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How do you describe, or what do you call, a test that uses two factors as independent variables but uses a dependent variable that is a difference between two measures taken repeatedly from the same individuals?

Usually, repeated measures means that multiple independent variables are measured on the same individuals. Here, it is the dependent, outcome variable that is the repeated measure. But, since the difference is taken for each individual, is this simply a two-way ANOVA?

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It could be called a change score design.

If you have only two repetitions of the measure, then it is one reasonable choice. However, it is very good to have more than two repetitions. If the dependent variable is measured with error (and which ones aren't?) then the change score is partly due to statistical error. e.g. suppose two people have identical true scores at both time 1 and time 2. But, for random reasons, they won't score the same at time 1 or time 2. Indeed, since

$O = T + E$

where O is observed score, T is true score and E is error, then, if $T_{11} = T_{21}$ and $T_{12} = T_{22}$ (where $T_{ij}$ is the score for person i at time j) then the change scores are just the changes in the errors and it's all regression to the mean.

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  • $\begingroup$ How about in the case where the two separate measures of the dependent variable are to measure an experimental effect that takes place in between measures? This is more akin to a paired t-test, but in this example there are two factors (or more). Most discussions/introductions to repeated measures ANOVA that I have found are about cases in which a dependent variable is predicted by a few to several independent variables that are measured on the same set of individuals (so the values are correlated within individuals). $\endgroup$ – ESS Oct 16 '13 at 1:07
  • $\begingroup$ That case isn't really different than the one I described. RM ANOVA is a generalization of a paired t-test but it makes some unrealistic assumptions such as sphericity. $\endgroup$ – Peter Flom Oct 16 '13 at 10:09
  • $\begingroup$ What I don't understand yet is this: In the paired t-test, it is an outcome variable that is "paired" or "correlated." In the case I described, it is the outcome variable that is correlated. In RM ANOVA, the outcome variable is predicted by independent variables, which are correlated. That is a different situation, no? $\endgroup$ – ESS Oct 16 '13 at 14:15
  • $\begingroup$ I do see the parallel between the paired t-test and RM ANOVA, that error variance is minimized. It is that aspect that generalizes. However, they differ in that paired t-test focuses on the differences between an outcome variable, whereas RM ANOVA focuses on repeated measures of predictor variables. $\endgroup$ – ESS Oct 16 '13 at 14:15
  • $\begingroup$ RM ANOVA can certainly be used when the DV is repeated. It generalizes the paired t because it is about more than 2 repeats. In the paired t-test the difference in the two outcomes is predicted by a single two-level categorical variable. $\endgroup$ – Peter Flom Oct 16 '13 at 14:18

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