Abstract:

Suppose random samples are drawn from each of n populations with unknown means and variances. Developing procedures to test the claim that the population variances are equal (homogeneity of variances) has frequently been a goal for statisticians. One widely used technique is due to Bartlett (1937) and is based on a slight modification of the generalized NeymanPearson likelihood ratio test. It is well known, however, that Bartlett's test is highly sensitive to the assumption that the underlying populations are normal (Box, 1953). If nonnormality is present, the use of Bartlett's test can often lead to the rejection of the claim of equal variances far more often than prescribed by its nominal level of significance. In other words, in this situation Bartlett's test is very conservative. In an attempt to circumvent this problem, several "robust" (relatively insensitive to nonnormality) tests for homogeneity of variances have been developed. The price paid for the robustness is less power than Bartlett's test when the populations are, in fact, normal. Consequently, it would seem appropriate to precede the possible use of Bartlett's test with a test for normality of the populations. If after such a test one concludes the normality assumption is not unreasonable, then Bartlett's test could follow. Otherwise, a robust test should probably be used. There are, at present, several tests for normality; however, the one in vogue is the ShapiroWilk (1965) test. This test, like its competitors, is designed to detect nonnormality in a variety of distributions and for moderate to large sample sizes. For sample sizes less than 10, there does not appear to be any reasonably powerful test for normality. As a matter of fact, when sample sizes are small, the power of the robust tests for homogeneity of variances is suspect. 