A Bayesian Local Gaussian‐Copula Allocation Model for Bivariate Discrete‐Time First‐Event Data With Time‐Dependent Covariates

ABSTRACT

Bivariate first‐event outcomes arise in longitudinal biomedical studies when two clinically related events are assessed on a shared visit schedule. We propose a Bayesian local Gaussian‐copula allocation model for bivariate discrete‐time first‐event data with baseline and time‐dependent covariates. Two outcome‐specific working discrete‐time decrement models define interval‐specific working decrement probabilities and thereby generate marginal survival coordinates on the common interval grid. Among subjects jointly at risk at the start of an interval, a Gaussian copula is used locally to allocate probability mass across four terminal outcomes: no event, event 1 only, event 2 only, and same‐interval co‐occurrence. The Gaussian copula accommodates negative local dependence, independence, and positive local dependence. We consider constant, unstructured interval‐specific, and hierarchical shrinkage specifications for the dependence parameter, and compare them with frequentist and Bayesian odds‐ratio benchmarks on the observed‐probability scale. Because the native copula‐model coefficients act on the working decrement scale, observed‐scale covariate effects are summarized by Kullback–Leibler projection. In simulations, the Gaussian‐copula models were competitive with odds‐ratio benchmarks for observed‐scale covariate‐effect estimation; the hierarchical model provided a regularized alternative to unstructured interval‐specific dependence. We illustrate the method using National Health and Aging Trends Study data on first onset of mobility help and self‐care help. The application gave broadly similar observed‐scale covariate‐effect conclusions across the odds‐ratio, constant Gaussian‐copula, and hierarchical Gaussian‐copula models.