Numerical Computing and Graphics for the Power Method Transformation Using Mathematica

This paper provides the requisite information and description of software that perform numerical computations and graphics for the power method polynomial transformation. The software developed is written in the Mathematica 5.2 package PowerMethod.m and is associated with fifth-order polynomials that are used for simulating univariate and multivariate non-normal distributions. The package is flexible enough to allow a user the choice to model theoretical pdfs, empirical data, or a user's own selected distribution(s). The primary functions perform the following (a) compute standardized cumulants and polynomial coefficients, (b) ensure that polynomial transformations yield valid pdfs, and (c) graph power method pdfs and cdfs. Other functions compute cumulative probabilities, modes, trimmed means, intermediate correlations, or perform the graphics associated with fitting power method pdfs to either empirical or theoretical distributions. Numerical examples and Monte Carlo results are provided to demonstrate and validate the use of the software package. The notebook Demo.nb is also provided as a guide for user of the power method.

JMASM27: An Algorithm for Implementing Gibbs Sampling for 2PNO IRT Models (Fortran)

A Fortran 77 subroutine is provided for implementing the Gibbs sampling procedure to a normal ogive IRT model for binary item response data with the choice of uniform and normal prior distributions for item parameters. The subroutine requires the user to have access to the IMSL library. The source code is available at http://www.siu.edu/~epse1/sheng/Fortran/, along with a stand alone executable file

Bayesian Hierarchical Modeling with 3PNO Item Response Models

Fully Bayesian estimation has been developed for unidimensional IRT models. In this context, prior distributions can be specified in a hierarchical manner so that item hyperparameters are unknown and yet still have their own priors. This type of hierarchical modeling is useful in terms of the three-parameter IRT model as it reduces the difficulty of specifying model hyperparameters that lead to adequate prior distributions. Further, hierarchical modeling ameliorates the noncovergence problem associated with nonhierarchical models when appropriate prior information is not available. As such, a Fortran subroutine is provided to implement a hierarchical modeling procedure associated with the three-parameter normal ogive model for binary item response data using Gibbs sampling. Model parameters can be estimated with the choice of noninformative and conjugate prior distributions for the hyperparameters

Gibbs Sampling for 2PNO Multi-unidimensional Item Response Theory Models (Fortran)

A Fortran 77 subroutine is provided for implementing the Gibbs sampling procedure to a multiunidimensional IRT model for binary item response data with the choice of uniform and normal prior distributions for item parameters. In addition to posterior estimates of the model parameters and their Monte Carlo standard errors, the algorithm also estimates the correlations between distinct latent traits.The subroutine requires the user to have access to the IMSL library

An Algorithm for Implementing Gibbs Sampling for 2PNO IRT Models (Fortran)

A Fortran 77 subroutine is provided for implementing the Gibbs sampling procedure to a normal ogive IRT model for binary item response data with the choice of uniform and normal prior distributions for item parameters. The subroutine requires the user to have access to the IMSL library

JMASM28: Gibbs Sampling for 2PNO Multi-unidimensional Item Response Theory Models (Fortran)

A Fortran 77 subroutine is provided for implementing the Gibbs sampling procedure to a multiunidimensional IRT model for binary item response data with the choice of uniform and normal prior distributions for item parameters. In addition to posterior estimates of the model parameters and their Monte Carlo standard errors, the algorithm also estimates the correlations between distinct latent traits. The subroutine requires the user to have access to the IMSL library. The source code is available at http://www.siuc.edu/~epse1/sheng/Fortran/MUIRT/GSMU2.FOR. An executable file is also provided for download at http://www.siuc.edu/~epse1/sheng/Fortran/MUIRT/EXAMPLE.zip to demonstrate the implementation of the algorithm on simulated data

An Alternative to Cronbach’s Alpha: A \u3ci\u3eL\u3c/i\u3e-Moment Based Measure of Internal-consistency Reliability

Data sets in the social and behavioral sciences are often small or heavytailed. Previous studies have demonstrated that small samples or leptokurtic distributions adversely affect the performance of Cronbach’s coefficient alpha. To address these concerns, we propose an alternative estimator of reliability based on Lcomoments. The empirical results of this study demonstrate that when sample sizes are small and distributions are heavy-tailed that the proposed coefficient L-alpha has substantial advantages over the conventional Cronbach estimator of reliability in terms of relative bias and relative standard error

On Simulating Univariate and Multivariate Burr Type III and Type XII Distributions

This paper describes a method for simulating univariate and multivariate Burr Type III and Type XII distributions with specied correlation matrices. The methodology is based on the derivation of the parametric forms of a pdf and cdf for this family of distributions. The paper shows how shape parameters can be computed for specied values of skew and kurtosis. It is also demonstrated how to compute percentage points and other measures of central tendency such as the mode, median, and trimmed mean. Examples are provided to demonstrate how this Burr family can be used in the context of distribution fitting using real data sets. The results of a Monte Carlo simulation are provided to confirm that the proposed method generates distributions with user specied values of skew, kurtosis, and intercorrelation. Tabled values of shape parameters and boundary values of kurtosis are also provided in the appendices for the user

Numerical Computing and Graphics for the Power Method Transformation Using Mathematica

This paper provides the requisite information and description of software that perform numerical computations and graphics for the power method polynomial transformation. The software developed is written in the Mathematica 5.2 package PowerMethod.m and is associated with fifth-order polynomials that are used for simulating univariate and multivariate non-normal distributions. The package is flexible enough to allow a user the choice to model theoretical pdfs, empirical data, or a user’s own selected distribution(s). The primary functions perform the following (a) compute standardized cumulants and polynomial coefficients, (b) ensure that polynomial transformations yield valid pdfs, and (c) graph power method pdfs and cdfs. Other functions compute cumulative probabilities, modes, trimmed means, intermediate correlations, or perform the graphics associated with fitting power method pdfs to either empirical or theoretical distributions. Numerical examples and Monte Carlo results are provided to demonstrate and validate the use of the software package. The notebook Demo.nb is also provided as a guide for user of the power method

Power Method Distributions through Conventional Moments and \u3ci\u3eL\u3c/i\u3e-Moments

This paper develops two families of power method (PM) distributions based on polynomial transformations of the (1) Uniform, (2) Triangular, (3) Normal, (4) D-Logistic, and (5) Logistic distributions. One family is developed in the context of conventional method of moments and the other family is derived through the method of L-moments. As such, each of the five conventional moment-based PM classes has an analogous L-moment based class. A primary focus of the development is on PM polynomial transformations of order three. Specifically, systems of equations are derived for computing polynomial coefficients for user specified values of skew (L-skew) and kurtosis (L-kurtosis). Boundary regions for determining feasible combinations of skew (L-skew) and kurtosis (L-kurtosis) are also derived for determining if a set of solved coefficients yields a valid PM probability density function. Further, the conventional moment-based family of PM distributions is compared with its L-moment based analog in terms of estimation, power, outliers, and distribution fitting. The results of the comparison demonstrate that the L-moment based PM family is superior to the conventional moment-based family in each of the categories considered