A Note on the Solution Multiplicity of the Vale–Maurelli Intermediate Correlation Equation

Abstract

The Vale and Maurelli algorithm is a widely used method that allows researchers to generate multivariate, nonnormal data with user-specified levels of skewness, excess kurtosis, and a correlation structure. Before obtaining the desired correlation structure, a transitional step requires the user to calculate the roots of a cubic polynomial referred to as the intermediate correlation equation. The Cardano method and a corollary of Rouché’s theorem were used to derive closed-form solutions to this equation. These solutions highlight the fact that three real-valued roots are possible, and solutions can either contain multiple roots within the allowable correlation range or can all be greater than 1 for many combinations of skewness and excess kurtosis. It was also found that large values of excess kurtosis exacerbate the issue of multiple solutions or solutions greater than 1, bringing further restrictions into the combinations of higher order moments that researchers can simulate from. A small computer study on the power of the t test for the correlation coefficient also uncovered that different values of the intermediate correlation can influence the results from Monte Carlos simulations. This note is intended to inform both researchers who conduct simulations with nonnormal data and users who inform their data analysis practice from simulation studies.