The F-Test and F-Distribution › Summary

The F-test and F-distribution are fundamental tools for comparing variances and testing model adequacy:

The F-distribution:

• Arises from the ratio of two independent variance estimates
• Parameterized by two degrees of freedom (numerator and denominator)
• Always non-negative, right-skewed, centered near 1 under the null hypothesis
• Converges to symmetry as degrees of freedom increase

F-test for comparing variances:

• Tests $H_0: \sigma_1^2 = \sigma_2^2$
• Statistic: $F = s_1^2 / s_2^2$ (larger variance in numerator)
• Sensitive to non-normality; prefer Levene's test in practice

F-test in regression:

• Overall significance: Tests whether any predictors have explanatory power
• Decomposes total variance into explained and unexplained components
• Large F indicates model explains more variation than expected by chance

Nested model comparison:

• Tests whether additional predictors significantly improve a model
• Compares reduction in residual variance to residual variance of full model
• Essential for variable selection and model building

Practical guidance:

• For comparing variances: Use Levene's test (less sensitive to non-normality) or visual inspection
• For regression: The F-test from standard output tests overall model significance
• For model selection: Nested model F-tests guide inclusion of predictors