**Sankhya: The Indian Journal of Statistics**

1995, Volume 57, Series B, Pt. 2, pp. 200--222

**RANDOMIZATION TESTS TO COMPARE MEANS WITH UNEQUAL VARIATION**

By

BRYAN F. J. MANLY, *University of Otago*

SUMMARY. A randomization framework for testing for significant differences between the means of two or more samples is proposed for the situation where the samples may be from distributions with different variances. This framework is based on the concept that the observed data arise from a random allocation of a fixed set of values to the samples, followed by linear transformations that are not necessarily the same for each sample. The null hypothesis is that, with respect to the distributions generated by the random allocation, the expected values of the sample means are equal but the expected values of sample variances may or may not be equal.

The model leads in an obvious way to a randomization test that is exact if the parameters for the linear transformations are known. When these parameters are not known (as is usually the case) three algorithms for approximate randomizations are proposed. The properties of these algorithms have been studied by simulation in comparison with Welch's test, the usual randomization F-test, and the usual F-test using tables. It has been found that two of the three algorithms for approximate randomization tests have better properties than the other four tests when the null hypothesis is true., for data from uniform and normal distributions. None of the tests perform well with data from exponential distributions, but one of the approximate randomization tests is superior to all of the other tests under most of the conditions simulated.

*AMS (1991) subject classification.* 62G09.

*Key words and phrases*. Computer intensive statistics, permutation test, Behrens-Fisher problem, comparison of means, analysis of variance.