Publications & Resources
Abstract: Automated Fitting of MIRT Models by a Simultaneous Perturbation Algorithm
Scott Monroe and Li Cai
Item factor analysis (IFA) is a useful technique for studying the structure of latent variables measured by psychological tests. Cai (2010) introduced the Metropolis-Hastings Robbins-Monro (MH-RM) algorithm to obtain the maximum likelihood (ML) solution for high-dimensional exploratory IFA. Although MH-RM substantially alleviates the challenge of high dimensionality in IFA, its reliance on analytic first- and second-order derivatives of the complete-data model constrains the types of multidimensional IRT (MIRT) models that a researcher can easily specify within a software implementation. This reliance on analytical derivatives may be problematic when such expressions are difficult to obtain or computationally intensive to evaluate. Other forms of stochastic approximation, however, can be used to address this lack of flexibility. One such form, utilized in this research, is simultaneous perturbation stochastic approximation (SPSA; Spall, 1992). SPSA efficiently approximates the gradient through only two function evaluations regardless of the number of free parameters to be estimated. Essentially, the approximation amounts to a form of two-sided numerical differentiation. Also, the SPSA approach can be extended to yield an approximation to the Hessian (Spall, 1997). Due to its relative efficiency and ease of implementation, SPSA has been used in the physical and social sciences, not to mention many areas of engineering. The proposed algorithm, Metropolized SPSA, retains the basic structure of MH-RM but computes the derivatives and iteratively updates parameters using SPSA. Consequently, a software implementation can accommodate arbitrary MIRT models more readily than an implementation of MH-RM. For Metropolized SPSA, a researcher only needs to specify the functional form of an MIRT model. This modest requirement is particularly attractive when uncommon or novel models are used. The implementation of the proposed algorithm is validated via a simulation study that compares the ML solutions with those obtained by MH-RM. Also, an application is provided.
Monroe, S., & Cai, L. (2014). Abstract: Automated fitting of MIRT models by a simultaneous perturbation algorithm. Multivariate Behavioral Research, 49(3), 291, doi:10.1080/00273171.2014.9129 21