Embracing Spillover: Spatial Effects in Experiments

Interventions delivered in space generate effects that spill over between experimental units.

We develop a framework for spatial experiments in which causal estimands, inclduing direct, indirect, total, and dose--response effects, are linear functionals of a spillover kernel.

Under cluster randomisation with a fixed exposure set size, the data identify the shape of the kernel but not its level: the level is aliased with the intercept and direct effect, at any sample size and under any outcome model.

Every estimator therefore rests on an anchoring assumption that fixes the level.

We derive an exact decomposition of the bias of any anchored estimator into a level term, a shape error from kernel misspecification, and a leakage term absorbed by the realised geometry, each computable from the design before data collection.

A single scalar, the level leverage, gives each estimand's exposure to the unidentified level, the exact level bias, and the variance cost of estimating the level instead.

The conventional cluster-trial analysis is the special case of an implicit anchor, with contamination bias in closed form.

Existing approaches, including identification through Bernoulli randomisation, elicited bounds on interference decay, and assumed compact support, are, within the linear exposure mapping, anchoring choices in this framework.

We give a taxonomy of design augmentations that purchase the level and a criterion for when to augment and when to anchor.