Bayesian Estimation of the DINA Model With Pólya-Gamma Gibbs Sampling

A sequential exploratory diagnostic model using a Pólya-gamma data augmentation strategy

Cognitive diagnostic models provide a framework for classifying individuals into latent proficiency classes, also known as attribute profiles. Recent research has examined the implementation of a Pólya-gamma data augmentation strategy binary response model using logistic item response functions within a Bayesian Gibbs sampling procedure. In this paper, we propose a sequential exploratory diagnostic model for ordinal response data using a logit-link parameterization at the category level and extend the Pólya-gamma data augmentation strategy to ordinal response processes. A Gibbs sampling procedure is presented for efficient Markov chain Monte Carlo (MCMC) estimation methods. We provide results from a Monte Carlo study for model performance and present an application of the model.

Exploratory Restricted Latent Class Models with Monotonicity Requirements under PÒLYA-GAMMA Data Augmentation

Restricted latent class models (RLCMs) provide an important framework for supporting diagnostic research in education and psychology. Recent research proposed fully exploratory methods for inferring the latent structure. However, prior research is limited by the use of restrictive monotonicity condition or prior formulations that are unable to incorporate prior information about the latent structure to validate expert knowledge. We develop new methods that relax existing monotonicity restrictions and provide greater insight about the latent structure. Furthermore, existing Bayesian methods only use a probit link function and we provide a new formulation for using the exploratory RLCM with a logit link function that has an additional advantage of being computationally more efficient for larger sample sizes. We present four new Bayesian formulations that employ different link functions (i.e., the logit using the Pòlya-gamma data augmentation versus the probit) and priors for inducing sparsity in the latent structure. We report Monte Carlo simulation studies to demonstrate accurate parameter recovery. Furthermore, we report results from an application to the Last Series of the Standard Progressive Matrices to illustrate our new methods.

Modeling Not-Reached Items in Cognitive Diagnostic Assessments

In cognitive diagnostic assessments with time limits, not-reached items (i.e., continuous nonresponses at the end of tests) frequently occur because examinees drop out of the test due to insufficient time. Oftentimes, the not-reached items are related to examinees’ specific cognitive attributes or knowledge structures. Thus, the underlying missing data mechanism of not-reached items is non-ignorable. In this study, a missing data model for not-reached items in cognitive diagnosis assessments was proposed. A sequential model with linear restrictions on item parameters for missing indicators was adopted; meanwhile, the deterministic inputs, noisy “and” gate model was used to model the responses. The higher-order structure was used to capture the correlation between higher-order ability parameters and dropping-out propensity parameters. A Bayesian Markov chain Monte Carlo method was used to estimate the model parameters. The simulation results showed that the proposed model improved diagnostic feedback results and produced accurate item parameters when the missing data mechanism was non-ignorable. The applicability of our model was demonstrated using a dataset from the Program for International Student Assessment 2018 computer-based mathematics cognitive test.

Bayesian Analysis of Aberrant Response and Response Time Data

In this article, a highly effective Bayesian sampling algorithm based on auxiliary variables is proposed to analyze aberrant response and response time data. The new algorithm not only avoids the calculation of multidimensional integrals by the marginal maximum likelihood method but also overcomes the dependence of the traditional Metropolis–Hastings algorithm on the tuning parameter in terms of acceptance probability. A simulation study shows that the new algorithm is accurate for parameter estimation under simulation conditions with different numbers of examinees, items, and speededness levels. Based on the sampling results, the powers of the two proposed Bayesian assessment criteria are tested in the simulation study. Finally, a detailed analysis of a high-state and large-scale computerized adaptive test dataset is carried out to illustrate the proposed methodology.

Estimating CDMs Using the Slice-Within-Gibbs Sampler

In this paper, the slice-within-Gibbs sampler has been introduced as a method for estimating cognitive diagnosis models (CDMs). Compared with other Bayesian methods, the slice-within-Gibbs sampler can employ a wide-range of prior specifications; moreover, it can also be applied to complex CDMs with the aid of auxiliary variables, especially when applying different identifiability constraints. To evaluate its performances, two simulation studies were conducted. The first study confirmed the viability of the slice-within-Gibbs sampler in estimating CDMs, mainly including G-DINA and DINA models. The second study compared the slice-within-Gibbs sampler with other commonly used Markov Chain Monte Carlo algorithms, and the results showed that the slice-within-Gibbs sampler converged much faster than the Metropolis-Hastings algorithm and more flexible than the Gibbs sampling in choosing the distributions of priors. Finally, a fraction subtraction dataset was analyzed to illustrate the use of the slice-within-Gibbs sampler in the context of CDMs.