A computer program to compute Scheffé’s homogeneity of variance statistic and estimate its power

Abstract

Many statistical tests assume homogeneity of variance, and may be affected by the violation of this assumption. This is particularly the case with analysis of variance for unequal cell size. For this reason, a powerful and robust test to detect heterogeneity of variance is de­ sirable. Unfortunately, most commonly used tests of homogeneity of variance are sensitive to departures from normality, especially with respect to departures from nonzero kurtosis (Box, 1953). A test that has been shown to be robust to violation of distribution assump­ tions is the Scheffe test, described by Winer (1971, pp.219·220). In the Scheffe test, each treatment group is randomly divided into a number of subsamples, and their variances are computed. The logarithms of the variances are taken, and an analysis of variance is performed. Two major drawbacks exist with the Scheffe pro­ cedure. First, because the grouping of subsamples is random, the test does not yield a unique solution. This has led some to reject it as unacceptable (Brown & Forsythe, 1974). Second, the test generally has poor power to detect existing heterogeneity of variance. Description. The present program uses a Monte Carlo sampling procedure to minimize the uniqueness problem and to provide information on the power to detect a specified degree of heterogeneity of variance. Essentially, the program computes the F statistic for a randomly generated group of subsamples and repeats this process a specified number of times, each time operating on a newly generated set of subsamples. The size of the loop is presently set at 100. In this manner, a mean F ratio, based on the series of replications, provides a stable estimate of the F ratio, with reduced influence from random subsample formation. To provide information on power to detect a given level of hetero­ geneity of variance, a series of runs can also be performed on a set of internally generated normal random deviates with a built-in degree of variance heterogeneity. In this way, a simulation, conforming to a researcher's design in terms of sample size and number of treatment groups, yields an empirical estimate of power to detect a degree of heterogeneity of variance. The number of replications performed to provide an estimate of power is currently set at 500.