The Bayesian Expectation-Maximization-Maximization for the 3PLM

Abstract

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).

Bayesian Modal Estimation for the One-Parameter Logistic Ability-Based Guessing (1PL-AG) Model

“…In contrast to the BEMM algorithm for 3PLM (Guo & Zheng, 2019), there is an obvious difference between two algorithms, even though both of them belong to the mixture-modeling framework, share the similar E-step and the same name. In the 3PLM, the first M-step is to calculate the guessing parameter via a closed-form solution, and then maximize the difficulty and discrimination parameters in the second M-step.…”

“…Next, a latent indicator variable W ij is defined to indicate whether examinee i knows the answer of item j only using the p-process (Béguin & Glas, 2001; Culpepper, 2015; Guo & Zheng, 2019). Specifically, let…”

“…The 1PL-AG model is applied to the flexMIRT example data set ''g341-19.dat'' which consists of the responses to 12 items from 2,844 examinees. In a prior study, Guo and Zheng (2019) calibrated the items using the 3PLM and found the difficulty of the whole test is relatively low. In this case, the authors believe this example can be a necessary complement to our simulation studies which are designed for a high difficulty situation.…”

Fast Bayesian Estimation for the Four-Parameter Logistic Model (4PLM)

“…Compared to MCMC methods, BME is a fast calibration method, but several issues need to be explored to evaluate the utility of BME and its implementation in the mirt package. For example, Guo and Zheng (2019) found that the item parameter estimates yielded by BME were unstable for the 3PLM when changing priors of guessing parameters. Moreover, in the mirt package of version 1.30, the standard errors of all item parameters cannot be calculated as a consequence of program errors in computing the Hessian matrix when a Beta distribution prior ("expbeta" in mirt) is imposed on guessing or slipping parameters.…”

IRTBEMM: An R Package for Estimating IRT Models With Guessing or Slipping Parameters

“…The R package IRTBEMM (Guo et al, 2020) is designed to implement the family of the Bayesian Expectation-Maximization-Maximization (BEMM) algorithm to estimate unidimensional dichotomous item response models with guessing or slipping parameters. The BEMM family includes (a) the BEMM algorithm (Guo & Zheng, 2019) for three-parameter logistic (3PL) model and one-parameter logistic guessing (1PL-G) model; (b) the Bayesian Expectation-Maximization-Maximization-Maximization (BE3M) algorithm (Zhang et al, 2018) for four-parameter logistic (4PL) model and one-parameter logistic ability-based guessing (1PL-AG) model; and (c) their maximum likelihood estimation versions developed by Zheng et al (2018). Both Bayesian modal estimates and maximum likelihood estimates are available.…”