Anova of Residuals and Ancova: Correcting an Illusion by Using Model Comparisons and Graphs

Analysis of covariance is often conceptualized as an analysis of variance of a single set of residual scores that are obtained by regressing the dependent variable on the covariate. Although this conceptualization of an equivalence between the two procedures may be intuitively appealing, it is mathematically incorrect. If residuals are obtained from the pooled within-groups regression coefficient ( b w ), an analysis of variance on the residuals results in an inflated α-level. If the regression coefficient for the total sample combined into one group ( b T ) is used, ANOVA on the residuals yields an inappropriately conservative test. In either case, analysis of variance of residuals fails to provide a correct test, because the significance test in analysis of covariance requires consideration of both b w and b T , unlike analysis of residuals. It is recommended that the significance test of treatment effects in analysis of covariance be conceptualized, not as an analysis of residuals, but as a comparison of models whose parameters are estimated by the principle of least squares. Focusing on model comparisons and their associated graphs can be used effectively here as in other cases to teach simply and correctly the logic of the statistical test.