An Improved Strategy for Bayesian Estimation of the Reduced Reparameterized Unified Model

Abstract

A Bayesian formulation for a popular conjunctive cognitive diagnosis model, the reduced reparameterized unified model (rRUM), is developed. The new Bayesian formulation of the rRUM employs a latent response data augmentation strategy that yields tractable full conditional distributions. A Gibbs sampling algorithm is described to approximate the posterior distribution of the rRUM parameters. A Monte Carlo study supports accurate parameter recovery and provides evidence that the Gibbs sampler tended to converge in fewer iterations and had a larger effective sample size than a commonly employed Metropolis–Hastings algorithm. The developed method is disseminated for applied researchers as an R package titled “rRUM.”

Development and Application of an Exploratory Reduced Reparameterized Unified Model

“…Unlike Culpepper and Hudson (2017), the conjunctive condensation rule in Equation 3 is not a function of the Q matrix. Instead, we fix qjk=1 in Equation 3 to address the confound between rjk* and qjk.…”

“…Third, in our preliminary investigations, we found that estimating Q is more difficult whenever the associations among the attributes satisfy a more parsimonious structure. Accordingly, there may be some instances where a parsimonious model for attributes provides better fit than an unstructured model (e.g., see Culpepper & Hudson, 2017). We incorporate a higher order model (de la Torre & Douglas, 2004; Maris, 1999) for attributes to provide a more parsimonious model for the 2 K latent classes and offer a new Gibbs sampling algorithm to estimate model parameters.…”

Diagnostic Classification Models for Ordinal Item Responses

“…The priors are considered less informative because a large standard deviation (i.e., 20) produces a relatively flat-shaped normal distribution, and a conjugate Dirichlet distribution with all equivalent parameter values (e.g., 2,2,2,2) is approximately a uniform distribution. Using less informative priors are recommended in similar DCM studies such as Chen et al (2018), Culpepper and Hudson (2018), and Jiang and Carter (2018).…”

Estimating CDMs Using the Slice-Within-Gibbs Sampler

“…However, the main motivation of CDMs is to identify the latent attribute profiles of examinees' and Bayesian methods are often more natural to reach the goal. The second most commonly used method is Markov chain Monte Carlo (MCMC) method (de la Torre and Douglas, 2004 ; Culpepper, 2015 ; Culpepper and Hudson, 2018 ; Zhan et al, 2018 , 2019 ; Jiang and Carter, 2019 ). Usually, to use the MH algorithm, it is necessary to choose a proposal distribution that can lead to optimal sampling efficiency.…”

“…Culpepper ( 2015 ) first introduced the Gibbs sampling to the DINA model and Zhang et al ( 2020 ) applied the Pólya-Gamma Gibbs sampling based on auxiliary variables to DINA model. Culpepper and Hudson ( 2018 ) introduced Bayesian method to the Reduced Reparameterized Unified Model (rRUM; DiBello et al, 1995 ; Roussos et al, 2007 ).…”