Markov Chain Monte Carlo Estimation of Normal Ogive IRT Models in MATLAB

Modeling the interaction between persons and items at the item level for binary response data, item response theory (IRT) models have been found useful in a wide variety of applications in various fields. This paper provides the requisite information and description of software that implements the Gibbs sampling procedures for the one-, two- and three-parameter normal ogive models. The software developed is written in the MATLAB package IRTuno. The package is flexible enough to allow a user the choice to simulate binary response data, set the number of total or burn-in iterations, specify starting values or prior distributions for model parameters, check convergence of the Markov chain, and obtain Bayesian fit statistics. Illustrative examples are provided to demonstrate and validate the use of the software package. The m-file v25i08.m is also provided as a guide for the user of the MCMC algorithms with the three dichotomous IRT models.

A MATLAB Package for Markov Chain Monte Carlo with a Multi-Unidimensional IRT Model

Unidimensional item response theory (IRT) models are useful when each item is designed to measure some facet of a unified latent trait. In practical applications, items are not necessarily measuring the same underlying trait, and hence the more general multi-unidimensional model should be considered. This paper provides the requisite information and description of software that implements the Gibbs sampler for such models with two item parameters and a normal ogive form. The software developed is written in the MATLAB package IRTmu2no. The package is flexible enough to allow a user the choice to simulate binary response data with multiple dimensions, set the number of total or burn-in iterations, specify starting values or prior distributions for model parameters, check convergence of the Markov chain, as well as obtain Bayesian fit statistics. Illustrative examples are provided to demonstrate and validate the use of the software package.

Bayesian Estimation of MIRT Models with General and Specific Latent Traits in MATLAB

Multidimensional item response models have been developed to incorporate a general trait and several specific trait dimensions. Depending on the structure of these latent traits, different models can be considered. This paper provides the requisite information and description of software that implement the Gibbs sampling procedures for three such models with a normal ogive form. The software developed is written in the MATLAB package IRTm2noHA. The package is flexible enough to allow a user the choice to simulate binary response data with a latent structure involving general and specific traits, specify prior distributions for model parameters, check convergence of the MCMC chain, and obtain Bayesian fit statistics. Illustrative examples are provided to demonstrate and validate the use of the software package.

Bayesian Estimation of MIRT Models with General and Specific Latent Traits in MATLAB

Multidimensional item response models have been developed to incorporate a general trait and several specific trait dimensions. Depending on the structure of these latent traits, different models can be considered. This paper provides the requisite information and description of software that implement the Gibbs sampling procedures for three such models with a normal ogive form. The software developed is written in the MATLAB package IRTm2noHA. The package is flexible enough to allow a user the choice to simulate binary response data with a latent structure involving general and specific traits, specify prior distributions for model parameters, check convergence of the MCMC chain, and obtain Bayesian fit statistics. Illustrative examples are provided to demonstrate and validate the use of the software package

Markov Chain Monte Carlo Estimation of Normal Ogive IRT Models in MATLAB

Modeling the interaction between persons and items at the item level for binary response data, item response theory (IRT) models have been found useful in a wide variety of applications in various fields. This paper provides the requisite information and description of software that implements the Gibbs sampling procedures for the one-, two- and three-parameter normal ogive models. The software developed is written in the MATLAB package IRTuno. The package is flexible enough to allow a user the choice to simulate binary response data, set the number of total or burn-in iterations, specify starting values or prior distributions for model parameters, check convergence of the Markov chain, and obtain Bayesian fit statistics. Illustrative examples are provided to demonstrate and validate the use of the software package. The m-file v25i08.m is also provided as a guide for the user of the MCMC algorithms with the three dichotomous IRT models

Is Coefficient Alpha Robust to Non-Normal Data?

Coefficient alpha has been a widely used measure by which internal consistency reliability is assessed. In addition to essential tau-equivalence and uncorrelated errors, normality has been noted as another important assumption for alpha. Earlier work on evaluating this assumption considered either exclusively non-normal error score distributions, or limited conditions. In view of this and the availability of advanced methods for generating univariate non-normal data, Monte Carlo simulations were conducted to show that non-normal distributions for true or error scores do create problems for using alpha to estimate the internal consistency reliability. The sample coefficient alpha is affected by leptokurtic true score distributions, or skewed and/or kurtotic error score distributions. Increased sample sizes, not test lengths, help improve the accuracy, bias, or precision of using it with non-normal data

High Performance Gibbs Sampling for IRTModels Using Row-Wise Decomposition

Item response theory (IRT) is a popular approach used for addressing statistical problems in psychometrics as well as in other fields. The fully Bayesian approach for estimating IRT models is computationally expensive. This limits the use of the procedure in real applications. In an effort to reduce the execution time, a previous study shows that high performance computing provides a solution by achieving a considerable speedup via the use of multiple processors. Given the high data dependencies in a single Markov chain for IRT models, it is not possible to avoid communication overhead among processors. This study is to reduce communication overhead via the use of a row-wise decomposition scheme. The results suggest that the proposed approach increased the speedup and the efficiency for each implementation while minimizing the cost and the total overhead. This further sheds light on developing high performance Gibbs samplers for more complicated IRT models

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