Testing equality of means in one-way ANOVA using three and four moment approximations

Abstract

In this study, we focus on two test statistics for testing the equality of treatment means in one-way analysis of variance (ANOVA). The first one is the well known Cochran ($C_{LS}$) test statistic based on least squares (LS) estimators and the second one is robust version of it ($RC_{MML}$) based on modified maximum likelihood (MML) estimators. These two test statistics are asymptotically distributed as chi-square. However, distributions of them are unknown for small samples. Therefore, three-moment chi-square and four moment $F$ approximations to the null distributions of $C_{LS}$ and $RC_{MML}$ are derived inspired by Tiku and Wong [19]. To investigate the small and moderate sample properties of these tests based on the mentioned approximations, an extensive Monte-Carlo simulation study is performed when the underlying distribution is long-tailed symmetric (LTS). Simulation results show that four-moment $F$ approximation provides better approximation than the three-moment chi-square approximation for both $C_{LS}$ and $RC_{MML}$ tests. Therefore, the simulated Type I error rates and powers of the $C_{LS}$ and $RC_{MML}$ test statistics are calculated using four-moment $F$ approximation. According to simulation results, $RC_{MML}$ test is more powerful than the corresponding $C_{LS}$ test.

Keywords

• Cochran test statistic
• three moment chi-square approximation
• four-moment F approximation
• Monte Carlo simulation

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