Probability Plots for Assessing Multivariate Normality

Abstract

A method of assessing multivariate normality by decomposition of measures of multivariate skewness and kurtosis into orthogonal components is described. The components can readily be inspected analytically, as well as graphically with probability plots. Examination of the individual components can often reveal departures from multivariate normality that may be masked in their underlying measures of skewness and kurtosis. There exist many techniques for assessing multivariate normality; see, for example, extensive reviews by Gnanadesikan (1977), Cox and Small (1978), Mardia (1980) and Koziol (1986). One graphical method devolves from the 'radius and angles' approach, which is based on the well-known transformation of Euclidean to spherical coordinates. Gnanadesikan (1977) in particular describes various informal tests for multivariate normal- ity with graphical procedures for examining the radii and angles following spherical coordinate transformation; Small (1978), Koziol (1982, 1983), Royston (1983), Szkutnik (1987) and Quiroz and Dudley (1991) among others have considered more formal procedures related to this approach. A different paradigm for assessing multivariate normality is based on the attributes of multivariate moments. In a series of seminal papers (Mardia, 1970, 1974, 1975; Mardia & Foster, 1983; Mardia & Kanazawa, 1983), Mardia introduced affine invariant measures of multivariate skewness and kurtosis, and comprehensively examined their properties under various circumstances. From a different perspective, Koziol (1986, 1987) considered measures of multivariate skewness and kurtosis suggested from the notion of Neyman's smooth tests; upon deriving smooth tests for assessing multivariate normality, he further found that the smooth test for multivariate skewness coincides with Mardia's measure of multivariate skewness, but the smooth test for multivariate kurtosis differed somewhat from Mardia's measure of multivariate kurtosis. Koziol noted that the smooth tests may be decomposed into individual components, which are distributed as independent %2 random variables, each with one degree of freedom. See also Mardia (1987) and Mardia and Kent (1991) for a unifying treatment devolving from Rao's score tests. The purpose of this note is to describe graphical techniques for depicting the components, and to demonstrate that examination of the individual components can be a valuable tool for assessing multivariate normality. For completeness, the smooth tests and their components are briefly reviewed in Section 2; examples of assessing multivariate normality with the components are found in