the dependent variable “job satisfaction” and the independent variable “age.”  This means that the independent variable “age” is not particularly useful in predicting the dependent variable.

At the 5% significance level, test the relationship between the dependent variable “job satisfaction” and the independent variable “income”.

[latex]\begin{eqnarray*} H_0: & & \beta_3=0 \\   H_a: & & \beta_3 \neq 0 \end{eqnarray*}[/latex]

The  p -value for the test on the income regression coefficient is in the bottom part of the table under the P-value column of the Income row .  So the  p -value=[latex]0.0060[/latex].

Because p -value[latex]=0.0060 \lt 0.05=\alpha[/latex], we reject the null hypothesis in favour of the alternative hypothesis.  At the 5% significance level there is enough evidence to suggest that there is a relationship between the dependent variable “job satisfaction” and the independent variable “income.”

• The null hypothesis [latex]\beta_3=0[/latex] is the claim that the regression coefficient for the independent variable [latex]x_3[/latex] is zero.  That is, the null hypothesis is the claim that there is no relationship between the dependent variable and the independent variable “income.”
• The alternative hypothesis is the claim that the regression coefficient for the independent variable [latex]x_3[/latex] is not zero.  The alternative hypothesis is the claim that there is a relationship between the dependent variable and the independent variable “income.”
• When conducting a test on a regression coefficient, make sure to use the correct subscript on [latex]\beta[/latex] to correspond to how the independent variables were defined in the regression model and which independent variable is being tested.  Here the subscript on [latex]\beta[/latex] is 3 because “income” is defined as [latex]x_3[/latex] in the regression model.
• The p -value of 0.0060 is a small probability compared to the significance level, and so is unlikely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis.  In other words, the regression coefficient [latex]\beta_3[/latex] is not zero, and so there is a relationship between the dependent variable “job satisfaction” and the independent variable “income.”  This means that the independent variable “income” is useful in predicting the dependent variable.

Concept Review

The test on a regression coefficient determines if there is a relationship between the dependent variable and the corresponding independent variable.  The p -value for the test is the sum of the area in tails of the [latex]t[/latex]-distribution.  The p -value can be found on the regression summary table generated by Excel.

The hypothesis test for a regression coefficient is a well established process:

• Write down the null and alternative hypotheses in terms of the regression coefficient being tested.  The null hypothesis is the claim that there is no relationship between the dependent variable and independent variable.  The alternative hypothesis is the claim that there is a relationship between the dependent variable and independent variable.
• Collect the sample information for the test and identify the significance level.
• The p -value is the sum of the area in the tails of the [latex]t[/latex]-distribution.  Use the regression summary table generated by Excel to find the p -value.
• Compare the  p -value to the significance level and state the outcome of the test.

Introduction to Statistics Copyright © 2022 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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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6.2.3 - more on model-fitting.

Suppose two models are under consideration, where one model is a special case or "reduced" form of the other obtained by setting \(k\) of the regression coefficients (parameters) equal to zero. The larger model is considered the "full" model, and the hypotheses would be

\(H_0\): reduced model versus \(H_A\): full model

Equivalently, the null hypothesis can be stated as the \(k\) predictor terms associated with the omitted coefficients have no relationship with the response, given the remaining predictor terms are already in the model. If we fit both models, we can compute the likelihood-ratio test (LRT) statistic:

\(G^2 = −2 (\log L_0 - \log L_1)\)

where \(L_0\) and \(L_1\) are the max likelihood values for the reduced and full models, respectively. The degrees of freedom would be \(k\), the number of coefficients in question. The p-value is the area under the \(\chi^2_k\) curve to the right of \( G^2)\).

To perform the test in SAS, we can look at the "Model Fit Statistics" section and examine the value of "−2 Log L" for "Intercept and Covariates." Here, the reduced model is the "intercept-only" model (i.e., no predictors), and "intercept and covariates" is the full model. For our running example, this would be equivalent to testing "intercept-only" model vs. full (saturated) model (since we have only one predictor).

Larger differences in the "-2 Log L" values lead to smaller p-values more evidence against the reduced model in favor of the full model. For our example, \( G^2 = 5176.510 − 5147.390 = 29.1207\) with \(2 − 1 = 1\) degree of freedom. Notice that this matches the deviance we got in the earlier text above.

Also, notice that the \(G^2\) we calculated for this example is equal to 29.1207 with 1df and p-value <.0001 from "Testing Global Hypothesis: BETA=0" section (the next part of the output, see below).

Testing the Joint Significance of All Predictors Section  

Testing the null hypothesis that the set of coefficients is simultaneously zero. For example, consider the full model

\(\log\left(\dfrac{\pi}{1-\pi}\right)=\beta_0+\beta_1 x_1+\cdots+\beta_k x_k\)

and the null hypothesis \(H_0\colon \beta_1=\beta_2=\cdots=\beta_k=0\) versus the alternative that at least one of the coefficients is not zero. This is like the overall F−test in linear regression. In other words, this is testing the null hypothesis of the intercept-only model:

\(\log\left(\dfrac{\pi}{1-\pi}\right)=\beta_0\)

versus the alternative that the current (full) model is correct. This corresponds to the test in our example because we have only a single predictor term, and the reduced model that removes the coefficient for that predictor is the intercept-only model.

In the SAS output, three different chi-square statistics for this test are displayed in the section "Testing Global Null Hypothesis: Beta=0," corresponding to the likelihood ratio, score, and Wald tests. Recall our brief encounter with them in our discussion of binomial inference in Lesson 2.

Large chi-square statistics lead to small p-values and provide evidence against the intercept-only model in favor of the current model. The Wald test is based on asymptotic normality of ML estimates of \(\beta\)s. Rather than using the Wald, most statisticians would prefer the LR test. If these three tests agree, that is evidence that the large-sample approximations are working well and the results are trustworthy. If the results from the three tests disagree, most statisticians would tend to trust the likelihood-ratio test more than the other two.

In our example, the "intercept only" model or the null model says that student's smoking is unrelated to parents' smoking habits. Thus the test of the global null hypothesis \(\beta_1=0\) is equivalent to the usual test for independence in the \(2\times2\) table. We will see that the estimated coefficients and standard errors are as we predicted before, as well as the estimated odds and odds ratios.

Residual deviance is the difference between −2 logL for the saturated model and −2 logL for the currently fit model. The high residual deviance shows that the model cannot be accepted. The null deviance is the difference between −2 logL for the saturated model and −2 logL for the intercept-only model. The high residual deviance shows that the intercept-only model does not fit.

In our \(2\times2\) table smoking example, the residual deviance is almost 0 because the model we built is the saturated model. And notice that the degree of freedom is 0 too. Regarding the null deviance, we could see it equivalent to the section "Testing Global Null Hypothesis: Beta=0," by likelihood ratio in SAS output.

For our example, Null deviance = 29.1207 with df = 1. Notice that this matches the deviance we got in the earlier text above.

The Homer-Lemeshow Statistic Section  

An alternative statistic for measuring overall goodness-of-fit is the  Hosmer-Lemeshow statistic .

This is a Pearson-like chi-square statistic that is computed after the data are grouped by having similar predicted probabilities. It is more useful when there is more than one predictor and/or continuous predictors in the model too. We will see more on this later.

\(H_0\): the current model fits well \(H_A\): the current model does not fit well

To calculate this statistic:

• Group the observations according to model-predicted probabilities ( \(\hat{\pi}_i\))
• The number of groups is typically determined such that there is roughly an equal number of observations per group
• The Hosmer-Lemeshow (HL) statistic, a Pearson-like chi-square statistic, is computed on the grouped data but does NOT have a limiting chi-square distribution because the observations in groups are not from identical trials. Simulations have shown that this statistic can be approximated by a chi-squared distribution with \(g − 2\) degrees of freedom, where \(g\) is the number of groups.

Warning about the Hosmer-Lemeshow goodness-of-fit test:

• It is a conservative statistic, i.e., its value is smaller than what it should be, and therefore the rejection probability of the null hypothesis is smaller.
• It has low power in predicting certain types of lack of fit such as nonlinearity in explanatory variables.
• It is highly dependent on how the observations are grouped.
• If too few groups are used (e.g., 5 or less), it almost always fails to reject the current model fit. This means that it's usually not a good measure if only one or two categorical predictor variables are involved, and it's best used for continuous predictors.

In the model statement, the option lackfit tells SAS to compute the HL statistic and print the partitioning. For our example, because we have a small number of groups (i.e., 2), this statistic gives a perfect fit (HL = 0, p-value = 1). Instead of deriving the diagnostics, we will look at them from a purely applied viewpoint. Recall the definitions and introductions to the regression residuals and Pearson and Deviance residuals.

Residuals Section  

The Pearson residuals are defined as

\(r_i=\dfrac{y_i-\hat{\mu}_i}{\sqrt{\hat{V}(\hat{\mu}_i)}}=\dfrac{y_i-n_i\hat{\pi}_i}{\sqrt{n_i\hat{\pi}_i(1-\hat{\pi}_i)}}\)

The contribution of the \(i\)th row to the Pearson statistic is

\(\dfrac{(y_i-\hat{\mu}_i)^2}{\hat{\mu}_i}+\dfrac{((n_i-y_i)-(n_i-\hat{\mu}_i))^2}{n_i-\hat{\mu}_i}=r^2_i\)

and the Pearson goodness-of fit statistic is

\(X^2=\sum\limits_{i=1}^N r^2_i\)

which we would compare to a \(\chi^2_{N-p}\) distribution. The deviance test statistic is

\(G^2=2\sum\limits_{i=1}^N \left\{ y_i\text{log}\left(\dfrac{y_i}{\hat{\mu}_i}\right)+(n_i-y_i)\text{log}\left(\dfrac{n_i-y_i}{n_i-\hat{\mu}_i}\right)\right\}\)

which we would again compare to \(\chi^2_{N-p}\), and the contribution of the \(i\)th row to the deviance is

\(2\left\{ y_i\log\left(\dfrac{y_i}{\hat{\mu}_i}\right)+(n_i-y_i)\log\left(\dfrac{n_i-y_i}{n_i-\hat{\mu}_i}\right)\right\}\)

We will note how these quantities are derived through appropriate software and how they provide useful information to understand and interpret the models.

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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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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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.

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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.

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14.4: Hypothesis Test for Simple Linear Regression

• Last updated
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• Page ID 20929

• Maurice A. Geraghty
• De Anza College

We will now describe a hypothesis test to determine if the regression model is meaningful; in other words, does the value of \(X\) in any way help predict the expected value of \(Y\)?

Simple Linear Regression ANOVA Hypothesis Test

Model Assumptions

• The residual errors are random and are normally distributed.
• The standard deviation of the residual error does not depend on \(X\)
• A linear relationship exists between \(X\) and \(Y\)
• The samples are randomly selected

Test Hypotheses

\(H_o\):  \(X\) and \(Y\) are not correlated   

\(H_a\):  \(X\) and \(Y\) are correlated   

\(H_o\):  \(\beta_1\) (slope) = 0   

\(H_a\):  \(\beta_1\) (slope) ≠ 0

Test Statistic

\(F=\dfrac{M S_{\text {Regression }}}{M S_{\text {Error }}}\)

\(d f_{\text {num }}=1\)

\(d f_{\text {den }}=n-2\)

Sum of Squares

\(S S_{\text {Total }}=\sum(Y-\bar{Y})^{2}\)

\(S S_{\text {Error }}=\sum(Y-\hat{Y})^{2}\)

\(S S_{\text {Regression }}=S S_{\text {Total }}-S S_{\text {Error }}\)

In simple linear regression, this is equivalent to saying “Are X an Y correlated?”

In reviewing the model, \(Y=\beta_{0}+\beta_{1} X+\varepsilon\), as long as the slope (\(\beta_{1}\)) has any non‐zero value, \(X\) will add value in helping predict the expected value of \(Y\). However, if there is no correlation between X and Y, the value of the slope (\(\beta_{1}\)) will be zero. The model we can use is very similar to One Factor ANOVA.

The Results of the test can be summarized in a special ANOVA table:

Example: Rainfall and sales of sunglasses

Design : Is there a significant correlation between rainfall and sales of sunglasses?

Research Hypothese s:

\(H_o\):  Sales and Rainfall are not correlated      \(H_o\):  1 (slope) = 0

\(H_a\):  Sales and Rainfall are correlated      \(H_a\):  1 (slope) ≠ 0

Type I error would be to reject the Null Hypothesis and \(t\) claim that rainfall is correlated with sales of sunglasses, when they are not correlated. The test will be run at a level of significance (\(\alpha\)) of 5%.

The test statistic from the table will be \(\mathrm{F}=\dfrac{\text { MSRegression }}{\text { MSError }}\). The degrees of freedom for the numerator will be 1, and the degrees of freedom for denominator will be 5‐2=3.  

Critical Value for \(F\) at  \(\alpha\)of 5% with \(df_{num}=1\) and \(df_{den}=3} is 10.13.  Reject \(H_o\) if \(F >10.13\). We will also run this test using the \(p\)‐value method with statistical software, such as Minitab.  

Data/Results

\(F=341.422 / 12.859=26.551\), which is more than the critical value of 10.13, so Reject \(H_o\). Also, the \(p\)‐value = 0.0142 < 0.05 which also supports rejecting \(H_o\).  

Sales of Sunglasses and Rainfall are negatively correlated.

Statistics Made Easy

A Simple Guide to Understanding the F-Test of Overall Significance in Regression

This tutorial explains how to identify the F-statistic in the output of a regression table as well as how to interpret this statistic and its corresponding p-value.

Understanding the F-Test of Overall Significance

The F-Test of overall significance in regression is a test of whether or not your linear regression model provides a better fit to a dataset than a model with no predictor variables. 

The F-Test of overall significance has the following two hypotheses:

Null hypothesis (H 0 ) : The model with no predictor variables (also known as an  intercept-only model ) fits the data as well as your regression model.

Alternative hypothesis (H A ) :  Your regression model fits the data better than the intercept-only model.

When you fit a regression model to a dataset, you will receive a regression table as output, which will tell you the F-statistic along with the corresponding p-value for that F-statistic.

If the p-value is less than the significance level you’ve chosen ( common choices are .01, .05, and .10 ), then you have sufficient evidence to conclude that your regression model fits the data better than the intercept-only model.

Example: F-Test in Regression

Suppose we have the following dataset that shows the total number of hours studied, total prep exams taken, and final exam score received for 12 different students:

To analyze the relationship between hours studied and prep exams taken with the final exam score that a student receives, we run a multiple linear regression using  hours studied and  prep  exams taken  as the predictor variables and  final exam score  as the response variable.

We receive the following output:

From these results, we will focus on the F-statistic given in the ANOVA table as well as the p-value of that F-statistic, which is labeled as  Significance F in the table. We will choose .05 as our significance level.

F-statistic: 5.090515

P-value: 0.0332

Technical note: The F-statistic is calculated as MS regression divided by MS residual. In this case MS regression / MS residual =273.2665 / 53.68151 = 5.090515 .

Since the p-value is less than the significance level, we can conclude that our regression model fits the data better than the intercept-only model.

In the context of this specific problem, it means that using our predictor variables Study Hours and  Prep Exams  in the model allows us to fit the data better than if we left them out and simply used the intercept-only model.

Notes on Interpreting the F-Test of Overall Significance

In general, if none of your predictor variables are statistically significant, the overall F-test will also not be statistically significant.

However, it’s possible on some occasions that this doesn’t hold because the F-test of overall significance tests whether all of the predictor variables are jointly  significant while the t-test of significance for each individual predictor variable merely tests whether each predictor variable is  individually  significant. 

Thus, the F-test determines whether or not  all  of the predictor variables are jointly significant.

It’s possible that each predictor variable is not significant and yet the F-test says that all of the predictor variables combined are jointly significant. 

Technical note:  In general, the more predictor variables you have in the model, the higher the likelihood that the The F-statistic and corresponding p-value will be statistically significant.

Another metric that you’ll likely see in the output of a regression is  R-squared , which measures the strength of the linear relationship between the predictor variables and the response variable is another.

Although R-squared can give you an idea of how strongly associated the predictor variables are with the response variable, it doesn’t provide a formal statistical test for this relationship.

This is why the F-Test is useful since it is a formal statistical test. In addition, if the overall F-test is significant, you can conclude that R-squared is not equal to zero and that the correlation between the predictor variable(s) and response variable is statistically significant.

Additional Resources

The following tutorials explain how to interpret other common values in regression models:

How to Read and Interpret a Regression Table Understanding the Standard Error of the Regression What is a Good R-squared Value?

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