Modifications to Bartlett’s and Levene’s tests for homogeneity of variance in factorial experiments

Abstract

In numerous investigations, whether practical or theoretical, a requirement for the Analysis of Variance (ANOVA), is that the variances of the measured groups are equal. Many researchers have compared various tests for testing HOV and drawn conclusions based on the statistical power and weaknesses of those tests without considering the consistency and further increase of efficiency of them, especially in factorial ANOVA. This thesis was devoted to looking at the following based on residuals: (1) compare the efficiency of some fourteen (14) HOV tests for testing Homogeneity of Variance, (2) improve the efficiency of the most widely and commonly used tests (i.e., Bartlett’s and Levene’s tests), by using residuals in different forms in single-factor and factorial ANOVA, (3) improve the efficiency by having modifications to the Bartlett’s and Levene’s tests, and (4) investigate consistency under suggested approaches.. All objectives were looked at through thousand simulations. The simulations were conducted under different scenarios such as different number of factor levels, different number of replicates, various mean differences, and various variance differences, to validate the proposed modifications. With regard to different forms of residuals, it was revealed that, the use of the absolute residuals was the best for both tests statistics irrespective of the increase in number of groups and replicates with regards to type I and type II error rates. The outcome was consistent with single factor ANOVA, and factorial ANOVA. Nevertheless, the use of the square of residuals were better than that of the residuals itself. The pattern of improvement in type I and type II error rates was consistent under each consideration.The results showed that, with the correction factor, even with 4 replicates, irrespective of the number of groups, one could reduce type I and type II error rates significantly even with three factor factorial ANOVA. The pattern of change in type I and type II errors, with or without the correction factor was the same as the conventional test of both Bartlett’s and Levene’s tests.