Multiple comparisons based on a modified one-step M-estimator

Although many methods are available for performing multiple comparisons based on some measure of location, most can be unsatisfactory in at least some situations, in simulations when sample sizes are small, say less than or equal to twenty. That is, the actual Type I error probability can substantially exceed the nominal level, and for some methods the actual Type I error probability can be well below the nominal level, suggesting that power might be relatively poor. In addition, all methods based on means can have relatively low power under arbitrarily small departures from normality. Currently, a method based on 20% trimmed means and a percentile bootstrap method performs relatively well (Wilcox, in press). However, symmetric trimming was used, even when sampling from a highly skewed distribution and a rigid adherence to 20% trimming can result in low efficiency when a distribution is sufficiently heavy-tailed. Robust M-estimators are more flexible but they can be unsatisfactory in terms of Type I errors when sample sizes are small. This paper describes an alternative approach based on a modified one-step M-estimator that introduces more flexibility than a trimmed mean but provides better control over Type I error probabilities compared with using a one-step M-estimator.