Covariate balance for no confounding in the sufficient-cause model

To show conditions of covariate balance for no confounding in the sufficient-cause model and discuss its relationship with exchangeability conditions.

We consider the link between the sufficient-cause model and the counterfactual model, emphasizing that the target population plays a key role when discussing these conditions. Furthermore, we incorporate sufficient causes within the directed acyclic graph framework. We propose to use each of the background factors in sufficient causes as representing a set of covariates of interest and discuss the presence of covariate balance by comparing joint distributions of the relevant background factors between the exposed and the unexposed groups.

We show conditions for partial covariate balance, covariate balance, and full covariate balance, each of which is stronger than partial exchangeability, exchangeability, and full exchangeability, respectively. This is consistent with the fact that the sufficient-cause model is a "finer" model than the counterfactual model.

Covariate balance is a sufficient, but not a necessary, condition for no confounding irrespective of the target population. Although our conceptualization of covariate imbalance is closely related to the recently proposed counterfactual-based definition of a confounder, the concepts of covariate balance and confounder should be clearly distinguished.