multiple regression analysis hypothesis testing

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Hypothesis Tests and Confidence Intervals in Multiple Regression

After completing this reading you should be able to:

• Construct, apply, and interpret hypothesis tests and confidence intervals for a single coefficient in a multiple regression.
• Construct, apply, and interpret joint hypothesis tests and confidence intervals for multiple coefficients in a multiple regression.
• Interpret the \(F\)-statistic.
• Interpret tests of a single restriction involving multiple coefficients.
• Interpret confidence sets for multiple coefficients.
• Identify examples of omitted variable bias in multiple regressions.
• Interpret the \({ R }^{ 2 }\) and adjusted \({ R }^{ 2 }\) in a multiple regression.

Hypothesis Tests and Confidence Intervals for a Single Coefficient

This section is about the calculation of the standard error, hypotheses testing, and confidence interval construction for a single regression in a multiple regression equation.

Introduction

In a previous chapter, we looked at simple linear regression where we deal with just one regressor (independent variable). The response (dependent variable) is assumed to be affected by just one independent variable.  M ultiple regression, on the other hand ,  simultaneously considers the influence of multiple explanatory variables on a response variable Y. We may want to establish the confidence interval of one of the independent variables. We may want to evaluate whether any particular independent variable has a significant effect on the dependent variable. Finally, We may also want to establish whether the independent variables as a group have a significant effect on the dependent variable. In this chapter, we delve into ways all this can be achieved.

Hypothesis Tests for a single coefficient

Suppose that we are testing the hypothesis that the true coefficient \({ \beta }_{ j }\) on the \(j\)th regressor takes on some specific value \({ \beta }_{ j,0 }\). Let the alternative hypothesis be two-sided. Therefore, the following is the mathematical expression of the two hypotheses:

$$ { H }_{ 0 }:{ \beta }_{ j }={ \beta }_{ j,0 }\quad vs.\quad { H }_{ 1 }:{ \beta }_{ j }\neq { \beta }_{ j,0 } $$

This expression represents the two-sided alternative. The following are the steps to follow while testing the null hypothesis:

• Computing the coefficient’s standard error.

$$ p-value=2\Phi \left( -|{ t }^{ act }| \right) $$

• Also, the \(t\)-statistic can be compared to the critical value corresponding to the significance level that is desired for the test.

Confidence Intervals for a Single Coefficient

The confidence interval for a regression coefficient in multiple regression is calculated and interpreted the same way as it is in simple linear regression. 

The t-statistic has n – k – 1 degrees of freedom where k = number of independents

Supposing that an interval contains the true value of \({ \beta }_{ j }\) with a probability of 95%. This is simply the 95% two-sided confidence interval for \({ \beta }_{ j }\). The implication here is that the true value of \({ \beta }_{ j }\) is contained in 95% of all possible randomly drawn variables.

Alternatively, the 95% two-sided confidence interval for \({ \beta }_{ j }\) is the set of values that are impossible to reject when a two-sided hypothesis test of 5% is applied. Therefore, with a large sample size:

$$ 95\%\quad confidence\quad interval\quad for\quad { \beta }_{ j }=\left[ { \hat { \beta } }_{ j }-1.96SE\left( { \hat { \beta } }_{ j } \right) ,{ \hat { \beta } }_{ j }+1.96SE\left( { \hat { \beta } }_{ j } \right) \right] $$

Tests of Joint Hypotheses

In this section, we consider the formulation of the joint hypotheses on multiple regression coefficients. We will further study the application of an \(F\)-statistic in their testing.

Hypotheses Testing on Two or More Coefficients

Joint null hypothesis.

In multiple regression, we canno t test the null hypothesis that all slope coefficients are equal 0 based on t -tests that each individual slope coefficient equals 0. Why? individual t-tests do not account for the effects of interactions among the independent variables.

For this reason, we conduct the F-test which uses the F-statistic .  The F-test tests the null hypothesis that all of the slope coefficients in the multiple regression model are jointly equal to 0, .i.e.,

\(F\)-Statistic

The F-statistic, which is always a one-tailed test , is calculated as:

To determine whether at least one of the coefficients is statistically significant, the calculated F-statistic is compared with the one-tailed critical F-value, at the appropriate level of significance.

Decision rule:

Rejection of the null hypothesis at a stated level of significance indicates that at least one of the coefficients is significantly different than zero, i.e, at least one of the independent variables in the regression model makes a significant contribution to the dependent variable.

An analyst runs a regression of monthly value-stock returns on four independent variables over 48 months.

The total sum of squares for the regression is 360, and the sum of squared errors is 120.

Test the null hypothesis at the 5% significance level (95% confidence) that all the four independent variables are equal to zero.

\({ H }_{ 0 }:{ \beta }_{ 1 }=0,{ \beta }_{ 2 }=0,\dots ,{ \beta }_{ 4 }=0 \)

\({ H }_{ 1 }:{ \beta }_{ j }\neq 0\) (at least one j is not equal to zero, j=1,2… k )

ESS = TSS – SSR = 360 – 120 = 240

The calculated test statistic = (ESS/k)/(SSR/(n-k-1))

=(240/4)/(120/43) = 21.5

\({ F }_{ 43 }^{ 4 }\) is approximately 2.44 at 5% significance level.

Decision: Reject H 0 .

Conclusion: at least one of the 4 independents is significantly different than zero.

Omitted Variable Bias in Multiple Regression

This is the bias in the OLS estimator arising when at least one included regressor gets collaborated with an omitted variable. The following conditions must be satisfied for an omitted variable bias to occur:

• There must be a correlation between at least one of the included regressors and the omitted variable.
• The dependent variable \(Y\) must be determined by the omitted variable.

Practical Interpretation of the \({ R }^{ 2 }\) and the adjusted \({ R }^{ 2 }\), \({ \bar { R } }^{ 2 }\)

To determine the accuracy within which the OLS regression line fits the data, we apply the coefficient of determination and the regression’s standard error . 

The coefficient of determination, represented by \({ R }^{ 2 }\), is a measure of the “goodness of fit” of the regression. It is interpreted as the percentage of variation in the dependent variable explained by the independent variables

\({ R }^{ 2 }\) is not a reliable indicator of the explanatory power of a multiple regression model.Why? \({ R }^{ 2 }\) almost always increases as new independent variables are added to the model, even if the marginal contribution of the new variable is not statistically significant. Thus, a high \({ R }^{ 2 }\) may reflect the impact of a large set of independents rather than how well the set explains the dependent.This problem is solved by the use of the adjusted \({ R }^{ 2 }\) (extensively covered in chapter 8)

The following are the factors to watch out when guarding against applying the \({ R }^{ 2 }\) or the \({ \bar { R } }^{ 2 }\):

• An added variable doesn’t have to be statistically significant just because the \({ R }^{ 2 }\) or the \({ \bar { R } }^{ 2 }\) has increased.
• It is not always true that the regressors are a true cause of the dependent variable, just because there is a high \({ R }^{ 2 }\) or \({ \bar { R } }^{ 2 }\).
• It is not necessary that there is no omitted variable bias just because we have a high \({ R }^{ 2 }\) or \({ \bar { R } }^{ 2 }\).
• It is not necessarily true that we have the most appropriate set of regressors just because we have a high \({ R }^{ 2 }\) or \({ \bar { R } }^{ 2 }\).
• It is not necessarily true that we have an inappropriate set of regressors just because we have a low \({ R }^{ 2 }\) or \({ \bar { R } }^{ 2 }\).

An economist tests the hypothesis that GDP growth in a certain country can be explained by interest rates and inflation.

Using some 30 observations, the analyst formulates the following regression equation:

$$ GDP growth = { \hat { \beta } }_{ 0 } + { \hat { \beta } }_{ 1 } Interest+ { \hat { \beta } }_{ 2 } Inflation $$

Regression estimates are as follows:

Is the coefficient for interest rates significant at 5%?

• Since the test statistic < t-critical, we accept H 0 ; the interest rate coefficient is  not   significant at the 5% level.
• Since the test statistic > t-critical, we reject H 0 ; the interest rate coefficient is not significant at the 5% level.
• Since the test statistic > t-critical, we reject H 0 ; the interest rate coefficient is significant at the 5% level.
• Since the test statistic < t-critical, we accept H 1 ; the interest rate coefficient is significant at the 5% level.

The correct answer is  C .

We have GDP growth = 0.10 + 0.20(Int) + 0.15(Inf)

Hypothesis:

$$ { H }_{ 0 }:{ \hat { \beta } }_{ 1 } = 0 \quad vs \quad { H }_{ 1 }:{ \hat { \beta } }_{ 1 }≠0 $$

The test statistic is:

$$ t = \left( \frac { 0.20 – 0 }{ 0.05 } \right)  = 4 $$

The critical value is t (α/2, n-k-1) = t 0.025,27  = 2.052 (which can be found on the t-table).

Conclusion : The interest rate coefficient is significant at the 5% level.

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Multiple Linear Regression | A Quick Guide (Examples)

Published on February 20, 2020 by Rebecca Bevans . Revised on June 22, 2023.

Regression models are used to describe relationships between variables by fitting a line to the observed data. Regression allows you to estimate how a dependent variable changes as the independent variable(s) change.

Multiple linear regression is used to estimate the relationship between  two or more independent variables and one dependent variable . You can use multiple linear regression when you want to know:

• How strong the relationship is between two or more independent variables and one dependent variable (e.g. how rainfall, temperature, and amount of fertilizer added affect crop growth).
• The value of the dependent variable at a certain value of the independent variables (e.g. the expected yield of a crop at certain levels of rainfall, temperature, and fertilizer addition).

Table of contents

Assumptions of multiple linear regression, how to perform a multiple linear regression, interpreting the results, presenting the results, other interesting articles, frequently asked questions about multiple linear regression.

Multiple linear regression makes all of the same assumptions as simple linear regression :

Homogeneity of variance (homoscedasticity) : the size of the error in our prediction doesn’t change significantly across the values of the independent variable.

Independence of observations : the observations in the dataset were collected using statistically valid sampling methods , and there are no hidden relationships among variables.

In multiple linear regression, it is possible that some of the independent variables are actually correlated with one another, so it is important to check these before developing the regression model. If two independent variables are too highly correlated (r2 > ~0.6), then only one of them should be used in the regression model.

Normality : The data follows a normal distribution .

Linearity : the line of best fit through the data points is a straight line, rather than a curve or some sort of grouping factor.

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See an example

Multiple linear regression formula

The formula for a multiple linear regression is:

• … = do the same for however many independent variables you are testing

To find the best-fit line for each independent variable, multiple linear regression calculates three things:

• The regression coefficients that lead to the smallest overall model error.
• The t statistic of the overall model.
• The associated p value (how likely it is that the t statistic would have occurred by chance if the null hypothesis of no relationship between the independent and dependent variables was true).

It then calculates the t statistic and p value for each regression coefficient in the model.

Multiple linear regression in R

While it is possible to do multiple linear regression by hand, it is much more commonly done via statistical software. We are going to use R for our examples because it is free, powerful, and widely available. Download the sample dataset to try it yourself.

Dataset for multiple linear regression (.csv)

Load the heart.data dataset into your R environment and run the following code:

This code takes the data set heart.data and calculates the effect that the independent variables biking and smoking have on the dependent variable heart disease using the equation for the linear model: lm() .

Learn more by following the full step-by-step guide to linear regression in R .

To view the results of the model, you can use the summary() function:

This function takes the most important parameters from the linear model and puts them into a table that looks like this:

The summary first prints out the formula (‘Call’), then the model residuals (‘Residuals’). If the residuals are roughly centered around zero and with similar spread on either side, as these do ( median 0.03, and min and max around -2 and 2) then the model probably fits the assumption of heteroscedasticity.

Next are the regression coefficients of the model (‘Coefficients’). Row 1 of the coefficients table is labeled (Intercept) – this is the y-intercept of the regression equation. It’s helpful to know the estimated intercept in order to plug it into the regression equation and predict values of the dependent variable:

The most important things to note in this output table are the next two tables – the estimates for the independent variables.

The Estimate column is the estimated effect , also called the regression coefficient or r 2 value. The estimates in the table tell us that for every one percent increase in biking to work there is an associated 0.2 percent decrease in heart disease, and that for every one percent increase in smoking there is an associated .17 percent increase in heart disease.

The Std.error column displays the standard error of the estimate. This number shows how much variation there is around the estimates of the regression coefficient.

The t value column displays the test statistic . Unless otherwise specified, the test statistic used in linear regression is the t value from a two-sided t test . The larger the test statistic, the less likely it is that the results occurred by chance.

The Pr( > | t | ) column shows the p value . This shows how likely the calculated t value would have occurred by chance if the null hypothesis of no effect of the parameter were true.

Because these values are so low ( p < 0.001 in both cases), we can reject the null hypothesis and conclude that both biking to work and smoking both likely influence rates of heart disease.

When reporting your results, include the estimated effect (i.e. the regression coefficient), the standard error of the estimate, and the p value. You should also interpret your numbers to make it clear to your readers what the regression coefficient means.

Visualizing the results in a graph

It can also be helpful to include a graph with your results. Multiple linear regression is somewhat more complicated than simple linear regression, because there are more parameters than will fit on a two-dimensional plot.

However, there are ways to display your results that include the effects of multiple independent variables on the dependent variable, even though only one independent variable can actually be plotted on the x-axis.

Here, we have calculated the predicted values of the dependent variable (heart disease) across the full range of observed values for the percentage of people biking to work.

To include the effect of smoking on the independent variable, we calculated these predicted values while holding smoking constant at the minimum, mean , and maximum observed rates of smoking.

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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.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).

A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.

Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line.

Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:

• measuring the distance of the observed y-values from the predicted y-values at each value of x;
• squaring each of these distances;
• calculating the mean of each of the squared distances.

Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.

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Lesson 5: Multiple Linear Regression (MLR) Model & Evaluation

Overview of this lesson.

In this lesson, we make our first (and last?!) major jump in the course. We move from the simple linear regression model with one predictor to the multiple linear regression model with two or more predictors. That is, we use the adjective "simple" to denote that our model has only predictor, and we use the adjective "multiple" to indicate that our model has at least two predictors.

In the multiple regression setting, because of the potentially large number of predictors, it is more efficient to use matrices to define the regression model and the subsequent analyses. This lesson considers some of the more important multiple regression formulas in matrix form. If you're unsure about any of this, it may be a good time to take a look at this Matrix Algebra Review .

The good news is that everything you learned about the simple linear regression model extends — with at most minor modification — to the multiple linear regression model. Think about it — you don't have to forget all of that good stuff you learned! In particular:

• The models have similar "LINE" assumptions. The only real difference is that whereas in simple linear regression we think of the distribution of errors at a fixed value of the single predictor, with multiple linear regression we have to think of the distribution of errors at a fixed set of values for all the predictors. All of the model checking procedures we learned earlier are useful in the multiple linear regression framework, although the process becomes more involved since we now have multiple predictors. We'll explore this issue further in Lesson 6.
• The use and interpretation of r 2 (which we'll denote R 2 in the context of multiple linear regression) remains the same. However, with multiple linear regression we can also make use of an "adjusted" R 2 value, which is useful for model building purposes. We'll explore this measure further in Lesson 11.
• With a minor generalization of the degrees of freedom, we use t -tests and t -intervals for the regression slope coefficients to assess whether a predictor is significantly linearly related to the response, after controlling for the effects of all the opther predictors in the model.
• With a minor generalization of the degrees of freedom, we use confidence intervals for estimating the mean response and prediction intervals for predicting an individual response. We'll explore these further in Lesson 6.

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:

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

In this lesson, we also learn how to perform each of the above three hypothesis tests.

• 5.1 - Example on IQ and Physical Characteristics
• 5.2 - Example on Underground Air Quality
• 5.3 - The Multiple Linear Regression Model
• 5.4 - A Matrix Formulation of the Multiple Regression Model
• 5.5 - Three Types of MLR Parameter Tests
• 5.6 - The General Linear F-Test
• 5.7 - MLR Parameter Tests
• 5.8 - Partial R-squared
• 5.9 - Further MLR Examples

Start Here!

• Welcome to STAT 462!
• Search Course Materials
• Lesson 1: Statistical Inference Foundations
• Lesson 2: Simple Linear Regression (SLR) Model
• Lesson 3: SLR Evaluation
• Lesson 4: SLR Assumptions, Estimation & Prediction
• 5.9- Further MLR Examples
• Lesson 6: MLR Assumptions, Estimation & Prediction
• Lesson 7: Transformations & Interactions
• Lesson 8: Categorical Predictors
• Lesson 9: Influential Points
• Lesson 10: Regression Pitfalls
• Lesson 11: Model Building
• Lesson 12: Logistic, Poisson & Nonlinear Regression
• Website for Applied Regression Modeling, 2nd edition
• Notation Used in this Course
• R Software Help
• Minitab Software Help

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