A Restricted Four-Parameter IRT Model: The Dyad Four-Parameter Normal Ogive (Dyad-4PNO) Model

Recently, there has been a renewed interest in the four-parameter item response theory model as a way to capture guessing and slipping behaviors in responses. Research has shown, however, that the nested three-parameter model suffers from issues of unidentifiability (San Martín et al. in Psychometrika 80:450-467, 2015), which places concern on the identifiability of the four-parameter model. Borrowing from recent advances in the identification of cognitive diagnostic models, in particular, the DINA model (Gu and Xu in Stat Sin https://doi.org/10.5705/ss.202018.0420 , 2019), a new model is proposed with restrictions inspired by this new literature to help with the identification issue. Specifically, we show conditions under which the four-parameter model is strictly and generically identified. These conditions inform the presentation of a new exploratory model, which we call the dyad four-parameter normal ogive (Dyad-4PNO) model. This model is developed by placing a hierarchical structure on the DINA model and imposing equality constraints on a priori unknown dyads of items. We present a Bayesian formulation of this model, and show that model parameters can be accurately recovered. Finally, we apply the model to a real dataset.

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

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

[3]  Steven Andrew Culpepper,et al.  Bayesian Estimation of the DINA Model With Gibbs Sampling , 2015 .

[4]  Shaoyan Guo,et al.  Expectation-Maximization-Maximization: A Feasible MLE Algorithm for the Three-Parameter Logistic Model Based on a Mixture Modeling Reformulation , 2018, Front. Psychol..

[5]  Francis Tuerlinckx,et al.  Identified Parameters, Parameters of Interest and Their Relationships , 2009 .

[6]  Janice A. Gifford,et al.  Bayesian estimation in the three-parameter logistic model , 1986 .

[7]  Yuan Tian,et al.  A Bayesian semi-parametric mixture model for bivariate extreme value analysis with application to precipitation forecasting , 2021 .

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

[9]  I. Partchev.  3PL: A Useful Model with a Mild Estimation Problem , 2009 .

[10]  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.

[11]  Furong Gao,et al.  Bayesian or Non-Bayesian: A Comparison Study of Item Parameter Estimation in the Three-Parameter Logistic Model , 2005 .

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

[13]  C. Matias,et al.  Identifiability of parameters in latent structure models with many observed variables , 2008, 0809.5032.

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

[15]  Kyung T. Han,et al.  Fixing the c Parameter in the Three-Parameter Logistic Model , 2012 .

[16]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[17]  S. Reise,et al.  Measuring Psychopathology With Nonstandard Item Response Theory Models: Fitting the Four-Parameter Model to the Minnesota Multiphasic Personality Inventory , 2010 .

[18]  Yinyin Chen,et al.  A Sparse Latent Class Model for Cognitive Diagnosis , 2020, Psychometrika.

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

[20]  Francis Tuerlinckx,et al.  On the Unidentifiability of the Fixed-Effects 3PL Model , 2015, Psychometrika.

[21]  Frederic M. Lord,et al.  An Upper Asymptote for the Three-Parameter Logistic Item-Response Model. , 1981 .

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

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

[24]  Gongjun Xu,et al.  Marginalized maximum a posteriori estimation for the four-parameter logistic model under a mixture modelling framework. , 2020, The British journal of mathematical and statistical psychology.

[25]  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.

[26]  F. Baker,et al.  Item response theory : parameter estimation techniques , 1993 .

[27]  Arne Gabrielsen.  Consistency and identifiability , 1978 .

[28]  Michael R. Harwell,et al.  Item Parameter Estimation Via Marginal Maximum Likelihood and an EM Algorithm: A Didactic , 1988 .

[29]  On Interpreting the Model Parameters for the Three Parameter Logistic Model , 2009 .

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

[31]  Jean-Marie Rolin,et al.  Identification of the 1PL Model with Guessing Parameter: Parametric and Semi-parametric Results , 2013, Psychometrika.

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

[33]  Jiajin Yuan,et al.  Individual Differences in Spontaneous Expressive Suppression Predict Amygdala Responses to Fearful Stimuli: The Role of Suppression Priming , 2017, Frontiers in psychology.

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

[35]  Sophia Rabe-Hesketh,et al.  Bayesian Comparison of Latent Variable Models: Conditional Versus Marginal Likelihoods , 2018, Psychometrika.

[36]  D. Thissen.  On Interpreting the Parameters for any Item Response Model , 2009 .

[37]  Gongjun Xu,et al.  Sufficient and Necessary Conditions for the Identifiability of the Q-matrix , 2018, Statistica Sinica.