Slice–Gibbs sampling algorithm for estimating the parameters of a multilevel item response model

Abstract In a fully Bayesian framework, a novel slice–Gibbs algorithm is developed to estimate a multilevel item response theory (IRT) model. The advantage of this algorithm is that it can recover parameters well based on various types of prior distributions of the item parameters, including informative and non-informative priors. In contrast to the traditional Metropolis–Hastings (M–H) within Gibbs algorithm, the slice–Gibbs algorithm is faster and more efficient, due to its drawing the sample with acceptance probability as one, rather than tuning the proposal distributions to achieve the reasonable acceptance probabilities, especially for the logistic model without conjugate distribution. In addition, based on the Markov chain Monte Carlo (MCMC) output, two model assessment methods are investigated concerning the goodness of fit between models. The information criterion method on the basis of marginal likelihood is proposed to assess the different structural multilevel models, and the cross-validation method is used to evaluate the overall multilevel IRT models. The feasibility and effectiveness of the slice–Gibbs algorithm are investigated in simulation studies. An application using a real data involving students’ mathematics test achievements is reported.

[1]  Kimberly S. Maier.  A Rasch Hierarchical Measurement Model , 2001 .

[2]  G. Masters.  A rasch model for partial credit scoring , 1982 .

[3]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[4]  Chun Wang,et al.  A mixture hierarchical model for response times and response accuracy. , 2015, The British journal of mathematical and statistical psychology.

[5]  Herbert Hoijtink,et al.  Posterior inference in the random intercept model based on samples obtained with Markov chain Monte Carlo methods , 2000, Comput. Stat..

[6]  Harvey Goldstein,et al.  A general model for the analysis of multilevel data , 1988 .

[7]  D. Hedeker,et al.  A random-effects ordinal regression model for multilevel analysis. , 1994, Biometrics.

[8]  Anthony Y. C. Kuk.  Laplace Importance Sampling for Generalized Linear Mixed Models , 1999 .

[9]  C. McCulloch,et al.  Generalized Linear Mixed Models , 2005 .

[10]  S. Rabe-Hesketh,et al.  Reliable Estimation of Generalized Linear Mixed Models using Adaptive Quadrature , 2002 .

[11]  Akihito Kamata,et al.  Item Analysis by the Hierarchical Generalized Linear Model. , 2001 .

[12]  H. Akaike.  Factor analysis and AIC , 1987 .

[13]  Nicholas T. Longford,et al.  Factor analysis for clustered observations , 1992 .

[14]  E. Muraki.  Information Functions of the Generalized Partial Credit Model , 1993 .

[15]  A. Gelfand,et al.  Bayesian Model Choice: Asymptotics and Exact Calculations , 1994 .

[16]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[17]  A. Béguin,et al.  MCMC estimation and some model-fit analysis of multidimensional IRT models , 2001 .

[18]  William J. Browne,et al.  Bayesian and likelihood-based methods in multilevel modeling 1 A comparison of Bayesian and likelihood-based methods for fitting multilevel models , 2006 .

[19]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[20]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[21]  S. Rabe-Hesketh,et al.  Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects , 2005 .

[22]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[23]  T. Ando.  Predictive Bayesian Model Selection , 2011 .

[24]  Bengt Muthén,et al.  Latent variable modeling in heterogeneous populations , 1989 .

[25]  Scott L. Zeger,et al.  Generalized linear models with random e ects: a Gibbs sampling approach , 1991 .

[26]  E. Muraki.  A Generalized Partial Credit Model: Application of an EM Algorithm , 1992 .

[27]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[28]  S. Geisser,et al.  A Predictive Approach to Model Selection , 1979 .

[29]  J. Albert.  Bayesian Estimation of Normal Ogive Item Response Curves Using Gibbs Sampling , 1992 .

[30]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[31]  Robert K. Tsutakawa,et al.  Estimation of Two-Parameter Logistic Item Response Curves , 1984 .

[32]  H. Skaug.  Automatic Differentiation to Facilitate Maximum Likelihood Estimation in Nonlinear Random Effects Models , 2002 .

[33]  Radford M. Neal.  Slice Sampling , 2003, The Annals of Statistics.

[34]  F. Vaida,et al.  Conditional Akaike information for mixed-effects models , 2005 .

[35]  Robert J. Mislevy,et al.  A HIERARCHICAL ITEM RESPONSE MODEL FOR EDUCATIONAL TESTING , 1989 .

[36]  A. Béguin,et al.  MCMC estimation of multidimensional IRT models , 1998 .

[37]  Raymond J. Adams,et al.  Multilevel Item Response Models: An Approach to Errors in Variables Regression , 1997 .

[38]  J. Fox,et al.  Bayesian estimation of a multilevel IRT model using gibbs sampling , 2001 .

[39]  Robert K. Tsutakawa,et al.  Approximation for Bayesian Ability Estimation , 1988 .

[40]  A. Raftery.  Bayesian Model Selection in Social Research , 1995 .

[41]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[42]  Richard J. Patz,et al.  A Straightforward Approach to Markov Chain Monte Carlo Methods for Item Response Models , 1999 .

[43]  Jean-Paul Fox,et al.  Bayesian longitudinal item response modeling with restricted covariance pattern structures , 2014, Statistics and Computing.

[44]  R. Darrell Bock,et al.  High-dimensional Full-information Item Factor Analysis , 1997 .

[45]  S. Raudenbush,et al.  Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order, Multivariate Laplace Approximation , 2000 .

[46]  T. Ando.  Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models , 2007 .

[47]  J-P Fox,et al.  Multilevel IRT using dichotomous and polytomous response data. , 2005, The British journal of mathematical and statistical psychology.

[48]  M. Newton.  Approximate Bayesian-inference With the Weighted Likelihood Bootstrap , 1994 .

[49]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[50]  Zucchini,et al.  An Introduction to Model Selection. , 2000, Journal of mathematical psychology.

[51]  Wasserman,et al.  Bayesian Model Selection and Model Averaging. , 2000, Journal of mathematical psychology.

[52]  R. D. Bock,et al.  Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm , 1981 .

[53]  H. Bozdogan.  Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions , 1987 .

[54]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[55]  W. Gilks,et al.  Adaptive Rejection Sampling for Gibbs Sampling , 1992 .