Comprehensive comparison of normality tests: Empirical study using many different types of data

Abstract

We compare many normality tests consisting of different sources of information extracted from the given data: Anderson-Darling test, Kolmogorov-Smirnov test, Cramervon Mises test, Shapiro-Wilk test, Shaprio-Francia test, Lilliefors, Jarque-Bera test, D'Agostino' D, Doornik-Hansen test, Energy test and Martinzez-Iglewicz test. For the purpose of comparison, those tests are applied to the various types of data generated from skewed distribution, unsymmetric distribution, and distribution with different length of support. We then summarize comparison results in terms of two things: type I error control and power. The selection of the best test depends on the shape of the distribution of the data, implying that there is no test which is the most powerful for all distributions.Keywords: Empirical power, empirical type I error, limiting distribution, normality tests. IntroductionMany statistical methodologies, which have been developed so far, work very well under the assumption that the data come from normal distribution and it is very important to check for normality before any statistical analysis is done. Over several decades, various tests including goodness-of-fit test (Kang et al., 2014;Lee, 2013) using different sources of information from data have been introduced by many researchers. More specifically, some methods exploit sample skewness or/and sample kurtosis (D'Agostino, 1970;D'Agostino and Pearson, 1973;Jarque and Bera, 1981) while other methods use the maximum distance between two distribution functions, i.e., target normal distribution function and empirical distribution function (Kolmogorov, 1933;Lilliefors, 1967Lilliefors, , 1969Smirnov, 1939). Other than that, some method uses the ratio of two estimators of variance (Martinez and Iglewicz, 1981 (Anderson, 1962;Cramer, 1928;von Mises, 1928;Shapiro and Wilk, 1965;Shapiro and Francia, 1972;Lilliefors, 1967;Jarque and Bera, 1981;D'Agostino, 1970;Doornik and Hansen, 2008;Szekely and Rizzo, 2005;Martinez and Iglewicz, 1981). Since each method makes use of different type of information extracted from the given data, the performance of each method depends heavily on the properties of the data: skewness, kurtosis, and the type of support. Accordingly, it is well known that there is no test which is the most powerful for all types of alternatives. In order to compare such methods, we apply those tests to the data generated from many different distributions and then find the best test for each alternative.In this paper, we focus on the one sample test. That is, given the samples X 1 , · · · , X n , we test for the normality of the given sample of data. Furthermore, for simplicity, our interest is restricted to the univariate normal data.

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“…In contrast, the fact that normality tests are not significant indicates that the data do not differ significantly from the normal distribution. In research studies, mostly Kolmogorov Smirnov (KS) Test (Smirnov, 1948) and Shapiro-Wilk Test (Shapiro & Wilk, 1965) (Lee et al, 2016;Marsaglia et al, 2003;Stephens, 1974). Sample size affects the results of normality tests.…”