We consider design-based causal inference in settings where randomized
treatments have effects that bleed out into space in complex ways that overlap
and in violation of the standard "no interference" assumption for many causal
inference methods. We define a spatial "average marginalized response," which
characterizes how, in expectation, units of observation that are a specified
distance from an intervention point are affected by treatments at that point,
averaging over effects emanating from other intervention points. We establish
conditions for non-parametric identification, asymptotic distributions of
estimators, and recovery of structural effects. We propose methods for both
sample-theoretic and permutation-based inference. We provide illustrations
using randomized field experiments on forest conservation and health