Ranking, estimation and hypothesis testing in unbalanced two-way additive models - a bayesian approach

For the usual 2-factor additive model, there has been comparatively little work in the ranking and selection literature for the case of unequal sample sizes (unequal variances). The existing papers (e.g., Huang and Panchapakesan (1976), Dudewicz (1977), Gupta and Hsu (1980), Taneja and Dudewicz (1982), and Bechhofer and Dunnett (1987)) do not give explicit procedures unless assuming an equal number of observations and equal variances. (In the 2 x 2 case, classical ranking and selection procedures do exist even for more complicated models. See Taneja and Dudewicz (1984, 1987).) However the case of unequal sample sizes may arise in many natural settings, say in the problem of designing an experiment for comparing treatments in the presence of blocks of different fixed sizes, where one may assign an equal number of experimental units to each treatment within the same block. Unequal sample sizes can be handled in the classical analysis of variance (AOV) model (see Bishop and Dudewicz (1978, 1981) and Dudewicz and Bishop (1981)), which may partly explain the popularity of AOV. A Bayesian approach to the problem is taken here, leading to computation of the posterior probabilities that each treatment mean is the largest. In addition, a Bayesian version of AOV will be considered. Calculation of the quantities of interest involves, at worst, 5-dimensional numerical integration, for which an efficient Monte Carlo method of evaluation is given. An example is presented to illustrate the methodology.