Development and Application of an Exploratory Reduced Reparameterized Unified Model

Abstract

Exploratory cognitive diagnosis models (CDMs) estimate the Q matrix, which is a binary matrix that indicates the attributes needed for affirmative responses to each item. Estimation of Q is an important next step for improving classifications and broadening application of CDMs. Prior research primarily focused on an exploratory version of the restrictive deterministic-input, noisy-and-gate model, and research is needed to develop exploratory methods for more flexible CDMs. We consider Bayesian methods for estimating an exploratory version of the more flexible reduced reparameterized unified model (rRUM). We show that estimating the rRUM Q matrix is complicated by a confound between elements of Q and the rRUM item parameters. A Bayesian framework is presented that accurately recovers Q using a spike–slab prior for item parameters to select the required attributes for each item. We present Monte Carlo simulation studies, demonstrating the developed algorithm improves upon prior Bayesian methods for estimating the rRUM Q matrix. We apply the developed method to the Examination for the Certificate of Proficiency in English data set. The results provide evidence of five attributes with a partially ordered attribute hierarchy.

Partial-Mastery Cognitive Diagnosis Models

“…We may consider a factor analysis model for the latent scores; for instance, the one-factor model is {Φ −1 (d k ); k = 1, • • • , K} = µ + Λη + , where Λ is a K × 1 loading vector, η is a subject specific factor following standard normal distribution and follows a multivariate Kdimensional normal distribution with zero mean and diagonal covariance E. Then this model corresponds to {Φ −1 (d k ); k = 1, • • • , K} ∼ N (µ, ΛΛ + E), which has fewer model parameters than the proposed model in Section 2 (when K ≥ 4). The above factor analysis modeling is similar to the higher-order factor models and higher-order CDMs proposed in de la Torre and Douglas ( 2004), Templin et al (2008), andCulpepper andChen (2019); this is also related to the multidimensional IRT model (Reckase, 2009). However, as discussed in Section 3.1, our method differs from these existing studies due to the different cognitive diagnosis modeling from the latent scores to responses.…”

Variational Bayes Inference for the DINA Model

“…Q-matrix estimation is becoming an important area of study, which has attracted researchers’ attention (Y. Chen, Culpepper, Chen, & Douglas, 2018; Culpepper & Chen, 2018; Liu et al, 2012). Future studies would address this extension.…”

Higher education innovation and reform model based on hierarchical probit

“…The ordered probit model uses observable ordered response data to establish a model to study the changing law of unobservable latent variables. It is a special case of the restricted dependent variable model [7]. The happiness degree variable studied in this article has no specific sample data, so it is also a kind of latent variable, and its influence equation is expressed in linear form as follows.…”

“…From this, the approximation function G * (ε, υ; θ ) of the joint distribution function G of ε and υ can be obtained. We replace Gin the likelihood function (7) with the approximate distribution function G * (ε, υ; θ ). The semi-parametric estimator of the parameter vector (β , γ 0 , γ, λ 1 , λ 2 , λ 3 , θ ) is obtained by solving the quasi-likelihood maximization problem.…”