Latent variable selection in multidimensional item response theory models using the expectation model selection algorithm

Abstract

The aim of latent variable selection in multidimensional item response theory (MIRT) models is to identify latent traits probed by test items of a multidimensional test. In this paper the expectation model selection (EMS) algorithm proposed by Jiang et al. (2015) is applied to minimize the Bayesian information criterion (BIC) for latent variable selection in MIRT models with a known number of latent traits. Under mild assumptions, we prove the numerical convergence of the EMS algorithm for model selection by minimizing the BIC of observed data in the presence of missing data. For the identification of MIRT models, we assume that the variances of all latent traits are unity and each latent trait has an item that is only related to it. Under this identifiability assumption, the convergence of the EMS algorithm for latent variable selection in the multidimensional two‐parameter logistic (M2PL) models can be verified. We give an efficient implementation of the EMS for the M2PL models. Simulation studies show that the EMS outperforms the EM‐based L1 regularization in terms of correctly selected latent variables and computation time. The EMS algorithm is applied to a real data set related to the Eysenck Personality Questionnaire.

Accelerating L1-penalized expectation maximization algorithm for latent variable selection in multidimensional two-parameter logistic models

“…In the literature, Xu et al [ 26 ] gives a similar approach to choose the naive augmented data ( y ij , θ i ) with larger weight for computing Eq (8) . In this paper, we however choose our new artificial data ( z , θ ( g ) ) with larger weight to compute Eq (15) .…”

“…The data set includes 754 Canadian females’ responses (after eliminating subjects with missing data) to 69 dichotomous items, where items 1–25 consist of the psychoticism (P), items 26–46 consist of the extraversion (E) and items 47–69 consist of the neuroticism (N). This data set was also analyzed in Xu et al [ 26 ]. In order to guarantee the psychometric properties of the items, we select those items whose corrected item-total correlation values are greater than 0.2 [ 39 ].…”

“…Further development for latent variable selection in MIRT models can be found in [ 25 , 26 ]. Zhang and Chen [ 25 ] proposed a stochastic proximal algorithm for optimizing the L 1 -penalized marginal likelihood.…”

“…However, the choice of several tuning parameters, such as a sequence of step size to ensure convergence and burn-in size, may affect the empirical performance of stochastic proximal algorithm. Xu et al [ 26 ] applied the expectation model selection (EMS) algorithm [ 27 ] to minimize the L 0 -penalized log-likelihood (for example, the Bayesian information criterion [ 28 ]) for latent variable selection in MIRT models. In their EMS framework, the model (i.e., structure of loading matrix) and parameters (i.e., item parameters and the covariance matrix of latent traits) are updated simultaneously in each iteration.…”

“…In their EMS framework, the model (i.e., structure of loading matrix) and parameters (i.e., item parameters and the covariance matrix of latent traits) are updated simultaneously in each iteration. In the simulation of Xu et al [ 26 ], the EMS algorithm runs significantly faster than EML1, but it still requires about one hour for MIRT with four latent traits.…”