Partial F-test for nested linear models ($F=\frac{({\mathrm{RSS}}_0-{\mathrm{RSS}}_1)/(p_1-p_0)}{{\mathrm{RSS}}_1/(n-p_1)}$)

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.