Conditional Probabilities of Rejecting H_0 by Pooled and Separate-Variances t Tests Given Heterogeneity of Sample Variances

摘要

It is known that the Type Ⅰ error probability of the Student t test is spuriously elevated or depressed by unequal variances combined with unequal sample sizes and that the Welch separate-variances version of the t test usually eliminates these effects. The present study found conditional probabilities of rejecting the null hypothesis, for both significance tests, given various conditions on the sample variances. The conditional probability of a Type Ⅰ error, given that sample variances are nearly equal, is also elevated or depressed, sometimes to an even greater extent than the unconditional probability. For various combinations of sample sizes and variance heterogeneity, similar results characterize the Welch t test. These findings imply that researchers cannot protect the significance level and power of the t test by deciding whether to use a pooled-variance or separate-variances version based solely on inspection of sample data.