Theory

 In the preceding sections, we have stayed relatively clear of mathematical notation. While it is hoped that the ideas are clear, it is probably best to write this all down notationally and generally in terms of the standard ANOVA model. The notation here is similar to that in Kuehl (1994), but similar developments can be found in most general texts in experimental design, such as Hinkelmann and Kempthorne (1994), Mason et al. (1989), Montgomery (1997), Kutner et al. (1996), or Winer (1991); or in more advanced linear-models texts like Christensen (1996), Graybill (1976), or Searle (1971).

Most design texts describe fixed-effects ANOVA models as a sum of effects that are constrained in various ways. For example, the two-factor model we have been discussing might be written as follows:

$$Y_{ijk} = \mu + \alpha_i + \beta_j + \alpha\beta_{ij} + E_{ijk}$$

where Y_ijk denotes the kth observation on the ith row and the jth column; $\mu, \alpha_i, \beta_j, \alpha\beta_{ij}$denote the grand mean and effects of rows, columns, and interaction, respectively; and E_ijk is the experimental error associated with Y_ijk; these are assumed to be mutually independent and normally distributed with mean 0 and variance $\sigma^2$. In the balanced design, subscripts i, j, and k range over the a rows, a columns, and n replications in the experiment.

The effects in this model are constrained to sum to zero over any subscript, holding all others fixed. Thus, in the above model, we assume that

$$\sum_{i=1}^{a}\alpha_i = 0, \quad \sum_{j=1}^{b}\beta_j = 0,$$

$$\sum_{i=1}^{a}\alpha\beta_{ij} = 0 \mbox{~~for each } j, \quad \sum_{j=1}^{b}\alpha\beta_{ij} = 0 \mbox{~~for each } i$$

These constraints make all the parameters uniquely identifiable.

The effect SDs are then defined as $$\begin{eqnarray*} \theta_\alpha &=& \sqrt{\sum_{i=1}^{a}\alpha_i^2\;/\;(a-1)} \... ...sum_{i=1}^{a}\sum_{j=1}^{b}\alpha\beta_{ij}^2\;/\;(a-1)(b-1)} \ \end{eqnarray*}$$

In general, a term $\tau_{ij\cdots p}$ in a fixed-effects model satisfies the constraint that $\sum_s \tau_{ij\cdots p} = 0$ over any subscript $s \in \{i,j,\ldots,p\}$, holding all other subscripts fixed. The effect SD for this term is defined as

$$\theta_\tau = \sqrt{\frac1\nu \sum_i\sum_j\cdots\sum_p \tau_{ij\cdots p}^2}$$

where $\nu$ is the associated degrees of freedom.