The procedures for testing hypotheses discussed in Chapters 8 and 9 apply to experiments in which the control and treatment groups contain different subjects (individuals). In contrast, it is often possible to design experiments in which each experimental subject can be observed before and after one or more treatments. Such experiments gain sensitivity because they make it possible to measure how the treatment affects each individual. When the control and treatment groups consist of different individuals, as in the experimental designs we have introduced so far, the changes brought about by the treatment may be masked by variability between individual experimental subjects. In this chapter we show how to analyze experiments in which each subject is observed repeatedly under different experimental conditions.
Experiments in which each subject receives more than one treatment are called repeated-measures designs and are analyzed using repeated-measures analysis of variance.* Repeated-measures analysis of variance is the generalization of a paired t test in the same sense that ordinary analysis of variance is a generalization of an unpaired t test. Repeated-measures analysis of variance permits testing hypotheses about any number of treatments whose effects are measured repeatedly in the same subjects. We will explicitly separate the total variability in the observations into three components: variability between individual experimental subjects, variability within each individual subject's response, and variability due to the treatments (Fig. 10-1). Then, depending on the experimental design, each of these components may be further subdivided into effects of interest for hypothesis testing.
Partitioning of the sums of squares for a single-factor repeated-measures analysis of variance. We first divide the variation into a between-subjects component that arises because of random differences between subjects and a within-subjects component. This procedure allows us to "discard" the variation between subjects and concentrate on the variation within experimental subjects to test hypotheses about the different treatment effects. The degrees of freedom partition in the same way.
In a repeated-measures design, we take into account the fact that different experimental subjects can have different mean responses because of between-subjects differences (as opposed to treatment effects). Doing so requires that we make assumptions in addition to those we already made to develop the test procedures for completely randomized designs in Chapters 8 and 9.
The treatment effects are fixed in the underlying mathematical model of the experiment. We assume that the treatments exert a fixed (but unknown) effect on the mean response of the dependent variable, and we use the data to estimate this fixed effect. In the regression implementation of analysis of variance, the regression coefficients associated with the treatment dummy variables provide the estimates of these fixed effects. As with earlier regression and analysis-of-variance models, we assume that