How to Use the linearHypothesis() Function in R

You can use the linearHypothesis() function from the car package in R to test linear hypotheses in a specific regression model.

This function uses the following basic syntax:

linearHypothesis(fit, c("var1=0", "var2=0"))

This particular example tests if the regression coefficients var1 and var2 in the model called fit are jointly equal to zero.

The following example shows how to use this function in practice.

Example: How to Use linearHypothesis() Function in R

Suppose we have the following data frame in R that shows the number of hours spent studying, number of practice exams taken, and final exam score for 10 students in some class:

#create data frame
df #view data frame
df

   score hours prac_exams
1     77     1          2
2     79     1          4
3     84     2          4
4     85     3          2
5     88     2          4
6     99     4          5
7     95     4          4
8     90     2          3
9     92     3          2
10    94     3          1

Now suppose we would like to fit the following multiple linear regression model in R:

Exam score = β₀ + β₁(hours) + β₂(practice exams)

We can use the lm() function to fit this model:

#fit multiple linear regression model
fit #view summary of model
summary(fit)

Call:
lm(formula = score ~ hours + prac_exams, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.8366 -2.0875  0.1381  2.0652  4.6381 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  72.7393     3.9455  18.436 3.42e-07 ***
hours         5.8093     1.1161   5.205  0.00125 ** 
prac_exams    0.3346     0.9369   0.357  0.73150    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.59 on 7 degrees of freedom
Multiple R-squared:  0.8004,	Adjusted R-squared:  0.7434 
F-statistic: 14.03 on 2 and 7 DF,  p-value: 0.003553

Now suppose we would like to test if the coefficient for hours and prac_exams are both equal to zero.

We can use the linearHypothesis() function to do so:

library(car)

#perform hypothesis test for hours=0 and prac_exams=0
linearHypothesis(fit, c("hours=0", "prac_exams=0"))

Linear hypothesis test

Hypothesis:
hours = 0
prac_exams = 0

Model 1: restricted model
Model 2: score ~ hours + prac_exams

  Res.Df    RSS Df Sum of Sq      F   Pr(>F)   
1      9 452.10                                
2      7  90.24  2    361.86 14.035 0.003553 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The hypothesis test returns the following values:

• F test statistic: 14.035
• p-value: .003553

This particular hypothesis test uses the following null and alternative hypotheses:

• H₀: Both regression coefficients are equal to zero.
• H_A: At least one regression coefficient is not equal to zero.

Since the p-value of the test (.003553) is less than .05, we reject the null hypothesis.

In other words, we don’t have sufficient evidence to say that the regression coefficients for hours and prac_exams are both equal to zero.

Additional Resources

The following tutorials provide additional information about linear regression in R:

How to Interpret Regression Output in R
How to Perform Simple Linear Regression in R
How to Perform Multiple Linear Regression in R
How to Perform Logistic Regression in R