A Gibbs Sampling Algorithm with Monotonicity Constraints for Diagnostic Classification Models

Abstract

Diagnostic classification models (DCMs) are restricted latent class models with a set of crossclass equality constraints and additional monotonicity constraints on their item parameters, both of which are needed to ensure the meaning of classes and model parameters. In this paper, we develop an efficient, Gibbs sampling-based Bayesian Markov chain Monte Carlo estimation method for general DCMs with monotonicity constraints. A simulation study was conducted to evaluate parameter recovery of the algorithm which showed accurate estimation of model parameters. Moreover, the proposed algorithm was compared to a previously developed Gibbs sampling algorithm which imposed constraints on only the main effect item parameters of the log-linear cognitive diagnosis model. The newly proposed algorithm showed less bias and faster convergence. An analysis of the 2000 Programme for International Student Assessment reading assessment data using this algorithm was also conducted.

Efficient Metropolis-Hastings Robbins-Monro Algorithm for High-Dimensional Diagnostic Classification Models

“…In recent decades, SDCMs have been actively developed, such as the LCDM. In the present study, the following SDCM is adopted (Yamaguchi & Templin, 2021) as the basis for developing the eMHRM:…”

“…This SDCM is formulated where y ni is the nth respondent's observed item response to item i (y ni = 1 for correct response and y ni = 0 for incorrect response), η ih denotes the hth probability parameter for the correct response for item i, α n is the nth respondent's latent attribute vector indicating his/her mastering status (0: not mastered vs. 1: mastered), q i represents the ith row vector of a Q-matrix for item i indicating which attributes are required, d ih is a binary design vector for η ih , and I(Á) denotes the indicator function which equals one if d ih = α n and zero otherwise. Taking an item with q = (1, 0, 1) for example (Yamaguchi & Templin, 2021), the 2…”

“…Hence, making use of this feature can easily build up the posterior distributions of model parameters with appropriate conjugate prior distributions. In addition, notice that once the η estimates are obtained, it is feasible to derive the corresponding one-to-one parameter estimates of other SDCMs such as the LCDM (see Yamaguchi & Templin, 2021). Take the LCDM for example, the η 1 , η 2 , η 3 , and η 4 in Table A1 are equated as…”

“…The SDCM needs monotonicity constraints on η for each item to identify the model parameters (Yamaguchi & Templin, 2021). For instance, the constraint table for an item with q = (1, 1, 1) is shown in Table B1 of Appendix B.…”

“…In addition to the EM and the eMHRM, a new MH algorithm is proposed here as a second baseline for comparison purposes. Yamaguchi and Templin (2021) proposed a Gibbs sampler for the parameter estimation of the SDCMs. However, their Gibbs sampler needs to evaluate the posterior distribution of α over 2 K latent classes, requiring O(2 K ) calculations.…”