= Partial F-test for nested linear models
{title2=$F=\frac{(\operatorname{RSS}_0-\operatorname{RSS}_1)/(p_1-p_0)}{\operatorname{RSS}_1/(n-p_1)}$}
= Partial F-test
{synonym}
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,
$$
F=\frac{(\operatorname{RSS}_0-\operatorname{RSS}_1)/(p_1-p_0)}{\operatorname{RSS}_1/(n-p_1)}\sim F_{p_1-p_0,n-p_1}.
$$
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.
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