| The problem of testing equality of several normal means when the variances are unknown and arbitrary is considered. Even though several tests are available in the literature, none of them perform well in terms of Type I error probability under various sample size and parameter combinations. In fact, the Type I errors can he highly inflated for some of the commonly used tests, a serious issue that appears to have been overlooked. We propose a parametric bootstrap (PB) approach and compare it with four existing location-scale invariant tests---the Welch test, the Johansen test, the James second-order test and the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test is the best among the five tests with respect to Type I error rates. The PB test performs very satisfactorily even for small samples while the Welch test, the Johansen test and the GF test exhibit poor Type I error properties when the sample sizes are small and/or the number of means to be compared is moderate to large. The James second-order test performs better than the Welch test, the Johansen test and the GF test. It is also noted that the same tests can he used to test the significance of the random effect variance component in a one-way random effect model under unequal error variances. Such models are widely used to analyze data from inter-laboratory studies. The proposed PB method is also extended to test equality of several normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. Comparison of the PB test with other existing tests show that the former is superior to others in terms of controlling Type I errors. All the methods are illustrated using practical examples. |
