The F-test of overall significance indicates whether your linear regression model provides a better fit to the data than a model that contains no independent variables. In this post, I look at how the F-test of overall significance fits in with other regression statistics, such as R-squared. R-squared tells you how well your model fits the data, and the F-test is related to it.
An F-test is a type of statistical test that is very flexible. You can use them in a wide variety of settings. F-tests can evaluate multiple model terms simultaneously, which allows them to compare the fits of different linear models. In contrast, t-tests can evaluate just one term at a time.
Read my blog post about how F-tests work in ANOVA.
To calculate the F-test of overall significance, your statistical software just needs to include the proper terms in the two models that it compares. The overall F-test compares the model that you specify to the model with no independent variables. This type of model is also known as an intercept-only model.
The F-test for overall significance has the following two hypotheses:
- The null hypothesis states that the model with no independent variables fits the data as well as your model.
- The alternative hypothesis says that your model fits the data better than the intercept-only model.
In statistical output, you can find the overall F-test in the ANOVA table. An example is below.
Interpreting the Overall F-test of Significance
Compare the p-value for the F-test to your significance level. If the p-value is less than the significance level, your sample data provide sufficient evidence to conclude that your regression model fits the data better than the model with no independent variables.
This finding is good news because it means that the independent variables in your model improve the fit!
Generally speaking, if none of your independent variables are statistically significant, the overall F-test is also not statistically significant. Occasionally, the tests can produce conflicting results. This disagreement can occur because the F-test of overall significance assesses all of the coefficients jointly whereas the t-test for each coefficient examines them individually. For example, the overall F-test can find that the coefficients are significant jointly while the t-tests can fail to find significance individually.
These conflicting test results can be hard to understand, but think about it this way. The F-test sums the predictive power of all independent variables and determines that it is unlikely that all of the coefficients equal zero. However, it’s possible that each variable isn’t predictive enough on its own to be statistically significant. In other words, your sample provides sufficient evidence to conclude that your model is significant, but not enough to conclude that any individual variable is significant.
Related post: How to Interpret Regression Coefficients and their P-values.
Additional Ways to Interpret the F-test of Overall Significance
If you have a statistically significant overall F-test, you can draw several other conclusions.
For the model with no independent variables, the intercept-only model, all of the model’s predictions equal the mean of the dependent variable. Consequently, if the overall F-test is statistically significant, your model’s predictions are an improvement over using the mean.
R-squared measures the strength of the relationship between your model and the dependent variable. However, it is not a formal test for the relationship. The F-test of overall significance is the hypothesis test for this relationship. If the overall F-test is significant, you can conclude that R-squared does not equal zero, and the correlation between the model and dependent variable is statistically significant.
It’s fabulous if your regression model is statistically significant! However, check your residual plots to determine whether the results are trustworthy! And, learn how to choose the correct regression model!