 Partial F-test for nested linear models

Let $M_0\subset M_1$ be nested normal linear models with full-rank design matrices of ranks $p_0,p_1$, where $0\le p_0<p_1<n$, 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.