Sample size determination for multiple comparison studies treating confidence interval width as random

Methods for optimal sample size determination are developed using four popu lar multiple comparison procedures (Scheffe's, Bonferroni's, Tukey's and Du nnett's procedures), where random samples of the same size n are to be sele cted from k(greater than or equal to 2) normal populations with common vari ance sigma(2), and where primary interest concerns inferences about a famil y of L linear contrasts among the k population means. For a simultaneous co verage probability of (1 - alpha), the optimal sample size is defined to be the smallest integer value n(m)* such that, simultaneously for all L confi dence intervals, the width of the lth confidence interval will be no greate r than tolerance 2 delta(t) (l = 1, 2, ..., L) with tolerance probability a t least (1 - gamma), treating the pooled sample variance S-p(2) as a random variable. Using Scheffe's procedure as an illustration, comparisons are ma de to usual sample size methods that incorrectly ignore the stochastic natu re of S-p(2). The latter approach can lead to serious underestimation of re quired sample sizes and hence to unacceptably low values of the actually to lerance probability (1 - gamma'). Our approach guarantees a lower bound of [1 - (alpha + gamma)] for the probability that the L confidence intervals w ill both cover the parametric functions of interest and also be sufficientl y narrow. Recommendations are provided regarding the choices among the four multiple comparison procedures for sample size determination and inference -making. Copyright (C) 1999 John Wiley & Sons, Ltd.