Latent Mixture Modeling-Based Expectation-3-Maximization And Bayesian Expectation-3-Maximization Algorithm For The 4-parameter Logistic Model

There is renewed interest in the four-parameter logistic model,but the lack of a user-friendly calibration method constitutes a major barrier to its widespread application.In the present study,this researcher reformulated the 4PL model from a latent mixture modeling view and developed the Expectation-3-Maximization(EMMM)algorithm.Combining the EMMM with the Bayesian approach,allowed the Bayesian Expectation-3-Maximization(BEMMM)algorithm to be proposed.By introducing a latent variable,the new algorithm separated the Maximization-Step in original EM algorithm into three steps,one for lower asymptote parameter,one for upper asymptote parameter,the other for discrimination and difficulty parameters.Then,the estimation of these parameters can be separated.That is the reason why the new method is accurate and fast.Through the definition of two new artificial data based on the original artificial data in EM,the researcher obtained a different interpretation of lower and upper asymptote parameters.In the first simulation study,the author compared the EMMM with BEMMM to confirm that the BEMMM method reduced the number of implausible estimates in EMMM.Next,when comparing the recovery of BEMMM with the Markov Chain Monte Carlo method(Culpepper,2016)and Bayesian Modal Estimation(Waller & Feuerstahler,2017),the results indicated that the BEMMM and the MCMC are more accurate than the BME.In the end,the results of a real data example for a socially desirability responding test showed the practicability of BEMMM.Meanwhile,the calculating speed of BEMMM is much faster than the MCMC.