On the solution multiplicity of the Fleishman method and its impact in simulation studies

Abstract

The Fleishman third-order polynomial algorithm is one of the most-often used non-normal data-generating methods in Monte Carlo simulations. At the crux of the Fleishman method is the solution of a non-linear system of equations needed to obtain the constants to transform data from normality to non-normality. A rarely acknowledged fact in the literature is that the solution to this system is not unique, and it is currently unknown what influence the different types of solutions have on the computer-generated data. To address this issue, analytical and empirical investigations were conducted, aimed at documenting the impact that each solution type has on the design of computer simulations. In the first study, it was found that certain types of solutions generate data with different multivariate properties and wider coverage of the theoretical range spanned by population correlations. In the second study, it was found that previously published recommendations from Monte Carlo simulations could change if different types of solutions were used to generate the data. A mathematical description of the multiple solutions to the Fleishman polynomials is provided, as well as recommendations for users of this method.

A method of generating multivariate non-normal random numbers with desired multivariate skewness and kurtosis

“…Astivia and Zumbo (2015) have shown that the Vale and Maurelli method has downward bias. In one of their later papers, Astivia and Zumbo (2018) also found the multiplicity solution issue of the Fleishman's polynomial-related method, which means that there are multiple possible solutions for the polynomial coefficients (a, b, c, and d). This issue might lead to the difference in the analysis even with the same inputs.…”

The Choice of Normal-Theory Weight Matrix When Computing Robust Standard Errors in Confirmatory Factor Analysis

“…Given the large variety of non-normal distributions, we deem it important to include other types of non-normality than the default type offered in many software packages (Vale & Maurelli, 1983). Also Xia et al (2016) used the Vale-Maurelli (VM) method for non-normality conditions, but it has its limitations (Astivia & Zumbo, 2018;Foldnes & Grønneberg, 2015). To extend the scope of non-normality, we therefore simulated non-normal data using three recently proposed alternatives to VM, namely the approaches by copula (Mair, Satorra, & Bentler, 2012), by independent generators (Foldnes & Olsson, 2016) and by regular vines (Bedford & Cooke, 2002;Grønneberg & Foldnes, 2017).…”