We develop a latent variable selection method for multidimensional item response theory models. The proposed method identifies latent traits probed by items of a multidimensional test. Its basic strategy is to impose an $L_1$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_{1}$$\end{document} penalty term to the log-likelihood. The computation is carried out by the expectation–maximization algorithm combined with the coordinate descent algorithm. Simulation studies show that the resulting estimator provides an effective way in correctly identifying the latent structures. The method is applied to a real dataset involving the Eysenck Personality Questionnaire.

In multidimensional tests, the identification of latent traits measured by each item is crucial. In addition to item–trait relationship, differential item functioning (DIF) is routinely evaluated to ensure valid comparison among different groups. The two problems are investigated separately in the literature. This paper uses a unified framework for detecting item–trait relationship and DIF in multidimensional item response theory (MIRT) models. By incorporating DIF effects in MIRT models, these problems can be considered as variable selection for latent/observed variables and their interactions. A Bayesian adaptive Lasso procedure is developed for variable selection, in which item–trait relationship and DIF effects can be obtained simultaneously. Simulation studies show the performance of our method for parameter estimation, the recovery of item–trait relationship and the detection of DIF effects. An application is presented using data from the Eysenck Personality Questionnaire.