8.5 Two-Way Classification (One Observation Per Mean): 8.5.1 Underlying Assumptions, 8.6 Higher-Way Classifications with Equal Numbers Per Mean: 8.6.1 Definition of Interactions, 8.6.2 Hypothesis Testing, 8.6.3 Missing Observations

Verify (8.28).

A three-factor experiment has population means $\mu_{i j k}(i = 1,2,3 ; j = 1,2,3 ; k = 1,2$), given by the following tables:

Show that the $A B C$ interactions are zero.

Let

where $\mathbf{X}$ has full rank. Suppose that the observations $\mathbf{Y}_{2}$ are missing. For the data observed, the least squares estimate of $\mathbf{\beta }$ is $\widehat{\mathbf{\beta }} = \left(\mathbf{X}_{1}^{T} \mathbf{X}_{1}\right)^{-1} \mathbf{X}_{1}^{T} \mathbf{Y}_{1}$. Let $\widehat{\mathbf{Y}}_{2} = \mathbf{X}_{2} \widehat{\mathbf{\beta }}$. Show that $\widehat{\mathbf{\beta }}$ can be obtained by applying least squares to the observed data augmented by $\widehat{\mathbf{Y}}_{2}$ and the regression matrix $\mathbf{X}$.

Suppose that we have a one-way classification

and suppose that $Y_{I J}$ is missing. Prove that the appropriate estimate of $Y_{I J}$ is the mean of the remaining observations on the mean $\mu_{I}$.