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).
Highlights
The field of educational testing has witnessed successful development and implementation of a great variety of test item formats, including multiple-choice questions, constructed response questions, and complex performance-based questions
A more intuitive conclusion can be drawn from Figure 2: The root mean squared error (RMSE) of c parameters from the right plot shows the Bayesian EMM has lower RMSE than the Bayesian expectation maximization (EM) in BILOG-MG
The difference of RMSEs produced by the Bayesian EMM between two prior conditions are much smaller than the Bayesian EM, which means the Bayesian EMM tends to be less affected by priors and yields more stable estimates than the Bayesian EM
Summary
The field of educational testing has witnessed successful development and implementation of a great variety of test item formats, including multiple-choice questions, constructed response questions, and complex performance-based questions. By ignoring the prior terms (αi − 1) and (βi − 1), the data component of the estimate offers a very intuitive interpretation of the guessing parameter: It is calculated as the proportion of examinees who answer item i correctly using the guessing strategy in the total sample This interpretation nicely fits into general philosophy of mixture modeling, drastically different from the traditional interpretation which is defined as the lower bound for the probability with which an examinee answers an item correctly. One prior in this study comes from BILOG-MG default setting, c ∼ Beta (4, 16) with μ = 0.2, σ 2 = 0.008, the other is a more non-informative prior from the flexMIRT (Houts and Cai, 2015) default setting, c ∼ Beta(1, 4) with μ = 0.2, σ 2 = 0.027 To implement these algorithms, we developed a MATLAB toolbox, IRTEMM, to obtain the BEMM estimates. We ran 50 replications for each condition in the fully crossed 4 (two BEMM methods vs. two BILOG-MG methods) × 3(1,000 vs. 1,500 vs. 2,000) × 2(10 vs. 20) design
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