How do I interpret the ANOVA F-test in a multiple regression?

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AcadiFi · Accepted Answer

The ANOVA F-test evaluates the joint significance of all slope coefficients in a multiple regression. The null hypothesis states that every slope is simultaneously zero, meaning the independent variables together explain nothing.

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The statistic is F = MSR / MSE, where MSR is the mean regression sum of squares (RSS/k) and MSE is the mean squared error (SSE/(n-k-1)). With k=4 predictors and n=80 observations, an F of 18.4 is compared against F(4, 75) at your chosen significance level. A p-value of 0.0001 means the probability of seeing this F under the null is 0.01%, so you reject joint insignificance.

This differs fundamentally from t-tests. Individual t-stats tell you whether a single slope is different from zero holding others constant, but multicollinearity can inflate standard errors and make every t-stat insignificant while the overall F remains large. Seeing a significant F with no significant t-stats is a classic multicollinearity red flag.

In our example, imagine regressing Brindlewood Corp's quarterly revenue on GDP growth, consumer sentiment, interest rates, and unemployment. The F-test tells you the macro model as a whole has explanatory power. You would then inspect adjusted R-squared and individual coefficients to determine which variables contribute most.

Always pair the F-test with residual diagnostics before trusting the conclusion.

How do I interpret the ANOVA F-test in a multiple regression?

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