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

Methods for optimal sample size determination are developed using four popular multiple comparison procedures (Scheffe's, Bonferroni's, Tukey's and Dunnett's procedures), where random samples of the same size n are to be selected from k (>/=2) normal populations with common variance sigma2, and where primary interest concerns inferences about a family of L linear contrasts among the k population means. For a simultaneous coverage probability of (1-alpha), the optimal sample size is defined to be the smallest integer value n*m such that, simultaneously for all L confidence intervals, the width of the lth confidence interval will be no greater than tolerance 2deltal (l=1,2,...,L) with tolerance probability at least (1-gamma), treating the pooled sample variance S2p as a random variable. Using Scheffe's procedure as an illustration, comparisons are made to usual sample size methods that incorrectly ignore the stochastic nature of S2p. The latter approach can lead to serious underestimation of required sample sizes and hence to unacceptably low values of the actually tolerance probability (1-gamma'). Our approach guarantees a lower bound of [1-(alpha+gamma)] for the probability that the L confidence intervals will both cover the parametric functions of interest and also be sufficiently narrow. Recommendations are provided regarding the choices among the four multiple comparison procedures for sample size determination and inference-making.