Modelling correlated proportion data: Bayesian and classical approaches to inference and diagnostics

Correlated proportions appear in many real-world applications and present a unique challenge in finding an appropriate probabilistic model and performing inference due to their constrained nature. The bivariate beta is a natural extension of the well-known beta distribution to the space of correlated quantities on [0, 1]² , but its construction is not unique. Over the years, many bivariate beta distributions have been proposed, ranging from three to eight or more parameters, and for which the joint density and distribution moments vary in terms of mathematical tractability. As a consequence, the literature on statistical inference for correlated proportion data is scattered. In this paper, we address this gap by thoroughly studying two rather different bivariate models from their construction to inference and diagnostics. We investigate two bivariate constructions namely the four-parameter bivariate beta proposed by Olkin and Trikalinos (2015) (OT) and a bivariate logistic normal distribution, which has one extra parameter. We provide a thorough account of inference, exploring both classical (frequentist) and Bayesian approaches to estimation, utilising the method of moments and latent variable augmentation coupled with Hamiltonian Monte Carlo, respectively. The elicitation of a bivariate beta distribution as a prior is also discussed. Further, we develop diagnostics for checking model fit and adequacy and test their performance with Monte Carlo experiments under well-specified and misspecified data-generating settings. We illustrate these methods using data on childhood vaccination coverage and on sensitivity/specificity from COVID-19 tests, comparing the OT bivariate beta model with a bivariate logit-normal model.