Ch. 25. Normal theory methods and their simple robust analogs for univariate and multivariate linear models

Abstract

The assumptions of univariate and multivariate normality inherent in most of the statistical procedures in routine use are rarely verified in practice. Several robustness studies have shown that that the failure of the underlying assumption of normality leads to either failure of type I error control, or to conservative test procedures. In univariate case failure of normality assumption is not uncommon, and in practice multivariate normality is elusive enough to be considered illusory. Alternatives to the normal theory based methods are nonparametric and robust methods. It is generally recognized that the nonparametric methods, which involve minimal assumptions, can be less efficient, and although the nonparametric test procedures are well developed for the univariate setting, their development for the multivariate setting is unsatisfactory. On a similar note, the development of the robust procedures in the univariate setting is more advanced than in the multivariate setting. Also, the multivariate nonparametric and robust test procedures are asymptotic in nature with a few exceptions. In this manuscript we focus on a few cases of the univariate and multivariate linear models which are of practical importance and present an overview of some relevant robust test procedures, which can be reasonably used with moderate size samples.