Bayesian Inference for Multivariate Ordinal Data Using Parameter Expansion

Bayesian analysis of longitudinal binary responses based on the multivariate probit model: A comparison of five methods

Dichotomous response data observed over multiple time points, especially data that exhibit longitudinal structures, are important in many applied fields. The multivariate probit model has been an attractive tool in such situations for its ability to handle correlations among the outcomes, typically by modeling the covariance (correlation) structure of the latent variables. In addition, a multivariate probit model facilitates controlled imputations for nonignorable dropout, a phenomenon commonly observed in clinical trials of experimental drugs or biologic products. While the model is relatively simple to specify, estimation, particularly from a Bayesian perspective that relies on Markov chain Monte Carlo sampling, is not as straightforward. Here we compare five sampling algorithms for the correlation matrix and discuss their merits: a parameter-expanded Metropolis-Hastings algorithm (Zhang et al., 2006), a parameter-expanded Gibbs sampling algorithm (Talhouk et al., 2012), a parameter-expanded Gibbs sampling algorithm with unit constraints on conditional variances (Tang, 2018), a partial autocorrelation parameterization approach (Gaskins et al., 2014), and a semi-partial correlation parameterization approach (Ghosh et al., 2021). We describe each algorithm, use simulation studies to evaluate their performance, and focus on comparison criteria such as computational cost, convergence time, robustness, and ease of implementations. We find that the parameter-expanded Gibbs sampling algorithm by Talhouk et al. (2012) often has the most efficient convergence with relatively low computational complexity, while the partial autocorrelation parameterization approach is more flexible for estimating the correlation matrix of latent variables for typical late phase longitudinal studies.

Bayes factor testing of equality and order constraints on measures of association in social research

Measures of association play a central role in the social sciences to quantify the strength of a linear relationship between the variables of interest. In many applications researchers can translate scientific expectations to hypotheses with equality and/or order constraints on these measures of association. In this paper a Bayes factor test is proposed for testing multiple hypotheses with constraints on the measures of association between ordinal and/or continuous variables, possibly after correcting for certain covariates. This test can be used to obtain a direct answer to the research question how much evidence there is in the data for a social science theory relative to competing theories. The stand-alone software package 'BCT' allows users to apply the methodology in an easy manner. The methodology will also be available in the R package 'BFpack'. An empirical application from leisure studies about the associations between life, leisure and relationship satisfaction and an application about the differences about egalitarian justice beliefs across countries are used to illustrate the methodology.

A Multivariate Probit Model for Learning Trajectories: A Fine-Grained Evaluation of an Educational Intervention

Advances in educational technology provide teachers and schools with a wealth of information about student performance. A critical direction for educational research is to harvest the available longitudinal data to provide teachers with real-time diagnoses about students' skill mastery. Cognitive diagnosis models (CDMs) offer educational researchers, policy makers, and practitioners a psychometric framework for designing instructionally relevant assessments and diagnoses about students' skill profiles. In this article, the authors contribute to the literature on the development of longitudinal CDMs, by proposing a multivariate latent growth curve model to describe student learning trajectories over time. The model offers several advantages. First, the learning trajectory space is high-dimensional and previously developed models may not be applicable to educational studies that have a modest sample size. In contrast, the method offers a lower dimensional approximation and is more applicable for typical educational studies. Second, practitioners and researchers are interested in identifying factors that cause or relate to student skill acquisition. The framework can easily incorporate covariates to assess theoretical questions about factors that promote learning. The authors demonstrate the utility of their approach with an application to a pre- or post-test educational intervention study and show how the longitudinal CDM framework can provide fine-grained assessment of experimental effects.

Likelihood Analysis of Multivariate Probit Models Using a Parameter Expanded MCEM Algorithm

Multivariate binary data arise in a variety of settings. In this paper, we propose a practical and efficient computational framework for maximum likelihood estimation of multivariate probit regression models. This approach uses the Monte Carlo EM (MCEM) algorithm, with parameter expansion to complete the M-step, to avoid the direct evaluation of the intractable multivariate normal orthant probabilities. The parameter expansion not only enables a closed-form solution in the M-step but also improves efficiency. Using the simulation studies, we compare the performance of our approach with the MCEM algorithms developed by Chib and Greenberg (1998) and Song and Lee (2005), as well as the iterative approach proposed by Li and Schafer (2008). Our approach is further illustrated using a real-world example.