The Factor Analytic Approach to Simultaneous Inference in the General Linear Model

Abstract

Consider the general linear model (GLM) Y = Xβ + ε. Suppose Θ₁,…, Θ_k, a subset of the β's, are of interest; Θ₁,…, Θ_k may be treatment contrasts in an ANOVA setting or regression coefficients in a response surface setting. Existing simultaneous confidence intervals for Θ₁,…, Θ_k are relatively conservative or, in the case of the MEANS option in PROC GLM of SAS, possibly misleading. The difficulty is with the multidimensionality of the integration required to compute exact coverage probability when X does not follow a nice textbook design. Noting that such exact coverage probabilities can be computed if the correlation matrix R of the estimators of Θ₁, …, Θ_k has a one-factor structure in the factor analytic sense, it is proposed that approximate simultaneous confidence intervals be computed by approximating R with the closest one-factor structure correlation matrix. Computer simulations of hundreds of randomly generated designs in the settings of regression, analysis of covariance, and unbalanced block designs show that the coverage probabilities are practically exact, more so than can be anticipated by even second-order probability bounds.