Causal interference occurs when a unit’s outcome depends on treatments assigned to other units. A spillover effect is a causal change transmitted through those other assignments. For a quant evaluating an execution rule, a treated order could change the conditions faced by a comparison order. An untreated order would then be an exposed control.
In the potential outcomes framework, write : own assignment is , and collects everyone else’s. No interference would allow the latter argument to be dropped. Shared market shocks can correlate outcomes without interference; interference specifically concerns the consequences of changing another unit’s treatment. Hudgens and Halloran, §2.1 formalise this distinction through assignment-indexed outcomes.
One pair, four possible outcomes
Consider a fictional paired-order experiment, with outcome measured in score points, higher being better. Stipulate perfect treatment receipt, no effects across pairs, and the same response rule for either member:
where is the peer’s treatment. These are model inputs, not fitted market effects:
| Own treatment | Peer treatment | Potential outcome |
|---|---|---|
| 0 | 0 | 10 |
| 1 | 0 | 12 |
| 0 | 1 | 13 |
| 1 | 1 | 15 |
Only one cell is observed for a particular unit under its realised assignment. The table does not reveal four observed outcomes for that order, and supplies no standard error.
Different treatment effects ask different questions. Use treatment-minus-comparison signs throughout:
- Direct effect at fixed peer exposure: .
- Spillover at fixed own assignment: . At , this is the effect on an untreated unit of treating its peer.
- Total effect, both treated versus neither: .
Here . The decomposition uses a specified intermediate cell. With an own–peer interaction, the direct effect could differ across peer exposures; one cannot combine arbitrary direct and spillover contrasts. These terms concern cross-unit effects; “direct” and “indirect” in mediation analysis instead distinguish causal pathways through a mediator. Hudgens and Halloran, §§3.2–3.5 define interference effects and their allocation-strategy averages.
Original paired-unit model. The same potential outcomes produce different treated–control comparisons because each design changes which peer exposures accompany own treatment.
The design changes the comparison
Imagine many pairs obeying that stipulated rule. Compare expected outcomes conditional on own assignment under three randomised designs:
| Assignment within each pair | Mean if own treated | Mean if own untreated | Difference |
|---|---|---|---|
| Independent fair coins | 2 | ||
| Exactly one treated, chosen fairly | 12 | 13 | −1 |
| Both treated or both untreated, fair choice | 15 | 10 | 5 |
The second design makes every untreated unit’s peer treated. Its negative comparison does not mean own treatment has a negative effect holding the peer fixed. The third design varies own and peer treatment together: it compares packages, and cannot separately recover the two effects without additional assumptions. These are exact conditional expectations under the model, not guaranteed differences in a finite realised sample.
A policy or overall effect compares population means under two allocation strategies, averaging over both own and others’ assignments. In this model, a policy independently treating each unit with probability gives
Changing from to changes the policy mean from to : an effect of points. It differs from the direct effect and the all-treated-versus-none effect . Specify the population weights and allocation rule: “the treatment effect” alone leaves the target ambiguous.
What identification needs
An exposure mapping compresses others’ assignments into relevant exposure, for example , giving . A treated-neighbour count assumes identities and more distant assignments do not matter beyond that count. Such compression is a substantive restriction, not a consequence of randomisation.
For design-based estimation, interventions and exposure must be well defined, observed outcomes must match the realised potential outcomes, and the assignment mechanism must be known. The exposure contrasts need positive assignment probabilities for the target units: positivity concerns the joint exposure, not only own treatment. Unequal exposure probabilities may require weighting; uncertainty must respect jointly generated exposures. Aronow and Samii, §§3–4 set out these conditions.
Cluster assignment does not automatically remove interference. Partial interference permits effects within groups while assuming none across groups. If orders in different assigned clusters interact through the same market, that assumption needs justification. Merely changing standard errors cannot turn an exposed control into an unexposed counterfactual. The Difference-in-Differences lesson examines this control-group requirement (public preview; full lesson requires membership).
Sources
- Hudgens, M. G. and Halloran, M. E. (2008), “Toward Causal Inference With Interference”, JASA 103(482), 832–842, §§2–3 and 4.1. Partial interference, allocation strategies and direct, indirect, total and overall effects. Their treatment-order convention differs from the positive treatment-minus-comparison convention used here.
- Aronow, P. M. and Samii, C. (2017), “Estimating Average Causal Effects Under General Interference, with Application to a Social Network Experiment”, Annals of Applied Statistics 11(4), 1912–1947, §§3–4. Exposure mappings, exposure probabilities, consistency and design-based average effects.