Modeling longitudinal data with non-random drop-outs using copulas

This paper develops a class of parametric models for longitudinal data with non-random drop-outs. Marginal regression models are fitted to the repeated measurements and the drop-out profiles to account for co-variate and time effects. The dependence between successive responses and between drop-out and response is also modeled using particular dependence functions, called copulas. Copulas are used to create a joint distribution with given marginal distributions. In our case, two copulas are used to obtain a parametric form for the joint density of the repeated responses and of the drop-out indicators. Parameters of the two marginal and copula models are jointly estimated using maximum likelihood. The method evenly applies to continuous or non-continuous ordinal responses and illustrations are provided for the two cases. A log-normal response (heart rate) is analysed in a toxicology study and a discrete ordinal variable (pain severity) is considered in a migraine trial. Three copula families are used: the product copula, the copula of Frank (1979) and a copula generated from the standard multivariate normal distribution. The use of copulas allows a consistent and flexible modeling of the dependence structure for continuous and non-continuous repeated measurements. Moreover, it permits a valid likelihood-based inference on marginal models for both the response and the drop-out process when the pattern of drop-out is not missing completely at random. In the two examples, serial dependence was detected in the data and the drop-out hazard was moderately depending on previous responses. Parameter estimates from the marginal models were found insensitive to the choice of the dependence model between drop-out and response