8.7 Designs with Simple Block Structure, 8.8 Analysis of Covariance

Exercises 3f, No. 2, in Section 3.7 gave the following two-stage method for finding least squares estimates for a general ANCOVA model $G$, say. First, we find the least squares estimate $\widehat{\mathbf{\beta }}$ of $\mathbf{\beta }$ for the model $\mathbb {E}\left[\mathbf{Y}\right] = \mathbf{X} \mathbf{\beta }$ and the residual sum of squares $\mathbf{Y}^{T} \mathbf{R} \mathbf{Y}$. Second, $\widehat{\gamma }_{G}$ is found by replacing $\mathbf{Y}$ by $\mathbf{Y}-\mathbf{Z} \mathbf{\gamma }$ in $\mathbf{Y}^{T} \mathbf{R Y}$ and minimizing with respect to $\mathbf{\gamma }$. This minimum value is the correct residual sum of squares for the model $G$. Third, referring to Theorem 3.6 , we see that $\widehat{\mathbf{\beta }}_{G}$ is obtained from $\widehat{\mathbf{\beta }}$ by replacing $\mathbf{Y}$ by $\mathbf{Y}-\mathbf{X} \widehat{\mathbf{\gamma }}$. This technique can be applied to the model for $H$ as well as $G$. Now apply this method to the following problem. Consider the model $Y_{i j} = \mu_{i}+\gamma_{i} x_{j}+\varepsilon_{i j}$, where $i = 1,2 ; j = 1,2, \ldots , J$; and the $\varepsilon_{i j}$ are independently distributed as $N\left(0, \sigma^{2}\right)$. Derive an $F$ statistic for testing the hypothesis that $\gamma_{1} = \gamma_{2}$ and show that this statistic is the square of the usual $t$-statistic for testing whether two lines are parallel.

Let $Y_{i j} = \mu_{i}+\gamma_{1} z_{i j}+\gamma_{2} w_{i j}+\varepsilon_{i j}$, where $i = 1,2, \ldots , I ; j = 1,2, \ldots , J$; and the $\varepsilon_{i j}$ are independently distributed as $N\left(0, \sigma^{2}\right)$.

Let $Y_{i j k} = \mu_{i j}+\gamma_{i j} z_{i j k}+\varepsilon_{i j k}$, where $i = 1,2, \ldots , I ; i = 1,2, \ldots , J$; $k = 1,2, \ldots , K$; and the $\varepsilon_{i j k}$ are independently distributed as $N\left(0, \sigma^{2}\right)$. Obtain a test statistic for testing the hypothesis