The two-sample trimmed t for unequal population variances

SUMMARY The effect of nonnormality on the Welch approximate degrees of freedom t test is demonstrated. A two-sample trimmed t statistic for unequal population variances is proposed and its performance is also evaluated in comparison to the Welch t test under normality and under long-tailed distributions. If the underlying distribution is long-tailed or contaminated with outliers, the trimmed t is strongly recommended. Some key word8: Behrens-Fisher problem; Robust test; t distribution; Trimming; Winsorization. Testing the equality of two means from independent samples is a common statistical problem. If the underlying distributions are normally distributed with equal population variances, it is well known that one would use the Student's t test. Unfortunately, this test statistic is sensitive to some nonnormal situations and that leads us to consider other more robust alternatives. Among the alternatives is the two-sample trimmed t, proposed and evaluated by Yuen & Dixon (1973); definitions of trimming will be given later. This statistic can be easily computed and its distribution is satisfactorily approximated by that of a Student's t with the degrees of freedom corresponding to the reduced sample. Results show that the loss of power efficiency in using trimmed t is small under exact normality, while the gain may be appreciable for long-tailed distributions. In cases where distributions are normal but population variances are unknown, the first exact solution was given by Behrens (1929) and later extended by Fisher (1939) as the fiducial solution. Among many others who have worked on this problem, Welch (1938, 1949) provided an approximate degrees of freedom t solution to his asymptotic series solution. Wang (1971) studied the probabilities of the type I errors of these two Welch solutions for selected cases and concluded that in practice, one can use the Welch approximate degrees of freedom t test without much loss of accuracy. For ease of reference in the rest of this paper, we shall refer to this approximate degrees of freedom t simply as the Welch t test. In this paper, the lack of robustness of the Welch t test when the underlying distribution is nonnormal is demonstrated. A two-sample trimmed t statistic for unequal population variances is suggested and its performance is evaluated in comparison to the Welch t test under normality and under long-tailed distributions.