The Ancova model for comparing two groups: a tutorial emphasizing statistical distribution theory

The analysis of covariance (Ancova) is a widely used statistical technique for the comparison of groups with respect to a quantitative dependent variable in such a way that the comparison takes into account concomitant differences in a quantitative covariate. Despite its widespread use, some of the main features of this technique have remained elusive, contentious, or misconceived in applied settings. For example, some authors claim that the validity of an Ancova depends on the assumption that the expected value of the covariate is the same for all participants, or that the adjusted mean difference evaluated in an Ancova has a useful interpretation as the difference between the mean change scores of each group, whereas these claims are disputed by other authors. I suggest that these issues are best addressed and settled in the context of the underlying exact sampling distribution theory since significance statements, effect size estimates, and statistical power all derive directly from the statistical sampling distribution theory implied by the Ancova model. The distributional approach also clarifies the central distinction between conditional and marginal means, and the way in which various study designs (controlled, randomized, observational) affect and modify conclusions derived from an Ancova. The tutorial provides an explicit distributional account of the standard Ancova model to compare two independent groups; it clarifies the assumptions underlying the Ancova model, the nature and limitations of the conclusions it provides, and corrects some common misconceptions associated with its applications.