Todd C. Headrick

Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributions

A general procedure is derived for simulating univariate and multivariate nonnormal distributions using polynomial transformations of order five. The procedure allows for the additional control of the fifth and sixth moments. The ability to control higher moments increases the precision in the approximations of nonnormal distributions and lowers the skew and kurtosis boundary relative to the competing procedures considered. Tabled values of constants are provided for approximating various probability density functions. A numerical example is worked to demonstrate the multivariate procedure. The results of a Monte Carlo simulation are provided to demonstrate that the procedure generates specified population parameters and intercorrelations.

Statistical Simulation : Power Method Polynomials and Other Transformations

Introduction The Power Method Transformation Univariate Theory Third-Order Systems Fifth-Order Systems Mathematica(R) Functions Limitations Multivariate Theory Using the Power Method Transformation Introduction Examples of Third- and Fifth-Order Polynomials Remediation Techniques Monte Carlo Simulation Some Further Considerations Simulating More Elaborate Correlation Structures Introduction Simulating Systems of Linear Statistical Models Methodology Numerical Example and Monte Carlo Simulation Some Additional Comments Simulating Intraclass Correlation Coefficients Methodology Numerical Example and Monte Carlo Simulation Simulating Correlated Continuous Variates and Ranks Methodology Numerical Example and Monte Carlo Simulation Some Additional Comments Other Transformations: The g-and-h and GLD Families of Distributions Introduction The g-and-h Family The Generalized Lambda Distributions (GLDs) Numerical Examples Multivariate Data Generation References Index

Properties of the rank transformation in factorial analysis of covariance

Real world data often fail to meet the underlying assumption of population normality. The Rank Transformation (RT) procedure has been recommended as an alternative to the parametric factorial analysis of covariance (ANCOVA). The purpose of this study was to compare the Type I error and power properties of the RT ANCOVA to the parametric procedure in the context of a completely randomized balanced 3 × 4 factorial layout with one covariate. This study was concerned with tests of homogeneity of regression coefficients and interaction under conditional (non)normality. Both procedures displayed erratic Type I error rates for the test of homogeneity of regression coefficients under conditional nonnormality. With all parametric assumptions valid, the simulation results demonstrated that the RT ANCOVA failed as a test for either homogeneity of regression coefficients or interaction due to severe Type I error inflation. The error inflation was most severe when departures from conditional normality were extreme. Also associated with the RT procedure was a loss of power. It is recommended that the RT procedure not be used as an alternative to factorial ANCOVA despite its encouragement from SAS, IMSL, and other respected sources.

On simulating multivariate non-normal distributions from the generalized lambda distribution

The class of generalized lambda distributions (GLDs) is primarily used for modeling univariate real-world data. The GLD has not been as popular as some other methods for simulating observations from multivariate distributions because of computational difficulties. In view of this, the methodology and algorithms are presented for extending the GLD from univariate to multivariate data generation with an emphasis on reducing computational difficulties. Algorithms written in Mathematica 5.1 and Fortran 77 are provided for implementing the procedure and are available from the authors. A numerical example is provided and a Monte Carlo simulation was conducted to confirm and demonstrate the methodology.

An investigation of the rank transformation in multiple regression

Real world data often fail to meet the underlying assumptions of normal statistical theory. The rank transformation (RT) procedure is recommended and used in the context of multiple regression analysis when the assumption of normality is violated. There is no general supporting theory of the RT. In view of this, the current study examined the Type I error and power properties of the RT in terms of multiple regression. The investigation included both additive and nonadditive models. Results indicated that there were severely in/ated Type I error rates associated with the RT procedure under both normal and nonnormal distributions (e.g., 0.772 with nominal alpha = 0:05). The RT also exhibited a substantial power loss relative to the usual ordinary least squares regression procedure. It is recommended that the RT be avoided in the context of multiple regression despite its encouragement from SAS and other well respected sources. c � 2001 Elsevier Science B.V. All rights reserved.

A Characterization of Power Method Transformations through L -Moments

Power method polynomial transformations are commonly used for simulating continuous nonnormal distributions with specified moments. However, conventional moment-based estimators can (a) be substantially biased, (b) have high variance, or (c) be influenced by outliers. In view of these concerns, a characterization of power method transformations by L-moments is introduced. Specifically, systems of equations are derived for determining coefficients for specified L-moment ratios, which are associated with standard normal and standard logistic-based polynomials of order five and three. Boundaries for L-moment ratios are also derived, and closed-formed formulae are provided for determining if a power method distribution has a valid probability density function. It is demonstrated that L-moment estimators are nearly unbiased and have relatively small variance in the context of the power method. Examples of fitting power method distributions to theoretical and empirical distributions based on the method of L-moments are also provided.

The power method transformation: its probability density function, distribution function, and its further use for fitting data

The power method polynomial transformation is a popular algorithm used for simulating non-normal distributions because of its simplicity and ease of execution. The primary limitations of the power method transformation are that its probability density function (pdf) and cumulative distribution function (cdf) are unknown. In view of this, the power methods pdf and cdf are derived in general form. More specific properties are also derived for determining if a given transformation will also have an associated pdf in the context of polynomials of order three and five. Numerical examples and para-metric plots of power method densities are provided to confirm and demonstrate the methodology. It is also shown how the power method transformation can be applied in the context of parameter estimation and distribution fitting using data from the National Institute on Alcohol Abuse and Alcoholism study Project MATCH.

Weighted Simplex Procedures for Determining Boundary Points and Constants for the Univariate and Multivariate Power Methods

The power methods are simple and efficient algorithms used to generate either univariate or multivariate nonnormal distributions with specified values of (marginal) mean, standard deviation, skew, and kurtosis. The power methods are bounded as are other transformation techniques. Given an exogenous value of skew, there is an associated lower bound of kurtosis. Previous approximations of the boundary for the power methods are either incorrect or inadequate. Data sets from education and psychology can be found to lie within, near, or outside tile boundary of the power methods. In view of this, we derived necessary and sufficient conditions using the Lagrange multiplier method to determine the boundary of the power methods. The conditions for locating and classifying modes for distributions on the boundary were also derived. Self-contained interactive Fortran programs using a Weighted Simplex Procedure were employed to generate tabled values of minimum kurtosis for a given value of skew and power constants for various (non)normal distributions.

Simulating multivariate g‐and‐h distributions

The Tukey family of g-and-h distributions is often used to model univariate real-world data. There is a paucity of research demonstrating appropriate multivariate data generation using the g-and-h family of distributions with specified correlations. Therefore, the methodology and algorithms are presented to extend the g-and-h family from univariate to multivariate data generation. An example is provided along with a Monte Carlo simulation demonstrating the methodology. In addition, algorithms written in Mathematica 7.0 are available from the authors for implementing the procedure.

A Gibbs Sampler for the Multidimensional Item Response Model

Current procedures for estimating compensatory multidimensional item response theory (MIRT) models using Markov chain Monte Carlo (MCMC) techniques are inadequate in that they do not directly model the interrelationship between latent traits. This limits the implementation of the model in various applications and further prevents the development of other types of IRT models that offer advantages not realized in existing models. In view of this, an MCMC algorithm is proposed for MIRT models so that the actual latent structure is directly modeled. It is demonstrated that the algorithm performs well in modeling parameters as well as intertrait correlations and that the MIRT model can be used to explore the relative importance of a latent trait in answering each test item.