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.