Parameter Estimation In The High-dimensional Cognitive Diagnosis Model

With the progress of society,people pay more and more attention to ed-ucation issues,and the education and psychological measurement is developing rapidly.The popularity of the Internet and computers have allowed us to col-lect large-scale data.Since people pay more and more attention to education issues,more sophisticated models are proposed to fit,predict,and estimate the latent ability of the examinees.At the beginning,researchers mainly focus on the continuous unidimensional latent ability.As the research interest shifts from ranking to classification,more research on the discrete multi-dimensional latent ability have emerged.The cognitive diagnosis model is a special type of the latent class model,which considers the discrete multi-dimensional latent ability in the modeling process and can be applied to the educational and psychological fields.The two main goals of the cognitive diagnosis model are to estimate the item parameters and classify the examinees.The classification of examinees is based on whether they have mastered the attributes or not.In the cognitive diagnosis models,the increase of attribute numbers will result in an exponential increase in model parameters.How to efficiently estimate the cognitive diagnosis model is a very important question,especially for the high-dimensional cognitive diag-nosis model.Aiming at this issue,this paper proposes two effective algorithms to obtain Bayesian estimation and maximum likelihood estimation,respectively.The main content of this article can be summarized into two parts.First,this paper proposes an efficient sequential Gibbs algorithm.In the previous literature,the Bayesian estimation in cognitive diagnosis models was to update all the attributes of an examinee at the same time.It is worth men-tioning that Culpepper&Hudson(2018)has updated the attributes one by one in the rRUM model,but his update process is severely dependent on the form of the rRUM model.In this paper,the sequential Gibbs algorithm is applied to the DINA and GDINA models.The advantages of this method can be sum-marized in the four points.First,the method doesn't depend on the forms of the cognitive diagnosis models.Second,JAGS is a widely used software for es-timating cognitive diagnosis models,and the calculation time of this method is much shorter than the time required by JAGS.Third,the GDINA model is a generalized framework for the cognitive diagnosis models,which can be degener-ated into many cognitive diagnosis models.This method can be applied to the GDINA model to show its good scalability.Fourth,the method is still efficient when estimating the high-dimensional cognitive diagnosis models.The simula-tion results show that the sequential Gibbs algorithm is consistent with JAGS in the low-dimensional case,but JAGS has failed in the high-dimensional case,and the sequential Gibbs algorithm still performs well.This paper also uses the sequential Gibbs algorithm analyze the Fraction-Subtraction Data and TIMSS The real data analysis results again confirm the reliability of the sequential Gibbs algorithm.Second,this paper proposes an improved SAEM algorithm to estimate the cognitive diagnosis model.The intention of the improvements is to make the EM algorithm more efficient to estimate the cognitive diagnosis model.The improvements include using Robbins-Monro update to complete approximation,generating attribute profiles by the sequential Gibbs algorithm,and an adaptive z-stop criterion to determine the stop time of the Markov chain.Robbins-Monro update uses more useful information in the approximation process to improve ac-curacy.The sequential Gibbs algorithm is used to generate a Markov chain,then realizations of attribute profiles can be sampled to approximate the E-step.The z-stop criterion is an adaptive criterion to determine the stop time of the Markov chain.The R package "CDM" is a common tool for estimating cognitive diag-nosis models,which uses the traditional EM algorithm.In the low-dimensional cognitive diagnosis model,"CDM" is more efficient than the improved SAEM algorithm,and in the high-dimensional cognitive diagnosis model,the improved SAEM algorithm is more efficient than“CDM”,so we suggest applying two methods depending on the dimensions.The contents of the simulation stud-ies are:discussing some characteristics of the improved SAEM algorithm itself;comparing the improved SAEM algorithm with the existing EM method;and the performance of the improved SAEM algorithm in high dimensions.In the real data analysis,we once again analyzed Fractional-Subtraction Data.This paper proposes two efficient methods,the sequential Gibbs algorithm and the improved SAEM algorithm,to estimate the high-dimensional cognitive diagnosis model.These two methods have good results when dealing with small sample sizes and correlated attributes.Although these two methods are based on the DINA and GDINA models,they mainly improve the process of dealing with attributes,which means that the methods in this paper can be extended to other cognitive diagnosis models.