Mixture-modelling-based Bayesian MH-RM algorithm for the multidimensional 4PLM.

Several recent works have tackled the estimation issue for the unidimensional four-parameter logistic model (4PLM). Despite these efforts, the issue remains a challenge for the multidimensional 4PLM (M4PLM). Fu et al. (2021) proposed a Gibbs sampler for the M4PLM, but it is time-consuming. In this paper, a mixture-modelling-based Bayesian MH-RM (MM-MH-RM) algorithm is proposed for the M4PLM to obtain the maximum a posteriori (MAP) estimates. In a comparison of the MM-MH-RM algorithm to the original MH-RM algorithm, two simulation studies and an empirical example demonstrated that the MM-MH-RM algorithm possessed the benefits of the mixture-modelling approach and could produce more robust estimates with guaranteed convergence rates and fast computation. The MATLAB codes for the MM-MH-RM algorithm are available in the online appendix.

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

The calibration of the one-parameter logistic ability-based guessing (1PL-AG) model in item response theory (IRT) with a modest sample size remains a challenge for its implausible estimates and difficulty in obtaining standard errors of estimates. This article proposes an alternative Bayesian modal estimation (BME) method, the Bayesian Expectation-Maximization-Maximization (BEMM) method, which is developed by combining an augmented variable formulation of the 1PL-AG model and a mixture model conceptualization of the three-parameter logistic model (3PLM). By comparing with marginal maximum likelihood estimation (MMLE) and Markov Chain Monte Carlo (MCMC) in JAGS, the simulation shows that BEMM can produce stable and accurate estimates in the modest sample size. A real data example and the MATLAB codes of BEMM are also provided.

Procedural fairness predicts cooperative behaviour for high-self-esteem individuals but not for low-self-esteem individuals.

Cooperation is vital for modern society. Previous studies showed that procedural fairness promotes cooperation; however, they mainly focused on cooperation intention, which may fail to reveal actual cooperative behaviour. Moreover, little is known regarding the personality boundary of the effect of procedural fairness on cooperation. Guided by previous findings that self-esteem increases sensitivity to procedural unfairness, we attempted to explore the moderating effect of self-esteem on the association between procedural fairness and cooperative behaviour. In Experiment 1, 160 participants' self-esteem was measured using the Rosenberg Self-Esteem Scale; procedural fairness was manipulated in two conditions, depending on whether money was allocated in an economic game by rolling the dice twice or an allocator's arbitrary choice. Cooperative behaviour was assessed using the chicken game paradigm. Experiment 2 (148 participants) aimed to replicate and extend the results of Experiment 1 using a more rigorous experimental design, in which the possible effect of outcome favourability was excluded. The results of both experiments consistently showed that procedural fairness positively predicted cooperative behaviour, and this association was significant in high-self-esteem individuals, but not in low-self-esteem individuals. These findings shed light on the vital role of self-esteem in understanding the relationship between procedural fairness and cooperative behaviour.

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

A recently released R package IRTBEMM is presented in this article. This package puts together several new estimation algorithms (Bayesian EMM, Bayesian E3M, and their maximum likelihood versions) for the Item Response Theory (IRT) models with guessing and slipping parameters (e.g., 3PL, 4PL, 1PL-G, and 1PL-AG models). IRTBEMM should be of interest to the researchers in IRT estimation and applying IRT models with the guessing and slipping effects to real datasets.

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).

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

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.