| R. A. Fisher's analysis of variance procedure to test the homogeneity of population means and its post hoc analysis is one of the most commonly used methodologies in statistics. The major assumption of this procedure is normality of the underlying populations. If this assumption is valid, then the classical test is optimal. If the assumption is incorrect, then any results obtained would be dubious. The only alternative currently available which does not require the assumption of normality is the nonparametric rank-based approach, which requires only minimal assumptions, such as continuity of the populations. However, the non-parametric procedures actually compare distribution functions, rather than means. Additionally, these tests lack clear intuitive appeal and, unlike the normal theory statistic, have not been adapted to the case of heteroscedasticity.; This dissertation focuses upon the one-way classification and proposes an approach which allows for relaxation of the normality assumption. In this approach, meant to yield robust alternatives valid in a broad neighborhood of a target family, the sample means and variances in the classical statistic for testing homogeneity of several population means are replaced by robust estimators of location and their corresponding studentizing factors. The asymptotic distributions of the robust estimators and their studentizing factors are empirically adjusted for use with small to moderate size samples, using the normal population as the target family. Both homoscedastic and heteroscedastic cases are addressed, along with the issue of post hoc multiple comparisons. The properties of the resulting methodologies are studied with a variety of Monte Carlo experiments. The results show that the new approach provides methods which have excellent type I error control, behave reasonably well for a broad spectrum of populations, including the Cauchy family, in a neighborhood of normality, and have considerable power advantages over the normal theory methods in the case of non-normal, especially heavy tailed, populations. |
