Directed Acyclic Graphs, Sufficient Causes, and the Properties of Conditioning on a Common Effect

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Abstract
In this paper, the authors incorporate sufficient-component causes into the directed acyclic graph (DAG) causal framework in order to make apparent several properties of conditioning on a common effect. By incorporating sufficient causes on a graph, it is possible to detect conditional independencies within strata of the conditioning variable which are not evident on DAGs without the representation of sufficient causes. It is also possible to determine the sign of the conditional covariance of two causes when conditioning on their common effect if some knowledge of the sufficient-cause mechanisms for the common effect is available. The incorporation of sufficient causes within the DAG framework also allows for the representation of interactions on DAGs and for the unification of several different causal frameworks. For illustration, the results are applied to an example concerning the familial coaggregation of two disorders.