Bayesian Longitudinal Multilevel Item Response Modeling Approach And Its Application

In the educational and psychological testing,how to determine the growth trend of examinees'latent traits over time is still an open question.To investigate such a growth trend,examinees have to attend many tests at different occasions.Item response theory?IRT?is widely used to model the relationship between the responses on the test and the latent trait.Interested readers can refer to Lord?1980?^[64].In the longitudinal data analysis,it is usually assumed that examinees'latent traits will change over time,which leads to more information than the one-time test.If a test consists of several subtests,the latent traits may have complex structure,and the traditional IRT models may not fit the longitudinal data well.In this paper,a longitudinal multilevel item response model with polynomial structure is proposed to measure the tendency of latent traits of examinees over time and the influence of covariables on this growth.A combined Bayesian procedure is developed to estimate model parameters.The validity of the proposed model and the precision of the combined Bayesian procedure are examined via simulation studies.The deviance information criterion?DIC?and the widely applicable information crite-rion?WAIC?are used as model selection indices.The simulation results show that the combined Bayesian estimation method can recover the model parameters well under various conditions.The results of DIC and WAIC are similar,and select the same best model among a set of candidate models.Finally,a longitudinal dataset about the de-velopment of achievement is analyzed to illustrate the significance and implementation of the proposed procedure.Moreover,in view of the possible dependence of some longitudinal data in the actual research,we focus on the test design and statistical modeling.We propose a longitudinal multilevel dichotomous item response theory model with antedependent?AD?residuals,and develop a Bayesian method to estimate parameters.The method can provide guarantee for accurate estimation of unknown parameters,so as to describe statistical rules of the true longitudinal data in the mathematics evaluation of students from grade 3 to grade 6.DIC and WAIC are used to evaluate all candidate models and choose the better model.As models become more complex,such as longitudinal,multilevel,graded re-sponse models,parameter estimation using the traditional Bayesian estimation algo-rithm?MCMC?will be time-consuming.Therefore,an improved estimation method,namely quick MCMC?QMCMC?,is proposed in this paper.The accuracy and effec-tiveness of the method will be illustrated via simulation studies.Under the simulation conditions,QMCMC improves its efficiency by nearly a quarter compared with M-CMC.When the whole simulation experiment needs to be done hundreds or thousands of times,QMCMC lead to obvious reduction of computational time with satisfactory accuracy.In summary,according to the dichotomous item response data,this paper propose two different longitudinal multilevel item response theory models to reveal the changes of examinees'latent traits over time,and MATLAB is used to implement the Bayesian estimation of model parameters.In this study,data from two empirical examples are respectively modeled by the proposed models,respectively,to show the longitudinal change trend of data.In addition,we propose a new Bayesian estimation method of a longitudinal multilevel polytomous item response theory model,which can greatly improve the estimation efficiency.