*48*

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.Â

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

**Null hypothesis (H _{0}) :** The model with no predictor variables (also known as anÂ

*intercept-only model*) fits the data as well as your regression model.

**Alternative hypothesis (H _{A}) :**Â 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 youâ€™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 usingÂ *hours studied *andÂ *prepÂ **exams takenÂ *as the predictor variables andÂ *final exam scoreÂ *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 asÂ *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 thatÂ 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 *andÂ *Prep ExamsÂ *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, itâ€™s possible on some occasions that this doesnâ€™t hold because the F-test of overall significance tests whether all of the predictor variables are *jointlyÂ *significant while the t-test of significance for each individual predictor variable merely tests whether each predictor variable isÂ *individuallyÂ *significant.Â

Thus, the F-test determines whether or notÂ *allÂ *of the predictor variables are jointly significant.

Itâ€™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.Â

Technical note:Â 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 youâ€™ll likely see in the output of a regression isÂ 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 doesnâ€™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?