Publications

A Method for Simulating Burr Type III and Type XII Distributions through 𝐿-Moments and 𝐿-Correlations

Mohan D. Pant et al. — Thu, 23 May 2013 08:02:30 PDT

This paper derives the Burr Type III and Type XII family of distributions in the contexts of univariate 𝐿-moments and the 𝐿- correlations. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of 𝐿-skew, 𝐿-kurtosis, and 𝐿-correlations. The procedure can be applied in a variety of settings such as statistical modeling (e.g., forestry, fracture roughness, life testing, operational risk, etc.) and Monte Carlo or simulation studies. Numerical examples are provided to demonstrate that 𝐿-moment-based Burr distributions are superior to their conventional moment-based analogs in terms of estimation and distribution fitting. Evaluation of the proposed procedure also demonstrates that the estimates of 𝐿-skew, 𝐿-kurtosis, and 𝐿-correlation are substantially superior to their conventional product moment-based counterparts of skew, kurtosis, and Pearson correlations in terms of relative bias and relative efficiency—most notably when heavy-tailed distributions are of concern.

High Performance Gibbs Sampling for IRTModels Using Row-Wise Decomposition

Yanyan Sheng et al. — Thu, 13 Dec 2012 11:38:14 PST

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.

A Logistic L-Moment-Based Analog for the Tukey g-h, g, h, and h-h System of Distributions

Todd C. Headrick et al. — Mon, 22 Oct 2012 11:59:06 PDT

This paper introduces a standard logistic L-moment-based system of distributions. The proposed system is an analog to the standard normal conventional moment-based Tukey g-h, g, h, and h-h system of distributions. The system also consists of four classes of distributions and is referred to as (i) asymmetric γ-κ, (ii) log-logistic γ, (iii) symmetric κ, and (iv) asymmetric κ_L-κ_R. The system can be used in a variety of settings such as simulation or modeling events—most notably when heavytailed distributions are of interest. A procedure is also described for simulating γ-κ, γ, κ, and κ_L-κ_R distributions with specified L-moments and L-correlations. The Monte Carlo results presented in this study indicate that estimates of L-skew, L-kurtosis, and L-correlation associated with the γ-κ, γ, κ, and κ_L-κ_R distributions are substantially superior to their corresponding conventional product-moment estimators in terms of relative bias and relative standard error.

An L-Moment-Based Analog for the Schmeiser-Deutsch Class of Distributions

Todd C. Headrick et al. — Wed, 17 Oct 2012 12:08:11 PDT

This paper characterizes the conventional moment-based Schmeiser-Deutsch (S-D) class of distributions through the method of L-moments. The system can be used in a variety of settings such as simulation or modeling various processes. A procedure is also described for simulating S-D distributions with specified L-moments and L-correlations. The Monte Carlo results presented in this study indicate that the estimates of L-skew, L-kurtosis, and L-correlation associated with the S-D class of distributions are substantially superior to their corresponding conventional product-moment estimators in terms of relative bias—most notably when sample sizes are small.

A Doubling Technique for the Power Method Transformations

Mohan D. Pant et al. — Fri, 05 Oct 2012 06:57:41 PDT

Power method polynomials are used for simulating non-normal distributions with specified product moments or L-moments. The power method is capable of producing distributions with extreme values of skew (L-skew) and kurtosis (L-kurtosis). However, these distributions can be extremely peaked and thus not representative of real-world data. To obviate this problem, two families of distributions are introduced based on a doubling technique with symmetric standard normal and logistic power method distributions. The primary focus of the methodology is in the context of L-moment theory. As such, L-moment based systems of equations are derived for simulating univariate and multivariate non-normal distributions with specified values of L-skew, L-kurtosis, and L-correlation. Evaluation of the proposed doubling technique indicates that estimates of L-skew, L-kurtosis, and L-correlation are superior to conventional product-moments in terms of relative bias and relative efficiency when extreme non-normal distributions are of concern.

A Method for Simulating Nonnormal Distributions with Specified L-Skew, L-Kurtosis, and L-Correlation

Todd C. Headrick et al. — Thu, 27 Sep 2012 07:40:38 PDT

This paper introduces two families of distributions referred to as the symmetric κ and asymmetric κ_L-κ_R distributions. The families are based on transformations of standard logistic pseudo-random deviates. The primary focus of the theoretical development is in the contexts of L-moments and the L-correlation. Also included is the development of a method for specifying distributions with controlled degrees of L-skew, L-kurtosis, and L-correlation. The method can be applied in a variety of settings such as Monte Carlo studies, simulation, or modeling events. It is also demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are superior to conventional productmoment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when moderate-to-heavy-tailed distributions are of concern.

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

Yanyan Sheng — Fri, 17 Aug 2012 14:06:37 PDT

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

Yanyan Sheng — Fri, 17 Aug 2012 14:06:36 PDT

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 IRT Models Incorporating General and Specific Abilities

Yanyan Sheng et al. — Fri, 17 Aug 2012 14:06:35 PDT

IRT-based models with a general ability and several specific ability dimensions are useful. Studies have looked at item response models where the general and specific ability dimensions form a hierarchical structure. It is also possible that the general and specific abilities directly underlie all test items. A multidimensional IRT model with such an additive structure is proposed under the Bayesian framework. Simulation studies were conducted to evaluate parameter recovery as well as model comparisons. A real data example is also provided. The results suggest that the proposed additive model offers a better way to represent the test situations not realized in existing models.

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

Yanyan Sheng — Fri, 17 Aug 2012 14:06:34 PDT

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.

A Sensitivity Analysis of Gibbs Sampling for 3PNO IRT Models

Yanyan Sheng — Fri, 17 Aug 2012 14:06:33 PDT

The performance of the Gibbs sampling procedure for the three-parameter normal ogive (3PNO) IRT model was investigated using Monte Carlo simulations. Model parameters were estimated for tests with 10, 20, and 40 items and samples of 100, 300, 500, and 1000 examinees, where different actual values and prior specifications were considered for the item parameters. Summary statistics showed that this procedure was more affected by the choice of the prior distributions for the three-parameter model than the two-parameter model. For the 3PNO model, appropriate informative priors with relatively small spread should be adopted for the slope and intercept parameters to obtain more efficient and accurate MCMC estimates when sample sizes are not large and/or tests are not long enough. When it is not clear whether the prior information is appropriate, informative priors with small prior variances are not recommended.

Parallel Computing with a Bayesian Item Response Model

Kyriakos Patsias et al. — Fri, 20 Jul 2012 07:45:09 PDT

Item response theory (IRT) is a modern test theory that has been used in various aspects of educational and psychological measurement. The fully Bayesian approach shows promise for estimating IRT models. Given that it is computationally expensive, the procedure is limited in practical applications. It is hence important to seek ways to reduce the execution time. A suitable solution is the use of high performance computing. This study focuses on the fully Bayesian algorithm for a conventional IRT model so that it can be implemented on a high performance parallel machine. Empirical results suggest that this parallel version of the algorithm achieves a considerable speedup and thus reduces the execution time considerably.

A Gibbs Sampler for the Multidimensional Item Response Model

Yanyan Sheng et al. — Wed, 13 Jun 2012 07:12:35 PDT

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.

On the Order Statistics of Standard Normal-Based Power Method Distributions

Todd C. Headrick et al. — Mon, 21 May 2012 12:01:22 PDT

This paper derives a procedure for determining the expectations of order statistics associated with the standard normal distribution Z and its powers of order three and five (Z³ and Z⁵). the procedure is demonstrated for sample sizes of n ≤ 9. It is shown that Z³ and Z⁵ have expectations of order statistics that are functions of the expectations for Z and can be expressed in terms of explicit elementary functions for sample sizes of n ≤ 5. For sample sizes of n = 6,7 the expectations of the order statistics for Z, Z³, and Z⁵ only require a single remainder term.

A Doubling Method for the Generalized Lambda Distribution

Todd C. Headrick et al. — Mon, 07 May 2012 07:03:04 PDT

This paper introduces a new family of generalized lambda distributions GLDs based on a method of doubling symmetric GLDs. The focus of the development is in the context of L-moments and L-correlation theory. As such, included is the development of a procedure for specifying double GLDs with controlled degrees of L-skew, L-kurtosis, and L-correlations. The procedure can be applied in a variety of settings such as modeling events and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy tailed distributions are of concern.

Is Coefficient Alpha Robust to Non-normal Data?

Yanyan Sheng et al. — Wed, 15 Feb 2012 07:39:49 PST

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 nonnormal error score distributions, or limited conditions. In view of this and the availability of advanced methods for generating univariate nonnormal data, Monte Carlo simulations were conducted to show that nonnormal 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 nonnormal data.

Characterizing Tukey h and hh-Distributions through L-Moments and the L-Correlation

Todd C. Headrick et al. — Mon, 13 Feb 2012 07:06:54 PST

This paper introduces the Tukey family of symmetric h and asymmetric hh-distributions in the contexts of univariate L-moments and the L-correlation. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of L-skew, L-kurtosis, and L-correlations. The procedure can be applied in a variety of settings such as modeling events (e.g., risk analysis, extreme events) and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy-tailed distributions are of concern.

Power Method Distributions through Conventional Moments and L-Moments

Flaviu A. Hodis et al. — Mon, 06 Feb 2012 10:47:32 PST

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.

Numerical Computing and Graphics for the Power Method Transformation Using Mathematica

Todd C. Headrick et al. — Wed, 16 Nov 2011 14:08:11 PST

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.

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

Todd C. Headrick et al. — Wed, 16 Nov 2011 14:01:48 PST

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.