A method of generating multivariate non-normal random numbers with desired multivariate skewness and kurtosis
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A method of generating multivariate non-normal random numbers with desired multivariate skewness and kurtosis Wen Qu1 · Haiyan Liu2 · Zhiyong Zhang1 © The Psychonomic Society, Inc. 2019
Abstract In social and behavioral sciences, data are typically not normally distributed, which can invalidate hypothesis testing and lead to unreliable results when being analyzed by methods developed for normal data. The existing methods of generating multivariate non-normal data typically create data according to specific univariate marginal measures such as the univariate skewness and kurtosis, but not multivariate measures such as Mardia’s skewness and kurtosis. In this study, we propose a new method of generating multivariate non-normal data with given multivariate skewness and kurtosis. Our approach allows researchers to better control their simulation designs in evaluating the influence of multivariate non-normality. Keywords Multivariate non-normal data · Multivariate skewness · Multivariate kurtosis · Random number generation
Introduction In social and behavioral sciences, the normality of data is assumed in most statistical methods. Nonetheless, data are rarely normally distributed in practice. Therefore, the statistical inferences may not be valid, and the results may not be reliable any more when procedures developed for normal data are used to analyze non-normal data (Cain, Zhang, & Yuan, 2017; Micceri, 1989). Many studies in the literature investigated the consequences of the violation of the normality assumption and proposed some alternative procedures to analyze non-normal data. For instance, Bradley (1980) showed that robustness of statistical procedures such as the classical Z, t, and F tests suffered from the non-normality of data. Non-parametric tests and procedures have won appreciation of researchers
Electronic supplementary material The online version of this article (https://doi.org/10.3758/s13428-019-01291-5) contains supplementary material, which is available to authorized users. Wen Qu
[email protected] 1
Department of Psychology, University of Notre Dame, Corbett Family Hall, Notre Dame, IN 46556, USA
2
Psychological Sciences, University of California, Merced, CA, USA
because they do not rely on the data distribution and therefore, the violation of normality does not directly disqualify data analysis (Hollander & Wolfe, 2015). In the literature, discussions on non-normality mainly focus on the univariate case; whereas the consequences of deviation from the multivariate normality are less explored. However, the analysis of multivariate data is routinely conducted in social and behavioral sciences research. Therefore, it is important to understand the influence of the multivariate non-normality on the multivariate analysis, which can be done through Monte Carlo simulations. To conduct such simulations, one needs to generate multivariate data with the control of the degree of nonnormality. In the literature, most non-normal data generators are developed for univariate data, such as the third-order polynom
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