A BAYESIAN APPROACH TO THE SYMMETRIC MULTIPLE COMPARISONS PROBLEM IN THE TWO-WAY BALANCED DESIGN

LIWEN ALICE SUN, Purdue University

Abstract

A class of Bayes rules is constructed for the comparison of all pairs of means resulting from combinations of treatment levels in a balanced two-factor design. For each pair of means u(,ij), u(,i'j'), it is decided whether (eta) = u(,ij) - u(,i'j') is positive, negative, or zero; losses for incorrect decision are linear in (VBAR)(eta)(VBAR). When considered jointly, decisions made in the component pairwise comparison problems must be consistent with one another in the sense that they produce non-circular rankings of the cell means. The loss for incorrect decision in the overall problem is taken to be the sum of the component losses. Finally, the usual assumptions of independent normally distributed observations X(,ijk) with means u(,ij) = E(X(,ijk)) and common error variance (sigma)(,e)('2), 1 (LESSTHEQ) i (LESSTHEQ) r, l (LESSTHEQ) j (LESSTHEQ) c, l (LESSTHEQ) k (LESSTHEQ) K, are made. The prior distributions selected for this problem have the useful conjugate property that prior information about the magnitudes of contrasts for main effects and interactions in the u(,ij)'s, and for the magnitude of the variance (sigma)(,e)('2), are pooled with similar information obtained from the data when forming the joint posterior density of the means u(,ij) and (sigma)(,e)('2). Each Bayes rule for the overall multiple comparisons problem is shown to result from the simultaneous applications of the corresponding Bayes rules for all component pairwise comparison problems. A computer program for simultaneously implementing the component Bayes rules is constructed. An important feature of this program is the use of bounds on the posterior Bayes risk function which in many cases enable the action with minimum posterior risk to be chosen without having to explicitly calculate these posterior risks. Since exact calculation of the posteror risks requires numerical evaluation of complicated double and/or triple integrals, computer time is drastically reduced by use of these bounds. Nevertheless, the exact Bayes rules are expensive in terms of computer time. For this reason, a large-sample approximation to the Bayes rule is proposed which is much easier (and cheaper) to apply, and which performs similarly to the exact Bayes rule on a diversity of examples of data of moderate sample size (error d.f. = 15,30) taken from standard statistical textbooks.

Degree

Ph.D.

Subject Area

Statistics

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