Estimating IRT Models Under Gaussian Mixture Modeling of Latent Traits: An Application of MSAEM Algorithm

The assumption of a normal distribution for latent traits is a common practice in item response theory (IRT) models. Numerous studies have demonstrated that this assumption is often inadequate, impacting the accuracy of statistical inferences in IRT models. To mitigate this issue, Gaussian mixture modeling (GMM) for latent traits, known as GMM-IRT, has been proposed. Moreover, the GMM-IRT models can also serve as powerful tools for exploring the heterogeneity of latent traits. However, the computation of GMM-IRT model estimation encounters several challenges, impeding its widespread application. The purpose of this paper is to propose a reliable and robust computing method for GMM-IRT model estimation. Specifically, we develop a mixed stochastic approximation EM (MSAEM) algorithm for estimating the three-parameter normal ogive model with GMM for latent traits (GMM-3PNO). Crucially, the GMM-3PNO is augmented to be a complete data model within the exponential family, thereby substantially streamlining the computation of the MSAEM algorithm. Furthermore, the MSAEM algorithm adeptly avoid the label-switching issue, ensuring its convergence. Finally, simulation and empirical studies are conducted to validate the performance of the MSAEM algorithm and demonstrate the superiority of the GMM-IRT models.