The unbiased estimate of a "treatment effect" reached by analysis of covariance in a nonrandomized experiment would often require that a different covariate be used in each treatment. A sufficient but unlikely condition for an unbiased estimate is that the covariate for each treatment is (1) the complete covariate that predicts the outcome as fully as possible from initial characteristics of the case, or (2) the complete discriminant that fully represents differences between group means on the initial characteristics. When the covariate for a treatment is a weighted composite of (1) and (2), the adjusted outcome mean may have positive or negative basis, the bias being a nonlinear function of the weights. If (1) and (2) do not wholly account for the covariate, the adjustment to the outcome mean is reduced in absolute magnitude. A procedure is offered to correct for error of measurement when there are two or more fallible covariables. Specification errors as well as errors of measurement have an attenuating effect, however. Since the parameters of the specification errors are unknown, no correction procedures can be counted on to provide an unbiased estimate of the treatment effect. (Author/BW)