Bayesian estimation of a multilevel IRT model using gibbs sampling

In this article, a two-level regression model is imposed on the ability parameters in an item response theory (IRT) model. The advantage of using latent rather than observed scores as dependent variables of a multilevel model is that it offers the possibility of separating the influence of item difficulty and ability level and modeling response variation and measurement error. Another advantage is that, contrary to observed scores, latent scores are test-independent, which offers the possibility of using results from different tests in one analysis where the parameters of the IRT model and the multilevel model can be concurrently estimated. The two-parameter normal ogive model is used for the IRT measurement model. It will be shown that the parameters of the two-parameter normal ogive model and the multilevel model can be estimated in a Bayesian framework using Gibbs sampling. Examples using simulated and real data are given.

[1]  S. R. Searle.  Linear Models , 1971 .

[2]  D. Lindley,et al.  Bayes Estimates for the Linear Model , 1972 .

[3]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[4]  D. Rubin.  Estimation in Parallel Randomized Experiments , 1981 .

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

[6]  R. D. Bock,et al.  Marginal maximum likelihood estimation of item parameters , 1982 .

[7]  C. Morris.  Parametric Empirical Bayes Inference: Theory and Applications , 1983 .

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

[9]  Robert J. Mislevy,et al.  Bayes modal estimation in item response models , 1986 .

[10]  Jan de Leeuw,et al.  Random Coefficient Models for Multilevel Analysis , 1986 .

[11]  H. Goldstein.  Multilevel Statistical Models , 2006 .

[12]  S. Raudenbush.  Educational Applications of Hierarchical Linear Models: A Review , 1988 .

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

[14]  G. C. Wei,et al.  A Monte Carlo Implementation of the EM Algorithm and the Poor Man's Data Augmentation Algorithms , 1990 .

[15]  S. E. Hills,et al.  Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling , 1990 .

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

[17]  Donald Hedeker,et al.  Full-information item bi-factor analysis , 1992 .

[18]  M. Seltzer,et al.  Sensitivity Analysis for Fixed Effects in the Hierarchical Model: A Gibbs Sampling Approach , 1993 .

[19]  Herbert Hoijtink,et al.  On person parameter estimation in the dichotomous Rasch model , 1995 .

[20]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[21]  A. O'Hagan,et al.  Fractional Bayes factors for model comparison , 1995 .

[22]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[23]  Wing Hung Wong,et al.  Bayesian Analysis in Applications of Hierarchical Models: Issues and Methods , 1996 .

[24]  D. S. Sivia,et al.  Data Analysis , 1996, Encyclopedia of Evolutionary Psychological Science.

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

[26]  G. Roberts,et al.  Updating Schemes, Correlation Structure, Blocking and Parameterization for the Gibbs Sampler , 1997 .

[27]  I. W. Molenaar,et al.  A multidimensional item response model: Constrained latent class analysis using the gibbs sampler and posterior predictive checks , 1997 .

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

[29]  Eric T. Bradlow,et al.  A Bayesian random effects model for testlets , 1999 .

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

[31]  Brian W. Junker,et al.  Applications and Extensions of MCMC in IRT: Multiple Item Types, Missing Data, and Rated Responses , 1999 .

[32]  Howard Wainer,et al.  Testlet Response Theory: An Analog for the 3PL Model Useful in Testlet-Based Adaptive Testing , 2000 .

[33]  Eric T. Bradlow,et al.  MML and EAP Estimation in Testlet-based Adaptive Testing , 2000 .