5.9 - Complexity Happens

For the simplest multi-factor treatment designs, those involving only two factors, we have now several situations to consider. With the possibility that either or both factors can be fixed or random effects in both crossed and nested designs, we end up with a lot of possibilities. The important thing to see here is that F tests will be constructed differently, depending on the nature of the treatment design. It emphasizes the need to look at expected mean squares (EMS) to determine the appropriate F test denominators.

Crossed
Source d.f. A fixed, B fixed A fixed, B random A random, B random
A a-1 \(\sigma^2+nb\frac{\sum\alpha_{i}^{2}}{a-1}\) \(\sigma^2+nb\frac{\sum\alpha_{i}^{2}}{a-1}+n\sigma_{\alpha\beta}^2\) \(\sigma^2 + nb\sigma_{\alpha}^2+n\sigma_{\alpha\beta}^2\)
B b-1 \(\sigma^2+na\frac{\sum\beta_{j}^{2}}{b-1}\) \(\sigma^2 + na\sigma_{\beta}^2\) \(\sigma^2 + na\sigma_{\beta}^2+n\sigma_{\alpha\beta}^2\)
A×B (a-1)(b-1) \(\sigma^2+n\frac{\sum\sum(\alpha\beta)_{ij}^{2}}{(a-1)(b-1)}\) \(\sigma^2 + n\sigma_{\alpha\beta}^2\) \(\sigma^2 + n\sigma_{\alpha\beta}^2\)
    \(\sigma^2\) \(\sigma^2\) \(\sigma^2\)
Nested
Source d.f. A fixed, B fixed A fixed, B random A random, B random
A a-1 \(\sigma^2+bn\frac{\sum\alpha_{i}^{2}}{a-1}\) \(\sigma^2+bn\frac{\sum\alpha_{i}^{2}}{a-1}+n\sigma_{\beta(\alpha)}^2\) \(\sigma^2 + bn\sigma_{\alpha}^2+n\sigma_{\beta(\alpha)}^2\)
B(A) a(b-1) \(\sigma^2+n\frac{\sum\sum\beta_{j(i)}^{2}}{a(b-1)}\)  \(\sigma^2 + n\sigma_{\beta(\alpha)}^2\)  \(\sigma^2 + n\sigma_{\beta(\alpha)}^2\)
Error ab(n-1) \(\sigma^2\) \(\sigma^2\) \(\sigma^2\)