A problem of considerable practical importance when applying multiple comparison procedures is that unequal variances can seriously affect power and the probability of a Type I error. A related problem is getting a precise indication of how many observations are required so that the length of the confidence intervals will be reasonably short. Two-stage procedures have been proposed that give an exact solution to these problems, the first stage being a pilot study for the purpose of obtaining sample estimates of the variances. However, the critical values of these procedures are available only when there are equal sample sizes in the first stage. This paper suggests a method of evaluating the experimentwise Type I error probability when the first stage has unequal sample sizes.
Consider k normal distributions having means μ1,..., μk and variances σ21,..., σ2 k. Let μ[1]≥...≥ μ[ k] be the means written in ascending order. Dudewicz and Dalai proposed a two-stage procedure for selecting the population having the largest mean μ[ k] where the variances are assumed to be unknown and unequal. This paper considers an approximate but conservative solution for situations where unequal sample sizes are used in the first stage. The paper also considers how to estimate the actual probability of selecting the “best” treatment; that is, the one having mean μ[ k], after a heteroscedastic ANOVA has been performed.
APPROPRIATE CRITICAL VALUES FOR PAIRWISE COMPARISONS OF MULTIPLE COMPARISON PROCEDURES FOR NORMAL VARIANCES WHEN SAMPLE SIZES ARE UNBALANCED