Monte Carlo evaluation of the ANOVA's F and Kruskal-Wallis tests under binomial distribution

Abstract

To verify the equality of more than two levels of a factor under interest in experiments conducted under a completely randomized design (CRD) it is common to use the F ANOVA test, which is considered the most powerful test for this purpose. However, the reliability of such results depends on the following assumptions: additivity of effects, independence, homoscedasticity and normality of the errors. The nonparametric Kruskal-Wallis test requires more moderate assumptions and therefore it is an alternative when the assumptions required by the F test are not met. However, the stronger the assumptions of a test, the better its performance. When the fundamental assumptions are met the F test is the best option. In this work, the normality of the errors is violated. Binomial response variables are simulated in order to compare the performances of the F and Kruskal-Wallis tests when one of the analysis of variance assumptions is not satisfied. Through Monte Carlo simulation, were simulated $3,150,000$ experiments to evaluate the type I error rate and power rate of the tests. In most situations, the power of the F test was superior to the Kruskal-Wallis and yet, the F test controlled the Type I error rates.