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A transformation of the Welch statistic to compare means is proposed to correct skewness and kurtosis of parent populations. The results show that this transformation seems to improve the performance of the test in heavy-tailed distributions more than other transformations focused only on skewness. The proposed test outperforms the Welch test in asymmetric heavy-tailed distributions with high heteroscedasticity and it behaves better than the Johnson’s transformation trimmed mean Welch test in normal, near-normal and light-tailed distributions. It may also be a better option when some of the distributions are heavy-tailed and some light-tailed.

Tables of optimal sample sizes are provided in this article for testing the equality of means of two normal populations when variances may be unequal. The Welch's Approximate t Test is used since it is increasingly recognized as the most practical solution to the Behrens–Fisher problem

Six procedures which convert tests of homogeneity of variance into tests for mean equality for independent groups are compared. The tests are the analysis of variance (ANOVA) and Welch F statistics. The Welch statistics are included since it was anticipated that ANOVA would not provide a robust test when samples of unequal sizes are obtained from non-normal populations. However, the Welch tests are not found to be uniformly preferrable. In addition, a prior recommendation for Miller's jackknife procedure is not supported for the unequal sample size case. The data indicates that the current tests for variance heterogeneity are either sensitive to non-normality or, if robust, lacking in power. Therefore, these tests cannot be recommended for the purpose of testing the validity of the ANOVA homogeneity assumption.

This paper examines the robustness of the Welch test, the James test as well as Tan's ANOVA test (to be referred as Fβ test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.