An Extension of the T-Method to General Unbalanced Models of Fixed Effects

Abstract

1. Introduction and Summary Among the various procedures for simultaneous interval estimation, Tukey's T-method gives the shortest confidence intervals for pairwise contrasts when it is applicable. (Cf. Scheffé, 1959, Ch. 3; Miller, 1966, Ch. 2.) However, the T-method in its exact formulation (excluding one way anova, see Spjøtvoll and Stoline, 1973) is restricted to well-balanced designs. In this paper we discuss a generalized T-type procedure for simultaneous interval estimation of all estimable parametric functions in general fixed effects, univariate linear models. The new method, which we subsequently abbreviate as the GT method, is an extension of a generalized T-procedure for the unbalanced one-way anova model given in Spjøtvoll and Stoline (1973). In Section 2 we derive the new procedure in full-rank and in less than full-rank models. Some important features of the procedure are discussed in Section 3. In order to gain insight into the types of models where the new procedure is expected to produce satisfactory confidence intervals for pairwise contrasts, we illustrate, in Section 4, the application of the GT method in some unbalanced designs. In Section 5 we discuss a modified GT procedure which constitutes a combination of the GT1 in Hochberg (1974) and the GT methods. The proofs of some of the theorems are given in an Appendix for purposes of a convenient reading.