Analysis of covariance (Ancova) can reduce variability due to individual differences, thereby yielding narrower confidence intervals and greater power. Prime opportunities arise with experiments that compare effectiveness of methods to produce some change: change in health, behavior, attitude, or skill. The change score, after–before, may seem the natural measure; this change score seems to adjust for individual differences in the before score. But change scores turn out to be full of pitfalls. Ancova not only avoids these pitfalls, but gives a superior analysis.
Furthermore, Ancova is more general than the change score. Ancova can utilize any correlate of the main response measure, even correlates on a different dimension for which a change score makes no sense.
Ancova rests on a simple idea: Combine Anova and regression. Let X denote an individual difference variable and Y the response measure. Run a regression of Y on X and get Ypred as shown in Chapter 9. Since Ypred is predictable error variability, (Y − Ypred) is less variable than Y alone, for it subtracts out that part of Y that is predictable from X. Take advantage of this lesser variability by (in effect) applying Anova to (Y − Ypred).
This chapter aims at a conceptual understanding of Ancova. Formulas and calculations, accordingly, are largely passed over in order to discuss empirical issues. With randomized subject groups, Ancova has considerable potential that seems to have been relatively neglected. If you can find an individual difference variable that is correlated with your response measure, you can decrease your response variability and increase your power, often at little cost.
With nonrandom subject groups, matters are very different. Nonrandom groups differ systematically on individual difference variables before the experiment begins. These systematic differences are confounded with experimental treatments. Many writers assert that Ancova “controls” or “partials out” these individual differences, thereby removing otherwise deadly confounds. Such assertions have little truth. The second part of this chapter shows why Ancova generally fails to “control” with nonrandom groups.