When a treatment influences both the primary response and the covariate, a standard analysis of covariance may misrepresent the real treatment effect by adjusting out that part of the effect which is manifest in the covariate. What parametric function should we examine if the treatments influence the covariate? An informative analysis would let the complete effects of the levels emerge, yet permit the effects to be adjusted to some standard conditions. To satisfy these requirements when the treatments have only one factor, treatment means should be adjusted to covariate values determined externally from the data. If instead the treatment structure has at least two factors, say columns and rows, which, respectively, affect and do not affect the covariate, an interpretable adjustment procedure would involve the response means in each column being adjusted to the column mean of the covariate. This change alters the computations of covariance, so the proposed analysis cannot easily be accommodated by widely-available computer packages. An example concerns beef cattle from various herds which are fed several diets. Herds influence initial weight, which is the covariate, but diets cannot affect the covariate.