Marginalized maximum a posteriori estimation for the four-parameter logistic model under a mixture modelling framework.

The four-parameter logistic model (4PLM) has recently attracted much interest in various applications. Motivated by recent studies that re-express the four-parameter model as a mixture model with two levels of latent variables, this paper develops a new expectation-maximization (EM) algorithm for marginalized maximum a posteriori estimation of the 4PLM parameters. The mixture modelling framework of the 4PLM not only makes the proposed EM algorithm easier to implement in practice, but also provides a natural connection with popular cognitive diagnosis models. Simulation studies were conducted to show the good performance of the proposed estimation method and to investigate the impact of the additional upper asymptote parameter on the estimation of other parameters. Moreover, a real data set was analysed using the 4PLM to show its improved performance over the three-parameter logistic model.

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

[2]  Paul De Boeck,et al.  IRT Models for Ability-Based Guessing , 2006 .

[3]  Tianyou Wang,et al.  Precision of Warm’s Weighted Likelihood Estimates for a Polytomous Model in Computerized Adaptive Testing , 2001 .

[4]  J. Butcher,et al.  Advances in clinical personality measurement : An item response theory analysis of the MMPI-2 PSY-5 scales , 1999 .

[5]  R. Hambleton,et al.  21 Assessing the Fit of Item Response Theory Models , 2006 .

[6]  Ronald K. Hambleton,et al.  Information Curves and Efficiency of Three Logistic Test Models. , 1971 .

[7]  Hua-Hua Chang,et al.  Item-Weighted Likelihood Method for Ability Estimation in Tests Composed of Both Dichotomous and Polytomous Items , 2012 .

[8]  H. Jane Rogers,et al.  A Monte Carlo Investigation of Several Person and Item Fit Statistics for Item Response Models , 1987 .

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

[10]  Kelly L. Rulison,et al.  I've Fallen and I Can't Get Up: Can High-Ability Students Recover From Early Mistakes in CAT? , 2009, Applied psychological measurement.

[11]  Craig N. Mills,et al.  A Comparison of Several Goodness-of-Fit Statistics , 1985 .

[12]  M. Davier.  Is There Need for the 3PL Model? Guess What? , 2009 .

[13]  Hua-Hua Chang,et al.  The linear transformation model with frailties for the analysis of item response times. , 2013, The British journal of mathematical and statistical psychology.

[14]  Haruhiko Ogasawara,et al.  Asymptotic expansions for the ability estimator in item response theory , 2012, Comput. Stat..

[15]  Steven Andrew Culpepper.  The Prevalence and Implications of Slipping on Low-Stakes, Large-Scale Assessments , 2017 .

[16]  André A. Rupp,et al.  Item Response Modeling With BILOG-MG and MULTILOG for Windows , 2003 .

[17]  T. A. Warm.  Weighted likelihood estimation of ability in item response theory , 1989 .

[18]  Niels G Waller,et al.  Bayesian Modal Estimation of the Four-Parameter Item Response Model in Real, Realistic, and Idealized Data Sets , 2017, Multivariate behavioral research.

[19]  Detection of determinant genes and diagnostic via Item Response Theory , 2004 .

[20]  Randall D. Penfield,et al.  Applying a Weighted Maximum Likelihood Latent Trait Estimator to the Generalized Partial Credit Model , 2005 .

[21]  Frank B. Baker,et al.  Item Response Theory : Parameter Estimation Techniques, Second Edition , 2004 .

[22]  N. Waller,et al.  Abstract: Estimation of the 4-Parameter Model with Marginal Maximum Likelihood , 2014, Multivariate behavioral research.

[23]  Eric Loken,et al.  Estimation of a four-parameter item response theory model. , 2010, The British journal of mathematical and statistical psychology.

[24]  D. Magis.  A Note on the Item Information Function of the Four-Parameter Logistic Model , 2013 .

[25]  J. Neyman,et al.  Consistent Estimates Based on Partially Consistent Observations , 1948 .

[26]  S. Reise,et al.  How many IRT parameters does it take to model psychopathology items? , 2003, Psychological methods.

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