Welch's Approximate Solution for the Behrens–Fisher Problem

Abstract

The Wald, likelihood ratio, and score statistics have been calculated for the Behrens–Fisher problem. These statistics provide asymptotically equivalent and weakly optimal tests. The test based on the Wald statistic was found to be very similar to the V test discussed by Welch (1937); hence the V test was compared with the likelihood ratio and score tests in a power study. Since the powers were very similar and the V test was more convenient to use from several points of view, we recommend this test. Sizes of the V test, as well as a power comparison with the usual pooled t test, were also obtained.

KEY WORDS:

• Welch test
• Likelihood ratio
• Score statistic
• Size
• Power