A Partially Confirmatory Approach to the Multidimensional Item Response Theory with the Bayesian Lasso
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A PARTIALLY CONFIRMATORY APPROACH TO THE MULTIDIMENSIONAL ITEM RESPONSE THEORY WITH THE BAYESIAN LASSO
Jinsong Chen THE UNIVERSITY OF HONG KONG
For test development in the setting of multidimensional item response theory, the exploratory and confirmatory approaches lie on two ends of a continuum in terms of the loading and residual structures. Inspired by the recent development of the Bayesian Lasso (least absolute shrinkage and selection operator), this research proposes a partially confirmatory approach to estimate both structures using Bayesian regression and a covariance Lasso within a unified framework. The Bayesian hierarchical formulation is implemented using Markov chain Monte Carlo estimation, and the shrinkage parameters are estimated simultaneously. The proposed approach with different model variants and constraints was found to be flexible in addressing loading selection and local dependence. Both simulated and real-life data were analyzed to evaluate the performance of the proposed model across different situations. Key words: MIRT, Bayesian Lasso, partially confirmatory, Lasso loading, local dependence.
It is common to encounter multiple latent traits with categorical data when developing educational and psychological tests. Multidimensional item response theory (MIRT) provides an important analytical modeling framework for this scenario. MIRT models are constructed mainly on the basis of two relationships: (1) the relationship between the items and traits which will be referred to as loading; and (2) the relationship between the items after controlling the traits or the residual covariance matrix with nonzero off-diagonal elements which is often referred to as local dependence (e.g., Embretson & Reise, 2000; Reckase, 2009). Two typical approaches of test development can be differentiated. In the confirmatory approach, both relationships are prespecified based on constraints or substantive knowledge and the main purpose is to confirm the structure of the relationships. In the exploratory approach, both relationships are unspecified without any constraint or substantive knowledge and the purpose is to explore an appropriate structure of the relationships. The exploratory approach is often referred to as item factor analysis when assuming local independence (Bock, Gibbons, & Muraki, 1988; Wirth & Edwards, 2007). The two typical approaches can be connected with exploratory factor analysis (1966) or confirmatory factor analysis (CFA; Jöreskog, 1969), respectively, in the case of continuous data. In general, however, different amounts of substantive input can be available, making the exploratory and confirmatory approaches two ends of a continuum. Recent development of regularization methods enables more flexibility to address the two relationships in both approaches. On the exploratory end, loading estimation is usually conceptualized as a regularized variableselection problem with the assumption of local independence under the latent variable modeling context (e.g., Lu, Chow, & Loken, 2016; Sun, Chen,
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