Modeling Partial Compliance Through Copulas in a Principal Stratification Framework

Within the principal stratification framework for causal inference, modeling partial compliance is challenging because the continuous nature of the principal strata raises subtle specification issues. In this context, we propose an approach based on the assumption that the joint distribution of the degree of compliance to the treatment and the degree of compliance to the control follows a Plackett copula, so that their association is modeled in a flexible way through a single parameter. Moreover, given the two compliances, the distribution of the outcomes is parameterized in a flexible way through a regression model which may include interaction and quadratic terms and may also be heteroscedastic. In order to estimate the parameters of the resulting model, and then the causal effect of the treatment, we adopt a maximum likelihood approach via the EM algorithm. In applying this approach, the marginal distributions of the two compliances are estimated by their empirical distribution functions, so that no constraints are posed on these distributions. Since the two compliances cannot be jointly observed, there is not direct empirical support for the association parameter. We describe a strategy for studying this parameter by a profile likelihood method and discuss an analysis of the sensitivity of the causal inference results to its value. We apply the proposed approach to data from a study about a drug for lowering cholesterol levels previously analyzed by Efron and Feldman and by Jin and Rubin. Estimated causal effects are in line with those of previous analyses, but the pattern of association between the compliances is qualitatively different, apparently due to the flexibility of the copula and to allowing regression equations in the proposed method to include interactions and heteroscedasticity.