Inward and Outward Spillover Effects of One Unit's Treatment on Network Neighbors under Partial Interference

Abstract: In settings where interference is present, direct effects are commonly defined as the average effect of a unit's treatment on their own outcome while fixing the treatment status or probability among interfering units, and spillover effects measure the average effect of a change in the latter while the individual's treatment status is kept fixed. Here, we define the average causal effect of a unit's treatment status on the outcome of their network neighbors, while fixing the treatment probability in the remaining interference set. We propose two different weighting schemes defining two causal effects: i) the outward spillover effect, which represents the average effect of a unit's treatment on their neighbors' potential outcomes, and ii) the inward spillover effect, which represents the impact of each neighbor's treatment on an individual's own potential outcome. We prove that outward and inward spillover effects generally differ, even in an undirected network. However, under specific conditions these two causal estimands become equivalent. We provide numerous examples illustrating the conditions for equivalence or discrepancy of the two spillover effects. We then compare their Horvitz-Thompson estimators, examining their relative variance under various graph structures and structural assumptions on potential outcomes.