Testing for Normality

This book is a compendium of results from many sources, covering several decades of research on hypothesis tests for the normal distribution. A surprisingly large number of proposals for testing normality have appeared over the last century, and a single source for many of them is quite useful. As stated in the Preface, the author’s intent in writing this book is “to focus on methodology and utility of the tests rather than [on] in-depth theoretical issues . and to make this text accessible to researchers with a minimal amount of theoretical statistical background . The emphasis here is on the calculation and performance of the tests” (p. vi). For the most part, the book achieves these broad purposes. Following a brief introduction, the book contains basically two parts: tests for univariate normality and tests for multivariate normality. (Additional topics on normal mixtures, robust estimation, and computational issues are mentioned brie y in a  nal chapter.) The  rst chapter in the univariate-normality part is “plots, probability plots, and regression tests. ” Because graphical assessment is an important precursor to formal hypothesis tests, the inclusion of this chapter is commendable; the author discusses stem-and-leaf plots, boxplots, quantile– quantile (Q–Q) plots, percentile–percentile (P–P) plots, and empirical cumulative distribution function (ECDF) plots. Table 2.1 (p. 20), “plotting positions for probability plots,” is useful, in my opinion. Commonly, a probability plot is constructed by plotting the order statistics X.i/ against F . pi/, where pi D . i ¡ a/=. nC 1 ¡ 2a/ and F is the distribution function of interest. Various values of a are considered in Table 2.1; for example, a D 3 , favored by Tukey (1977), is designated as method XII. It can be shown that a can be selected so that F . pi/ is a good estimate of the median of the distribution of X.i/ for a wide variety of distributions F . Regression tests include those by Wilk-Shapiro and tests on the correlation coef cient of a probability plot (e.g., Filliben 1975). The other chapter in this part of the book discusses tests using moments (popular, although not suf cient for uniquely characterizing normality), likelihood ratio tests for normality (versus speci c alternative distributions), U statistics and statistics based on tail weight (e.g., Hogg 1967), and goodness-of- t tests (e.g., those based on the ECDF and on normalized spacings; see, e.g., D’Agostino and Stephens 1986). The author also discusses Box–Cox transformations to normality as well as tests for univariate outliers (e.g., Grubbs 1969; Dixon 1950). In the 1980s, research on tests for outliers in regression settings was extremely popular, and more recently, tests for multivariate outliers have come into vogue. Because data collection and storage are increasingly convenient, multivariate data are common. Thus the section on testing for multivariate normality is especially valuable for many of today’s problems. Unfortunately, this part of the book is much shorter, because the problem is much harder, and consequently less research has been conducted. The range and variety of multivariate tests include those on the apparent distribution of the Mahalanobis distances and transformations of them, multivariate Q–Q plots, test of multivariate skewness and kurtosis, extensions of the Shapiro–Wilk tests, the probability plot correlation coef cient, ECDF tests, tests on projections, transformations, and nearest distances. The chapter on normal mixtures describes tests mostly in the context of estimating the components  rst, and then determining the signi cance of the added components. (Another approach to this problem was given in Scott and Szewczyk 2002), involving estimation of normal mixtures using nonlinear minimization instead of using the expectation-maximization algorithm. ) The chapter on robust estimation concentrates mostly on M and L estimates and includes only a brief paragraph on robust multivariate estimation. However, robust estimation is only tangentially related to the subject of the book, and an in-depth treatment would repeat material available in other sources (e.g., Huber 1981; Hoaglin, Mosteller, and Tukey 1984; Hampel, Ronchetti, Rousseeuw, and Stahel 1986). Some useful aspects of this book include the sections on choosing normality tests appropriate for the circumstances and the rather complete set of references. The book strikes a comfortable balance between simple presentation of methods and complete theoretical treatment. As the author recognizes, the latter approach would have resulted in an excessively long book. Suf cient references and acknowledgement of theoretical aspects and underlying assumptions have been provided. Researchers who want to know speci cs and more details of the underlying theory will have to consult the references. The book contains no problems, making it more useful as a reference rather than as a textbook for a graduate course. One minor quibble is the appearance of the cited references; they appear at the end of the chapter in which they were cited rather than at the end of the book. This choice makes it easier to locate a reference while reading the text, but more dif cult later. Overall, I think most researchers will  nd Testing for Normality very useful. Interested readers can track down further results on the methods if desired, whereas others will be content to see the variety and assumptions in tests that they may presently recognize only from the computer output of a statistics package. Unfortunately, the book’s high price will probably discourage many people from adding it to their personal libraries.