  Classical causal inference assumes treatments meant for a given unit do not
have an effect on other units. This assumption is violated in interference
problems, where new types of spillover causal effects arise, and causal
inference becomes much more difficult. In addition, interference introduces a
unique complication where variables may transmit treatment influences to each
other, which is a relationship that has some features of a causal one, but is
symmetric. In settings where a natural causal ordering on variables is not
available, addressing this complication using statistical inference methods
based on Directed Acyclic Graphs (DAGs) leads to conceptual difficulties. In
this paper, we develop a new approach to decomposing the spillover effect into
unit-specific components that extends the DAG based treatment decomposition
approach to mediation of Robins and Richardson. We give conditions for these
components of the spillover effect to be identified in a natural type of causal
model that permits stable symmetric relations among outcomes induced by a
process in equilibrium. We discuss statistical inference for identified
components of the spillover effect, including a maximum likelihood estimator,
and a doubly robust estimator for the special case of two interacting outcomes.
We verify consistency and robustness of our estimators via a simulation study,
and illustrate our method by assessing the causal effect of education
attainment on depressive symptoms using the data on households from the
Wisconsin Longitudinal Study.
