Empirical Power Comparison Of Goodness of Fit Tests for Normality In The Presence of Outliers

Most statistical tests such as t-tests, linear regression analysis and Analysis of Variance (ANOVA) require the normality assumptions. When the normality assumption is violated, interpretation and inferences may not be reliable. Therefore it is important to assess such assumption before using any appropriate statistical test. One of the commonly used procedures in determining whether a random sample of size n comes from a normal population are the goodness-of-fit tests for normality. Several studies have already been conducted on the comparison of the different goodness-of-fit(see, for example [2]) but it is generally limited to the sample size or to the number of GOF tests being compared(see, for example [2] [5] [6] [7] [8]). This paper compares the power of six formal tests of normality: Kolmogorov-Smirnov test (see [3]), Anderson-Darling test, Shapiro-Wilk test, Lilliefors test, Chi-Square test (see [1]) and D'Agostino-Pearson test. Small, moderate and large sample sizes and various contamination levels were used to obtain the power of each test via Monte Carlo simulation. Ten thousand samples of each sample size and contamination level at a fixed type I error rate α were generated from the given alternative distribution. The power of each test was then obtained by comparing the normality test statistics with the respective critical values. Results show that the power of all six tests is low for small sample size(see, for example [2]). But for n = 20, the Shapiro-Wilk test and Anderson – Darling test have achieved high power. For n = 60, Shapiro-Wilk test and Liliefors test are most powerful. For large sample size, Shapiro-Wilk test is most powerful (see, for example [5]). However, the test that achieves the highest power under all conditions for large sample size is D'Agostino-Pearson test (see, for example [9]).