6.7 - Complexity Happens
6.7 - Complexity HappensFrom what we have discussed so far we see that even for the simplest multi-factor studies (i.e. those involving only two factors) there are many possibilities of treatment designs; each factor is either fixed or random as well as either crossed or nested.
For any of these possibilities, we can carry out the hypothesis tests using the EMS expressions to identify the correct denominator for the relevant F-statistics. The possible EMS expressions are summarized in the following tables.
| Crossed | ||||
|---|---|---|---|---|
| Source | d.f. | A fixed, B fixed | A fixed, B random | A random, B random |
| A | a-1 | \(\sigma^2+nb\frac{\sum\alpha_{i}^{2}}{a-1}\) | \(\sigma^2+n\sigma_{\alpha\beta}^2+nb\frac{\sum\alpha_{i}^{2}}{a-1}\) | \(\sigma^2 + nb\sigma_{\alpha}^2+n\sigma_{\alpha\beta}^2\) |
| B | b-1 | \(\sigma^2+na\frac{\sum\beta_{j}^{2}}{b-1}\) | \(\sigma^2 + n\sigma_{\alpha\beta}^2+ na\sigma_{\beta}^2\) | \(\sigma^2 + na\sigma_{\beta}^2+n\sigma_{\alpha\beta}^2\) |
| A×B | (a-1)(b-1) | \(\sigma^2+n\frac{\sum\sum(\alpha\beta)_{ij}^{2}}{(a-1)(b-1)}\) | \(\sigma^2 + n\sigma_{\alpha\beta}^2\) | \(\sigma^2 + n\sigma_{\alpha\beta}^2\) |
| Error | ab(n-1) | \(\sigma^2\) | \(\sigma^2\) | \(\sigma^2\) |
| 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) | \(\sigma^2\) | \(\sigma^2\) | \(\sigma^2\) |