Spillover (experiment)
In
Analysis of spillover effects involves relaxing the non-interference assumption, or
One solution to this problem is to redefine the causal
Once the potential outcomes are redefined, the rest of the
Examples of spillover effects
Spillover effects can occur in a variety of different ways. Common applications include the analysis of social network spillovers and geographic spillovers. Examples include the following:
- Communication: An intervention that conveys information about a technology or product can influence the take-up decisions of others in their network if it diffuses beyond the initial user.[1]
- Competition: Job placement assistance for young job seekers may influence the job market prospects of individuals who did not receive the training but are competing for the same jobs.[2]
- Contagion: Receiving deworming drugs can decrease other's likelihood of contracting the disease.[3]
- Deterrence: Information about government audits in specific municipalities can spread to nearby municipalities.[4]
- Displacement: A hotspot policing intervention that increases policing presence on a given street can lead to the displacement of crime onto nearby untreated streets.[5]
- Reallocation of resources: A hotspot policing intervention that increases policing presence on a given street can decrease police presence on nearby streets.
- Social comparison: A program that randomizes individuals to receive a voucher to move to a new neighborhood can additionally influence the control group's beliefs about their housing conditions.[6]
In such examples, treatment in a randomized-control trial can have a direct effect on those who receive the intervention and also a spillover effect on those who were not directly treated.
Statistical issues
Estimating
Relaxing the non-interference assumption
One key assumption for
Estimating spillover effects requires relaxing the non-interference assumption. This is because a unit's outcomes depend not only on its treatment assignment but also on the treatment assignment of its neighbors. The researcher must posit a set of potential outcomes that limit the type of interference. As an example, consider an
- Y0,0 refers to an individual's potential outcomes when they are not treated (0) and neither was their roommate (0).
- Y0,1 refers to an individual's potential outcome when they are not treated (0) but their roommate was treated (1).
- Y1,0 refers to an individual's potential outcome when they are treated (1) but their roommate was not treated (0).
- Y1,1 refers to an individual's potential outcome when they are treated (1) and their roommate was treated (1).
Now an individual's outcomes are influenced by both whether they received the treatment and whether their roommate received the treatment. We can estimate one type of
While researchers typically embrace
Exposure mappings
The next step after redefining the causal estimand of interest is to characterize the probability of spillover exposure for each subject in the analysis, given some vector of treatment assignment. Aronow and Samii (2017)[12] present a method for obtaining a matrix of exposure probabilities for each unit in the analysis.
First, define a diagonal matrix with a vector of treatment assignment probabilities
Second, define an indicator matrix of whether the unit is exposed to spillover or not. This is done by using an adjacency matrix as shown on the right, where information regarding a network can be transformed into an indicator matrix. This resulting indicator matrix will contain values of , the realized values of a random binary variable , indicating whether that unit has been exposed to spillover or not.
Third, obtain the sandwich product , an N × N matrix which contains two elements: the individual probability of exposure on the diagonal, and the joint exposure probabilities on the off diagonals:
- In a similar fashion, the joint probability of exposure of i being in exposure condition and j being in a different exposure condition can be obtained by calculating :
The obtained exposure probabilities then can be used for inverse probability weighting (IPW, described below), in an estimator such as the Horvitz–Thompson estimator.
One important caveat is that this procedure excludes all units whose probability of exposure is zero (ex. a unit that is not connected to any other units), since these numbers end up in the denominator of the IPW regression.
Need for inverse probability weights
Estimating
Figure 1 displays an example where units have varying probabilities of being assigned to the spillover condition. Subfigure A displays a network of 25 nodes where the units in green are eligible to receive treatment. Spillovers are defined as sharing at least one edge with a treated unit. For example, if node 16 is treated, nodes 11, 17, and 21 would be classified as spillover units. Suppose three of these six green units are selected randomly to be treated, so that different sets of treatment assignments are possible. In this case, subfigure B displays each node's probability of being assigned to the spillover condition. Node 3 is assigned to spillover in 95% of the randomizations because it shares edges with three units that are treated. This node will only be a control node in 5% of randomizations: that is, when the three treated nodes are 14, 16, and 18. Meanwhile, node 15 is assigned to spillover only 50% of the time—if node 14 is not directly treated, node 15 will not be assigned to spillover.
Using inverse probability weights
When analyzing
Using randomization inference for hypothesis testing
In some settings, estimating the
See also
References
- ^ "Diffusion of Technologies within Social Networks: Evidence from a Coffee Training Program in Rwanda". IGC. 31 March 2010. Retrieved 2018-12-11.
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- ^ "Worms: Identifying Impacts on Education and Health in the Presence of Treatment Externalities | Edward Miguel, Professor of Economics, University of California, Berkeley". emiguel.econ.berkeley.edu. Retrieved 2018-12-11.
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- ^ "PsycNET". psycnet.apa.org. Retrieved 2018-12-11.
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- ^ Kao, Edward; Toulis, Panos (2013-02-13). "Estimation of Causal Peer Influence Effects". International Conference on Machine Learning: 1489–1497.
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- ^ A. Colin Cameron; Douglas L. Miller. "A Practitioner's Guide to Cluster-Robust Inference" (PDF). Cameron.econ.ucdavis.edu. Retrieved 19 December 2018.
- ^ "10 Things to Know About Randomization Inference". Egap.org. Retrieved 2018-12-11.