Multiple comparisons in the general linear model

Whereas multiple comparisons computations in a one-way model are well understood, multiple comparisons computations in a general linear model (GLM) are not. For models with the so-called “one-way structure,” no new technique is needed beyond proper substitution of terms. Examples of designs that guarantee a one-way structure include variance balanced designs and orthogonal designs. For models without a one-way structure, more sophisticated computational techniques are needed. Approximations based on the probabilistic inequalities of Bonferroni, Šidák, and Slepian are too conservative. Even the second-order Hunter—Worsley inequality is rather conservative. The so-called factor analytic approximation is quite accurate for multiple comparison with a control (MCC) and multiple comparison with the best (MCB), but conditions for it to be conservative are not known. This article describes a highly accurate, deterministic, conservative approximation that is applicable to a popular class of general linear models, and a fast, stochastic, conservative approximation that is generally applicable.