Identification and Estimation of Causal Mechanisms in Clustered Encouragement Designs: a Bayesian Principal Stratification Approach

Encouragement designs arise frequently when the treatment cannot be enforced because of ethical or practical constrains and an encouragement is conceived with the purpose of increasing the uptake of the treatment of interest. By design, encouragements always entail the complication of non-compliance. Furthermore, they can give rise to a variety of mechanisms, especially when they are assigned at cluster level. In fact, social interactions among units receiving different treatments within the same cluster can result in spillover effects, i.e. one subject's outcome is also affected by the treatment received by other subjects belonging to the same social group. Disentangling the effect of encouragement through spillover effects and other mechanisms from that through the enhancement of the treatment would give a better insight into the intervention and it could be crucial for improving the program and planning the scale-up. Over the last decade, statistical analysis of mechanisms has focused on the estimation of effects involving hypothetical intervention on intermediate variables. Identification of these causal estimands requires sequential ignorability assumptions, which are usually questioned when the intermediate variable arises from non-compliance. We capitalize on the principal stratification framework to define stratum-specific causal effects, that is, effects for specific latent subpopulations, defined by the joint potential compliance statuses under both encouragement conditions. We provide alternative assumptions under which an extrapolation across principal strata allows to disentangle the effects. To face identification issues, estimation can be performed with Bayesian machinery using hierarchical models to account for clustering. We illustrate the proposed methodology on a cluster randomized experiment implemented in Zambia and designed to evaluate the impact on malaria prevalence of an agricultural loan program intended to increase bed net coverage.