Bayesian Estimation of the DINA Model With Gibbs Sampling

A Bayesian model formulation of the deterministic inputs, noisy “and” gate (DINA) model is presented. Gibbs sampling is employed to simulate from the joint posterior distribution of item guessing and slipping parameters, subject attribute parameters, and latent class probabilities. The procedure extends concepts in Béguin and Glas, Culpepper, and Sahu for estimating the guessing and slipping parameters in the three- and four-parameter normal-ogive models. The ability of the model to recover parameters is demonstrated in a simulation study. The technique is applied to a mental rotation test. The algorithm and vignettes are freely available to researchers as the “dina” R package.

[1]  Dirk Eddelbuettel,et al.  Rcpp: Seamless R and C++ Integration , 2011 .

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

[3]  J. Templin,et al.  Skills Diagnosis Using IRT-Based Latent Class Models , 2007 .

[4]  J. D. L. Torre,et al.  The Generalized DINA Model Framework. , 2011 .

[5]  Jingchen Liu,et al.  Theory of the Self-learning Q-Matrix. , 2010, Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability.

[6]  C. Mitchell Dayton,et al.  The Use of Probabilistic Models in the Assessment of Mastery , 1977 .

[7]  Jimmy de la Torre,et al.  An Empirically Based Method of Q‐Matrix Validation for the DINA Model: Development and Applications , 2008 .

[8]  Andrew Gelman,et al.  General methods for monitoring convergence of iterative simulations , 1998 .

[9]  Louis A. Roussos,et al.  The fusion model skills diagnosis system , 2007 .

[10]  Matthias von Davier,et al.  A general diagnostic model applied to language testing data. , 2008, The British journal of mathematical and statistical psychology.

[11]  Dirk Eddelbuettel,et al.  Seamless R and C++ Integration with Rcpp , 2013 .

[12]  J. D. L. Torre,et al.  DINA Model and Parameter Estimation: A Didactic , 2009 .

[13]  Jonathan Templin,et al.  Diagnostic Measurement: Theory, Methods, and Applications , 2010 .

[14]  J. Templin,et al.  Measurement of psychological disorders using cognitive diagnosis models. , 2006, Psychological methods.

[15]  Edward H. Haertel.  Using restricted latent class models to map the skill structure of achievement items , 1989 .

[16]  E. Maris.  Estimating multiple classification latent class models , 1999 .

[17]  B. Junker,et al.  Cognitive Assessment Models with Few Assumptions, and Connections with Nonparametric Item Response Theory , 2001 .

[18]  Matthias von Davier.  The DINA model as a constrained general diagnostic model: Two variants of a model equivalency. , 2014, The British journal of mathematical and statistical psychology.

[19]  S. Culpepper.  Revisiting the 4-Parameter Item Response Model: Bayesian Estimation and Application , 2016, Psychometrika.

[20]  Jean-Claude Falmagne,et al.  Knowledge spaces , 1998 .

[21]  Sarah M. Hartz,et al.  A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality. , 2002 .

[22]  T. Pham-Gia,et al.  Distribution of the linear combination of two general beta variables and applications , 1998 .

[23]  S. Sahu.  Bayesian Estimation and Model Choice in Item Response Models , 2002 .

[24]  P. K. Imbrie,et al.  Psychometric properties of the revised PSVT:R for measuring first year engineering students’ spatial ability , 2013 .

[25]  L. T. DeCarlo.  Recognizing Uncertainty in the Q-Matrix via a Bayesian Extension of the DINA Model , 2012 .

[26]  Alan Huebner,et al.  A Note on Comparing Examinee Classification Methods for Cognitive Diagnosis Models , 2011 .

[27]  Jimmy de la Torre,et al.  Model Evaluation and Multiple Strategies in Cognitive Diagnosis: An Analysis of Fraction Subtraction Data , 2008 .

[28]  Mary Kathryn Cowles,et al.  Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models , 1996, Stat. Comput..

[29]  J. Templin,et al.  The Effects of Q-Matrix Misspecification on Parameter Estimates and Classification Accuracy in the DINA Model , 2008 .

[30]  Chia-Yi Chiu.  Statistical Refinement of the Q-Matrix in Cognitive Diagnosis , 2013 .

[31]  Descriptors Educational.  Meeting of the American Educational Research , 1995 .

[32]  Jingchen Liu,et al.  Data-Driven Learning of Q-Matrix , 2012, Applied psychological measurement.

[33]  Z. Ying,et al.  Statistical Analysis of Q-Matrix Based Diagnostic Classification Models , 2015, Journal of the American Statistical Association.

[34]  M. Plummer,et al.  CODA: convergence diagnosis and output analysis for MCMC , 2006 .

[35]  John T. Willse,et al.  Defining a Family of Cognitive Diagnosis Models Using Log-Linear Models with Latent Variables , 2009 .

[36]  D. Uttal,et al.  The malleability of spatial skills: a meta-analysis of training studies. , 2013, Psychological bulletin.

[37]  Jeffrey A Douglas,et al.  Higher-order latent trait models for cognitive diagnosis , 2004 .

[38]  So Yoon Yoon,et al.  Psychometric properties of the Revised Purdue Spatial Visualization Tests: Visualization of Rotations (the Revised PSVT:R) , 2011 .

[39]  Christian P. Robert,et al.  Introducing Monte Carlo Methods with R , 2009 .

[40]  So Yoon Yoon,et al.  A Meta-Analysis on Gender Differences in Mental Rotation Ability Measured by the Purdue Spatial Visualization Tests: Visualization of Rotations (PSVT:R) , 2013 .