Proposal and Pilot Study: A Generalization of the W or W' Statistic for Multivariate Normality

Abstract

The aim of this paper is to present a generalization of the Shapiro-Wilk W-test or Shapiro-Francia W ' -test for application to two or more variables. It consists of calculating all the unweighted linear combinations of the variables and their W- or W ' -statistics with the Royston’s log-transformation and standardization, z ln(1-W) or z ln(1-W ' ) . Because the calculation of the probability of z ln(1-W) or z ln(1-W ' ) is to the right tail, negative values are truncated to 0 before doing their sum of squares. Independence in the sequence of these half-normally distributed values is required for the test statistic to follow a chi-square distribution. This assumption is checked using the robust Ljung-Box test. One degree of freedom is lost for each cancelled value. Defined the new test with its two variants (Q-test or Q ' -test), 50 random samples with 4 variables and 20 participants were generated, 20% following a multivariate normal distribution and 80% deviating from this distribution. The new test was compared with Mardia’s, runs, and Royston’s tests. Central tendency differences in type II error and statistical power were tested using the Friedman’s test and pairwise comparisons using the Wilcoxon’s test. Differences in the frequency of successes in statistical decision making were compared using the Cochran’s Q test and pairwise comparisons using the McNemar’s test. Sensitivity, specificity and efficiency proportions were compared using the McNemar’s Z test. The generated 50 samples were classified into five ordered categories of deviation from multivariate normality, the correlation between this variable and p-value of each test was calculated using the Spearman’s coefficient and these correlations were compared. Family-wise error rate corrections were applied. The new test and the Royston’s test were the best choices, with a very slight advantage Q-test over Q ' -test. Based on these promising results, further study and use of this new sensitive, specific and effective test are suggested .