# Robustness of Multiple Comparisons Against Variance Heterogeneity

## Abstract

_{0}:μ

_{1}= … = μ

_{k}is rejected for normal populations with classical one way analysis of variance, it is usually of interest to know where the differences may be. If the population variances are equal there are several approaches one might consider:

- 1.
Least Significant Difference test (Fisher, 1935)

- 2.
Multiple Range test for equal sample sizes (Newman, 1939)

- 3.
An adaptation for unequal sample sizes (Kramer, 1956)

- 4.
Multiple F-test (Duncan, 1951)

- 5.
Multiple Comparisons test (Duncan, 1952).

For all these methods (including the one way analysis of variance) alternatives exist that are robust against variance heterogeneity. A modification of (3) has some unattractive properties if the variances and the sample size differ greatly. The adaptations for unequal variances of (4) and (5) seem better than (1) for cases with many samples. Test (2) is rather robust in itself if the variances are not too much different. Modifications exist that allow slight unequalities in the sample sizes.

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