We consider the two way layout with factors $A$ and $B$, each with two levels. We denote $y_{ijk}$ as the response, with $k=1,\ldots,n$. Consider the model $$y_{ijk}=\eta + \alpha_i + \beta_j + w_{ij} + \epsilon$$ under the zero sum constraints: $\sum_{i=1}^2\alpha_i = \sum_{j=1}^2 \beta_j = \sum_{i=1}^2 w_{ij} = \sum_{j=1}^2 w_{ij}=0$ and we assume that $\epsilon\sim \mathcal{N}(0,1)$.
If $t_\alpha$ is the test statistic for testing $H_0:\alpha_1=0$ and $F_\alpha$ be the partial F-test statistic for testing $H_0:\alpha_1=\alpha_2=0$. How can I show that $t_\alpha^2$=$F_\alpha$?
I know that $t_\alpha=\frac{\hat{\alpha_1}}{\sqrt{\frac{SSR}{IJ(n-1)}}}$ with $I=J=2$ and that $F_\alpha = \frac{SS_\alpha/(I-1)}{SSR/(IJ(n-1))}$. Clearly the problem of $t_\alpha^2=F_\alpha$ simplifies to $\hat{\alpha_1} = SS_\alpha$, but $SS_\alpha=\sum_{ijk}\hat{\alpha_i}$ but I don't think this is equal to $\hat{\alpha_1}$. If anyone has any pointers on where I went wrong it would be greatly appreciated. Thanks!
