Some Techniques for Assessing Multivarate Normality Based on the Shapiro‐Wilk W

SUMMARY Shapiro and Wilk's (1965) W test is a powerful procedure for detecting departures from univariate normality. The present paper extends the application of W to testing multivariate normality, and also to Healy's (1968) test ba'sed on squared radii. Three examples illustrate the approach, and also the utility of careful scrutiny of lower- dimensional subsets of the data where otherwise unsuspected departures from normality may appear. A fair number of test procedures for multivariate normality have been proposed in the literature. Cox and Small (1978) reviewed most of these recently, and I shall not cover the same ground again. Procedures may be said to concentrate either on combinations of univariate tests of normality (Small, 1979; Malkovich and Afifi, 1973), or on the geometrical properties in Rm of two or more variates taken together (Healy, 1968; Cox and Small, 1978), in particular, probability plots of squared radii from the centroid of the data in the metric defined by the sample covariance matrix (Healy, 1968; Small, 1978). Cox and Small (1978) considered departures represented by curvature in the variate-variate plots. Certain general difficulties appear. Quite frequently, suggested test statistics have intractable null hypothesis distributions. Convincing power studies are rare, due to the large variety of possible types of departure from normality and the associated expense of Monte Carlo computer studies. Outliers, too, have unpredictable effects-for example, Campbell (1980) pointed out the sensitivity of estimated covariance matrices to outliers, and recommended robust estimation of these matrices. Plotting is a valuable aid, but geometry in m dimensions (m > 3) is hard to depict graphically; also, if m is large, the number of possible plots of subsets of variates multiplies rapidly, making for difficulties in interpretation (and heavy computation). The present paper does not pretend to solve all these problems. Fortunately, however, many of the above conditions present related symptoms; for example curvature in the xl V. x2 plot is quite likely to manifest itself as a skew departure from 1-normality in xl or x2 (or both), and a data transformation may correct matters. I shall show that Shapiro and Wilk's (1965) W test furnishes a set of univariate test statistics which show low correlation even when the parent variates are quite highly correlated, which makes the computation of a combined test fairly simple. I also apply the W test to plots of squared radii, and extend the treatment to subsets of variates. Three detailed illustrative examples are given.