Joint and marginal causal effects for binary non-independent outcomes

Causal inference on multiple non-independent outcomes raises serious challenges, because multivariate techniques that properly account for the outcome's dependence structure need to be considered. We focus on the case of binary outcomes framing our discussion in the potential outcome approach to causal inference. We define causal effects of treatment on joint outcomes introducing the notion of product outcomes. We also discuss a decomposition of the causal effect on product outcomes into marginal and joint causal effects, which respectively provide information on treatment effect on the marginal (product) structure of the product outcomes and on the outcomes’ dependence structure. We propose to model the distribution of the potential outcomes conditional on pre-treatment variables using the class of log-mean linear regression models, which is particularly appealing because all the causal estimands of interest and the decomposition into marginal and joint causal effects can be easily derived by model parameters. The method is illustrated in two randomized experiments concerning (i) the effect of the administration of oral pre-surgery morphine on pain intensity after surgery; and (ii) the effect of honey on nocturnal cough and sleep difficulty associated with childhood upper respiratory tract infections.