Scientific African (Sep 2022)
The Efficiency of Bartlett's Test using Different forms of Residuals for Testing Homogeneity of Variance in Single and Factorial Experiments-A Simulation Study
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 HOV and drawn conclusions based on the statistical power and weaknesses of those tests without considering the further increase in efficiency, especially for factorial ANOVA. This study was devoted to looking at different forms of residuals to improve the efficiency of Bartlett's test in single-factor and factorial ANOVA. The forms of residuals evaluated were; (i) the residuals, i.e., uij=Xij−X^ij, (ii) the absolute residuals, i.e., |uij|=|Xij−X^ij|, (iii) the square of residuals, i.e., uij2=(Xij−X^ij)2, and (iv) the square root of the absolute residuals, i.e., |uij|=|Xij−X^ij|. The consistency of each residual form was investigated under different scenarios, viz., different number of factor levels, different number of replicates, various mean differences, and various variance differences with respect to type I and type II errors through 1000 simulations. The study revealed that, the use of the absolute residuals was the best for the test statistic and it was consistent across all scenarios with both single factor ANOVA, and factorial ANOVA. Nevertheless, the use of the square of residuals was better than that of the residuals itself. However, it was observed that with all forms of residuals, the power of the test reduces consistently as the number of factors and levels of factors increase irrespective of the number of replicates. The study recommends that absolute residuals as a form of residuals can improve the efficiency of the Bartlett's test.
