You can use the **coeftest()** function from the **lmtest** package in R to perform a t-test for each estimated coefficient in a regression model.

This function uses the following basic syntax:

**coeftest(x)**

where:

**x**: Name of the fitted regression model

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

**Example: How to Use coeftest() 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 <- data.frame(score=c(77, 79, 84, 85, 88, 99, 95, 90, 92, 94), hours=c(1, 1, 2, 3, 2, 4, 4, 2, 3, 3), prac_exams=c(2, 3, 3, 2, 4, 5, 4, 3, 5, 4)) #view data frame df score hours prac_exams 1 77 1 2 2 79 1 3 3 84 2 3 4 85 3 2 5 88 2 4 6 99 4 5 7 95 4 4 8 90 2 3 9 92 3 5 10 94 3 4

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

Exam score = β_{0} + β_{1}(hours) + β_{2}(practice exams)

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

#fit multiple linear regression model fit <- lm(score ~ hours + prac_exams, data=df)

We can then use the **coeftest()** function to perform a t-test for each fitted regression coefficient in the model:

library(lmtest) #perform t-test for each coefficient in model coeftest(fit) t test of coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 68.40294 2.87227 23.8150 5.851e-08 *** hours 4.19118 0.99612 4.2075 0.003998 ** prac_exams 2.69118 0.99612 2.7017 0.030566 * --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The t test statistic and corresponding p-value is shown for each t-test:

**Intercept**: t = 23.8150, p = <0.000**hours**: t = 4.2075, p = .003998**prac_exams**: t = 2.7017, p = .030566

Note that we use the following null and alternative hypotheses for each t-test:

**H**: β_{0}_{i}= 0 (the slope is equal to zero)**H**: β_{A}_{i}≠ 0 (the slope is not equal to zero)

If the p-value of the t-test is less than some threshold (e.g. α = .05) then we reject the null hypothesis and conclude that there is a statistically significant relationship between the predictor variable and the response variable.

Since the p-value for each t-test is less than .05, we would conclude that each predictor variable in the model has a statistically significant relationship with the response variable.

In the context of this example, we would say that hours spent studying and number of practice exams taken are both statistically significant predictors of final exam score for students.

**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