  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 intrinsic and extrinsic causal
effects, which respectively provide information on treatment effect on the
intrinsic (product) structure of the product outcomes and on the outcomes'
dependence structure. We propose a log-mean linear regression approach for
modeling the distribution of the potential outcomes, which is particularly
appealing because all the causal estimands of interest and the decomposition
into intrinsic and extrinsic 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.
