A Bayesian Approach to the One Way ANOVA under Unequal Variance

This study involves testing the equality of several normal means under unequal variances, which is the setup of one-way analysis of variances (one-way ANOVA). Several tests are available in the literature, however, most of them perform poorly in terms of type I error rate under unequal variances. In fact, Type I errors can be highly inflated for some of the commonly used tests, a serious issue that seems to have been overlooked. Even though several tests have been proposed to overcome the problem, most of them show difficulty in calculation. Accordingly, the test for ANOVA with estimation of parameters using Bayesian approach is proposed as an alternative to such tests. The proposed test is compared with four existing tests such as the original test, James's test, Welch's test and the parametric bootstrap (PB) test. Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our results indicated that the performance of the proposed test is superior to the original test and is comparable to James's test, Welch's test and the PB test, controlling Type I error rate quite well and showing high power of the test. Our study suggested that the proposed test has high performance and should be used as an alternative to the four existing tests due to its simple formula.