Variational Bayes inference for hidden Markov diagnostic classification models

Abstract

Diagnostic classification models (DCMs) can be used to track the cognitive learning states of students across multiple time points or over repeated measurements. This study developed an effective variational Bayes (VB) inference method for hidden Markov longitudinal general DCMs. The simulations performed in this study verified the validity of the proposed algorithm for satisfactorily recovering true parameters. Simulation and applied data analyses were conducted to compare the proposed VB method to Markov chain Monte Carlo (MCMC) sampling. The results revealed that the parameter estimates provided by the VB method were consistent with the MCMC method with the additional benefit of a faster estimation time. The comparative simulation also indicated differences between the two methods in terms of posterior standard deviation and coverage of 95% credible intervals. Thus, with limited computational resources and time, the proposed VB method can output estimations comparable to that of MCMC.

variationalDCM: An R package for variational Bayesian inference in diagnostic classification models

“…In the HM-DCM, model parameters depend on the choice of a measurement model. For example, Yamaguchi and Martinez (2023) consider the general DCM as the measurement model. In the setting, the probability of observing X ijt = x ijt is given as…”

“…lkt and g j2tl = K k=1 α q jkt lkt . Under this setting, the guessing and slip parameters correspond to the item parameter θ jht with θ j1t = guessing jt and θ j2t = 1 − slip jt (Yamaguchi and Martinez 2023).…”

“…For the DINO model, MC-DINA model, saturated DCM, and HM-DCM, the VB algorithm is developed in a similar manner as the above algorithm in the paper shown in Table 1. For more details, refer to Section 3 of Yamaguchi (2020), Section 3 of Yamaguchi and Okada (2020a), and Section 4 of Yamaguchi and Martinez (2023).…”

“…Furthermore, DCMs have recently been extended to the longitudinal item response data to model the time course of learning multiple attributes. The hidden Markov DCMs (HM-DCMs), which allow researchers to model the transition in mastery statuses over time, have been developed and applied to achieve this goal (e.g., Yamaguchi and Martinez 2023;Chen, Culpepper, Wang, and Douglas 2018b).…”

Recent Development of Parameter Estimation Methods in Diagnostic Classification Models

“…et al, 2017, pp.668-670) ． また，正則化推定法はアトリビュート階層構造の推定(Wang and Lu, 2021;Ma et al, 2023b) や縦断的 DCM(e.g.,Yamaguchi and Martinez, 2023)におけるアトリビュート習得パ タンの遷移で表される学習軌跡の推定(Wang, 2021) ，Q 行列の推定(Chen et al, 2020(Chen et al, , 2015Xu and Shang, 2018) Θ, π) + log(p(Θ)) + log(p(π))} とパラメタの推定値を得る方法である．ここで，log(p(Θ)) と log(p(π)) はそれぞれ項目 パラメタと混合比率パラメタの対数事前確率密度である．例えば，θ jh の事前分布と してパラメタが a 0 jh > 0, b 0 jh > 0 であるベータ分布，π の事前分布としてパラメタが…”