Ftests

F-tests are statistical procedures that use the F-distribution to assess hypotheses about population variances or linear models. They are most commonly employed in analysis of variance (ANOVA) and in linear regression to test whether a set of model terms contributes significantly to explaining variation in the data.

The F-statistic is a ratio of two mean squares. In ANOVA, F = MS_between / MS_within, where MS_between reflects variation explained by group means and MS_within reflects unexplained variation. In regression, F = MS_regression / MS_residual, where MS_regression measures explained variance by the predictors and MS_residual is the unexplained variance. Under the null hypothesis and standard assumptions, the statistic follows an F-distribution with df1 and df2 equal to the numerator and denominator degrees of freedom.

Hypotheses: In regression, H0: a set of regression coefficients is zero; in ANOVA, H0: all group means are equal.

Calculation and interpretation: Given data, compute the relevant sums of squares, derive the mean squares, form F, and obtain a p-value from the F-distribution. A small p-value leads to rejection of H0, implying that the model or the group means explain a significant portion of variation.

For model selection, F-tests compare nested models using the statistic F = (SS_full - SS_reduced)/df_diff divided by SS_full/df_full. In experimental design, they test main effects and interactions.

Assumptions and limitations: You need independent observations, normally distributed residuals, and equal variances across groups. Violations can undermine the test; alternatives include Welch's test or nonparametric methods.