The Bayesian Expectation-Maximization-Maximization for the 3PLM

The current study proposes an alternative feasible Bayesian algorithm for the three-parameter logistic model (3PLM) from a mixture-modeling perspective, namely, the Bayesian Expectation-Maximization-Maximization (Bayesian EMM, or BEMM). As a new maximum likelihood estimation (MLE) alternative to the marginal MLE EM (MMLE/EM) for the 3PLM, the EMM can explore the likelihood function much better, but it might still suffer from the unidentifiability problem indicated by occasional extremely large item parameter estimates. Traditionally, this problem was remedied by the Bayesian approach which led to the Bayes modal estimation (BME) in IRT estimation. The current study attempts to mimic the Bayes modal estimation method and develop the BEMM which, as a combination of the EMM and the Bayesian approach, can bring in the benefits of the two methods. The study also devised a supplemented EM method to estimate the standard errors (SEs). A simulation study and two real data examples indicate that the BEMM can be more robust against the change in the priors than the Bayes modal estimation. The mixture modeling idea and this algorithm can be naturally extended to other IRT with guessing parameters and the four-parameter logistic models (4PLM).