Investigating the Performance of Procedures for Correlations Under Nonnormality

Abstract

This dissertation takes a detailed look at two seemingly well-established procedures in Monte Carlo research. Part I closely examines the 3rd and 5th order polynomial transforms and the g-and-h distribution, all designed to simulate multivariate nonnormal distributions. The general method common to all three techniques has been in use for over three decades, yet, some fundamental properties and behavior are uncovered and discussed for the first time in this study. These properties include: Non-uniqueness of transformation coefficients, restrictions of range for the correlation between the resulting nonnormal variables, expected values for sample skewness and kurtosis at a given sample size, and distributional shape. It is demonstrated that these issues are not only of theoretical interest, but can have fundamental effects on outcomes of Monte Carlo studies. The second part investigates the interpretation of results from Monte Carlo studies themselves, focusing on performance evaluation for confidence intervals. Central to the discussion in Part II is an empirically constructed exact confidence interval for correlations drawn from a nonnormal population. This exact CI is used to evaluate five approximate CIs: The Fisher Z CI, two asymptotically distribution-free CIs, and two bootstrap CIs. Scatterplots of the confidence interval limits for several CIs against sample correlations and bivariate distributions of CI limit difference scores between the exact and the approximate CIs are created. Thereby, the behavior of the approximate confidence intervals is scrutinized and a foundation for a practical definition of confidence interval bias developed.