Expectation-Maximization-Maximization: A Feasible MLE Algorithm for the Three-Parameter Logistic Model Based on a Mixture Modeling Reformulation

Abstract

Stable maximum likelihood estimation (MLE) of item parameters in 3PLM with a modest sample size remains a challenge. The current study presents a mixture-modeling approach to 3PLM based on which a feasible Expectation-Maximization-Maximization (EMM) MLE algorithm is proposed. The simulation study indicates that EMM is comparable to the Bayesian EM in terms of bias and RMSE. EMM also produces smaller standard errors (SEs) than MMLE/EM. In order to further demonstrate the feasibility, the method has also been applied to two real-world data sets. The point estimates in EMM are close to those from the commercial programs, BILOG-MG and flexMIRT, but the SEs are smaller.

The Bayesian Expectation-Maximization-Maximization for the 3PLM

“…A reformulation of the ability-based 3PLM, similar to Zheng et al (2017), can be derived readily. Following Culpepper (2015), we may introduce a latent variable v ij ∈ V , and vij~ Bernoulli(Pi*(θj)):…”

“…Two arrangements of these processes can be identified: the g-process comes first or the way around, and thus two different versions of reformulation of the 3PLM can be developed. Interesting enough, the one proposed by Zheng et al (2017) corresponds to the one with the g-process coming first and Béguin and Glas (2001) proposed an ability-based reformulation for the three-parameter normal ogive model (3PNO) which coincides with the one with the p-process coming first. The two reformulations are briefly reviewed here as the starting point of the two BEMM algorithms…”

“…Zheng et al (2017) proposed the EMM algorithm based on the new likelihood function derived from the reformulation with the g-process coming first (see Appendix A). The EMM can solve the convergence problem of MMLE/EM in a modest sample size of about 1,000 examinees, but it cannot eliminate occasional improbably large estimates.…”

“…Recently, Zheng et al (2017) pursued a different direction and developed a more powerful MLE algorithm based on a mixture modeling reformation of 3PLM, namely the Expectation-Maximization-Maximization (EMM). The EMM essentially is modified variant of MMLE/EM and its major difference from the traditional method is to expand the complete data from ( U , θ) to ( U , Z , θ) where U and θ are the response and examinee matrices, and Z is the latent indicator for whether some examinees use the guessing strategy for one item.…”

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.…”