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!

Duc-Anh Luong says

Hi Jim,

Thank you so much for your interesting and easily understandable post. However, I have a question when we have the conflict between overall F-test and significant t-test for each predictor. What should we do if the t-test for some of prediction is non-significant? Should we remove this predictors and fit the model again?

Many thanks,

Duc Anh

Jim Frost says

Hi Duc-Anh,

Frequently you do remove an independent variable from a model if it is not statistically significant. There are some exceptions to this rule. If you believe that theoretical considerations suggest that the variable should be in the model despite an insignificant p-value, you could consider leaving it in. Additionally, if it is a variable that you are specifically testing in an experiment, you would leave it in to demonstrate the test results.

But, yes, frequently you would consider removing the predictor from the model if it is not statistically significant. Your dataset provides insufficient evidence to conclude that there is a relationship between that predictor and the response.

One more point, be sure to check the residual plots. There might be a curvilinear relationship.

I hope this helps,

Jim

Duc-Anh Luong says

Hi Jim,

Thank you so much for your reply. In case we keep one or more predictors that are not statistically significant based on some except rule you mentioned in the previous comment, how can we interpret the results now?

Best regards,

Duc Anh

Jim Frost says

Hi again Duc Anh,

It depends on why you leave the predictor in the model. If you’re leaving it in the model because it’s the specific term you are testing for your experiment, then you state that you have insufficient evidence to conclude that there is a relationship between this variable and the response.

However, if you’re leaving the variable in for theoretical reasons, that’s what you should state. The variable wasn’t statistically significant but theory/other studies suggest it belongs in the model. You might even investigate possible reasons for why it is not significant, such as a small sample size, noisy data, a fluky sample, etc. Even though you suspect the variable belongs in the model, your sample still provides insufficient evidence to conclude that the relationship exists. You really have to make sure you have a good strong reason for this approach and state clearly why you are doing so.

I hope this helps,

Jim

Duc-Anh Luong says

Hi Jim,

Thank you so much for your very specific response. I think that it is very true when we interpret the model parameter. How’s about when we use the model with one or more non-statistically significant variables to make prediction? Sorry for my stupid questions!

Best regards,

Duc Anh

Jim Frost says

Hi Duc Anh,

I was referring to the case where you leave a predictor in the model when it is not significant. If you’re using the model to make predictions, you have the additional consideration of the precision of the predictions. Leaving an insignificant predictor in the model might reduce the precision.

What you want to do is to compare the predicted R-squared and width of the prediction intervals between the model with the insignificant predictors and the model with only significant predictors. Read my post about using regression to make predictions for more information!

And, there really is no such thing as a stupid question! ðŸ™‚

Jim