The techniques of analysis of covariance are employed in three mathematically similar but conceptually very different kinds of problem. Examples of all three kinds arise in connection with the development of pharmaceutical products.

In the first case, a regression model is expected to fit the data well enough to serve as the basis for prediction. In testing the stability of a drug product, for example, the potency may be modeled as a linear function of time, and the possibility of different lines for different batches of the product needs to be allowed for. The purpose of the statistical analysis is to ensure, with a stated degree of confidence, that the potency at a given time will be within given limits.

The second and perhaps widest application of analysis of covariance is in observational studies, such as arise in the postmarketing phase of drug development. It may be desired, for example, to study the association of some outcome with exposure to a drug. It is necessary to adjust for covariates that may be systematically associated both with the outcome and with the exposure and so induce a spurious relationship between the outcome and the exposure. In such studies the unexplained variation is typically high, so the model is not expected to fit the individual observations well. It must, however, include all the important potential confounders and must have at least approximately the right functional form, if a causal relationship, or the absence of one, between the outcome and the exposure is to be inferred.

The third kind of application of analysis of covariance, although the first historically,1 is to randomized, controlled experiments such as clinical trials of the efficacy of new drugs. In such experiments, adjustment for covariates is optional in a sense, because the validity of unadjusted comparisons is ensured by randomization. Adjustments properly planned and executed, however, can reduce the probabilities of inferential errors and so help to control the size, cost, and time of clinical trials.

The modeling problem is straightforward, well covered in textbooks, and, strictly speaking, not a matter of “adjustment.” The observational problem, in contrast, is essentially intractable from the standpoint of formal statistical inference; but heuristic methods have had wide application and discussion. We focus here on the adjustment for covariates in the experimental setting. This problem has had relatively little attention in the literature, partly because early writings1 are largely complete, correct, and still sufficient. Unfortunately, the more recent literature on modeling and on observational studies has been misapplied to the experimental problem. Either a well-fitting model is thought to be required, as in the first problem, or the analysis is supposed to be heuristic, as in the second. In fact, a rigorous theory of analysis of covariance in controlled experiments can be developed, even in the absence of a good model for the covariate effects.