F-test

Let $M_0\subset M_1$ be nested normal linear models with $p_0,p_1$ independent coefficients and $n$ observations. Under the reduced model,The difference of the orthogonal projection matrices onto the two model spaces is an orthogonal projection of rank $p_1-p_0$, orthogonal to the full-model residual projection. Independent normal distribution projections therefore give independent chi-squared numerators and denominators after division by ${\sigma }^2$. This proves the ratio law. The test evaluates the extra terms conditional on all reduced-model terms, and estimates the unknown common variance from the full model.

An F-statistic is a ratio of two variance estimates. Under the null hypothesis of an F-test, suitably independent scaled sums of squares make this ratio follow an F-distribution.

For nested Gaussian linear models differing by $r$ parameters, the statistichas an $F_{r,n-p_1}$ distribution under the reduced model.

For nested normal linear models, the decrease in residual sum of squares obtained by adding coefficients is their extra sum of squares. Dividing by the number of added coefficients and by the full-model residual mean square gives the partial F-test. A sequential analysis of variance attributes this decrease according to the specified order of entry; it need not equal the partial comparison when other terms are retained.