Partial F-Test

Definition

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ and the hypothesis $H_0:{\beta }_r={\beta }_{r+1}=\cdots ={\beta }_{p-1}=0$

Define the full model as $Y_i={\beta }_0+{\beta }_1{X}_{i1}+{\beta }_2{X}_{i2}+\cdots +{\beta }_{p-1}{X}_{i,p-1}+{ϵ}_i,i=1,2,\dots ,n$ and under $H_0$, define the reduced model as $Y_i={\beta }_0+{\beta }_1{X}_{i1}+{\beta }_2{X}_{i2}+\cdots +{\beta }_{r-1}{X}_{i,r-1}+{ϵ}_i,i=1,2,\dots ,n$

We reject $H_0$ if $SSR(F)-SSR(R)$ is large, where $SSR(F)$ and $SSR(R)$ are the regression sum of squares of the full and reduced model, respectively. By the equality $SST=SSE(F)+SSR(F)=SSE(R)+SSR(R)$, we can test the $SSE(R)-SSE(F)$ instead of the $SSR(F)-SSR(R)$

$\frac{SSE(R)-SSE(F)}{{\sigma }^2}\sim {\chi }^2(p-r)$ and $\frac{SSE(F)}{{\sigma }^2}\sim {\chi }^2(n-p)\sim {\chi }^2(n-p)$. Therefore, $F=\frac{(SSE(R)-SSE(F))/({\text{df}}_R-{\text{df}}_F)}{SSE(F)/{\text{df}}_F}\sim F(p-r,n-p)$ where ${\text{df}}_R=n-r$ and ${\text{df}}_F=n-p$

Hence, we reject $H_0$ if $F>F_{\alpha }(p-r,n-p)$ and we call it partial F-test