Abstract
The four-parameter logistic (4PL) model has recently attracted much interest in educational testing and psychological measurement. This paper develops a new Gibbs-slice sampling algorithm for estimating the 4PL model parameters in a fully Bayesian framework. Here, the Gibbs algorithm is employed to improve the sampling efficiency by using the conjugate prior distributions in updating asymptote parameters. A slice sampling algorithm is used to update the 2PL model parameters, which overcomes the dependence of the Metropolis–Hastings algorithm on the proposal distribution (tuning parameters). In fact, the Gibbs-slice sampling algorithm not only improves the accuracy of parameter estimation, but also enhances sampling efficiency. Simulation studies are conducted to show the good performance of the proposed Gibbs-slice sampling algorithm and to investigate the impact of different choices of prior distribution on the accuracy of parameter estimation. Based on Markov chain Monte Carlo samples from the posterior distributions, the deviance information criterion and the logarithm of the pseudomarginal likelihood are considered to assess the model fittings. Moreover, a detailed analysis of PISA data is carried out to illustrate the proposed methodology.
Highlights
Over the past four decades, item response theory (IRT) models have been extensively used in educational testing and psychological measurement (Lord and Novick, 1968; Van der Linden and Hambleton, 1997; Embretson and Reise, 2000; Baker and Kim, 2004)
When the number of examinees is fixed at 500, 1,000, or 2,000, and the number of items is fixed at 40, the average MSE and SD show that the recovery results of the discrimination, difficulty, guessing, and slipping parameters are close to those in the case where the total test length is 20, which indicates that the Gibbs-slice sampling algorithm is stable and there is no reduction in accuracy owing to an increase in the number of items
It is shown again that the Gibbs-slice sampling algorithm is effective and that the estimated results are accurate under various simulation conditions
Summary
Over the past four decades, item response theory (IRT) models have been extensively used in educational testing and psychological measurement (Lord and Novick, 1968; Van der Linden and Hambleton, 1997; Embretson and Reise, 2000; Baker and Kim, 2004). In this case, using WinBUGS to infer the model parameters may lead to biased estimates when the sample size (the number of examinees) is small and the prior distributions play an important role.
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