Why is the Tukey method not appropriate for covariance analysis when estimating treatment effects?

Question

In the book 'Applied Linear Statistical Models' chapter 22 it states that the Tukey method is not appropriate for covariance analysis. The context is in estimation of treatment effects. Instead, when a family of interval estimates is desired, the authors recommend using either the Scheffe or the Bonferroni multiple comparison procedures.

Why does the presence of a covariate make the Tukey method an inappropriate choice? In the text book example on page 930, the analyst wished to obtain all pairwise comparisons so and they proceeded with the Scheffe procedure. I am not sure why Tukey is inappropriate in this instance.

Answer 1

It's not the presence of a covariate that makes the Tukey method inappropriate, it's the fact the book's authors formulate the covariance model using sum contrasts. With sum contrasts, as opposed to treatment contrasts, you're not simply comparing group means. You're comparing a group mean to the grand mean of the group means. This page does a nice job of explaining sum contrasts, or as they call it, "deviation coding". Hence the need for the Scheffe procedure, which generalizes to all possible contrasts, not just pairwise comparisons of means. The Tukey procedure is specifically for pairwise comparisons of means.

It's not clear to me why the authors specified the ANCOVA model using sum contrasts. I suppose it's because they're orthogonal. I think it would have been better if they said "The Tukey method is not appropriate for covariance analysis as we have defined it." I see no reason why you can't use treatment contrasts and follow-up with a pairwise comparison of means.