Decomposing Direct and Spillover Contrasts

Why Decompose a Joint Contrast?

When outcomes can depend on both a person's own treatment and other people's treatment, a comparison that changes both at once combines two kinds of change. One component changes the person's own treatment while holding peer exposure fixed; the other changes peer exposure while holding own treatment fixed. The numerical values assigned to these components can depend on which change is made first.

This tutorial gives an algebraic decomposition for a specified pair of own-treatment and peer-exposure conditions. It focuses on what the joint contrast means along each path and why the total is invariant even when its components are not. The decomposition is defined in terms of potential outcomes; by itself, it does not establish that those outcomes or contrasts can be identified from observed data.

Potential Outcomes and Contrasts

Suppose own treatment \(z\) and peer exposure \(g\) are each binary, with values 0 and 1. For unit \(i\), let \(Y_i(z,g)\) be the potential outcome under own-treatment level \(z\) and peer-exposure level \(g\). The peer exposure is the summary of others' treatment relevant under the chosen exposure mapping. The four potential outcomes describe the four combinations of these two conditions.

Assume that these potential outcomes are well-defined for the conditions being compared and use the same exposure mapping throughout. This is the setup needed to define the contrasts below. No assumption of additivity or absence of interaction is imposed.

The direct contrast compares own-treatment conditions while holding peer exposure fixed. The spillover contrast compares peer-exposure conditions while holding own treatment fixed. These are contrasts between potential outcomes, not necessarily estimates from observed data. When averaging over a population, the same definitions apply to mean potential outcomes or, equivalently, to the averages of the unit-level contrasts.

Two Paths Between the Same Conditions

There are two natural paths from \((0,0)\) to \((1,1)\). On the first path, change own treatment from 0 to 1 while peer exposure remains 0, then change peer exposure from 0 to 1 while own treatment is 1. On the second path, change peer exposure first, then own treatment while peer exposure is 1.

Each line is a telescoping sum: the intermediate potential outcome is added and subtracted, so it cancels. The first decomposition uses a direct contrast at peer exposure 0 and a spillover contrast at own treatment 1. The second uses a spillover contrast at own treatment 0 and a direct contrast at peer exposure 1. Both sum to the same joint contrast because both start at \(Y_i(0,0)\) and end at \(Y_i(1,1)\).

Why the Components Depend on the Path

The difference between the two direct contrasts is an additive interaction between own treatment and peer exposure. Define it for unit \(i\) as \(I_i=D_i(1)-D_i(0)\). Expanding the definition gives an equivalent expression in terms of spillover contrasts.

Thus the interaction is both the change in the direct contrast when peer exposure moves from 0 to 1 and the change in the spillover contrast when own treatment moves from 0 to 1. This identity follows by rearranging the four potential outcomes; it does not require a statistical model.

The sign depends on the order in which paths are compared. In particular, direct(first path) minus direct(second path) is \(-I_i\), while spillover(first path) minus spillover(second path) is \(I_i\). Reversing the comparison reverses both signs. This explicit direction matters: saying only that the components differ by the interaction can conceal a sign error.

Population-Average Decompositions

For a target population of \(N\) units, define the average potential outcome \(\bar{Y}(z,g)=N^{-1}\sum_{i=1}^{N}Y_i(z,g)\). Define the population joint contrast as \(\bar{T}=\bar{Y}(1,1)-\bar{Y}(0,0)\), with average direct and spillover contrasts defined in the same way. Because averaging is linear, the unit-level identities also hold for the population averages. The population interaction is \(\bar{I}=N^{-1}\sum_i I_i\).

A zero average interaction makes the two average direct components equal and the two average spillover components equal. It does not imply that every unit has zero interaction: positive and negative unit-level interactions may cancel when averaged.

Interpreting a Path-Specific Decomposition

A path-specific decomposition answers a conditional accounting question: how much of a specified joint contrast is assigned to each step along the chosen sequence? It does not establish that one sequence is uniquely correct. Choosing a path changes which fixed condition is used to define each component. If the interaction is nonzero, the two paths allocate the same joint contrast differently.

This distinction is important when communicating results. A direct contrast must state the peer-exposure level at which it is defined, and a spillover contrast must state the own-treatment level. Reporting a direct or spillover component without those conditions leaves its meaning incomplete. The additive interaction also depends on the outcome scale: the identities above concern differences on the stated outcome scale.

The algebra uses potential outcomes for all four conditions. Observed data generally reveal only the outcome under the condition a unit actually received. Consequently, the equalities prove how the defined potential-outcome contrasts decompose; they do not by themselves supply an estimator, an identification argument, or evidence that the required contrasts are empirically recoverable.

Common Mistake

A frequent error is to compare the path-specific direct components in the wrong signed direction. If \(I_i=D_i(1)-D_i(0)\), it is incorrect to say that direct(first path) minus direct(second path) equals \(I_i\). The first path's direct component is \(D_i(0)\), whereas the second path's is \(D_i(1)\); therefore the difference is \(D_i(0)-D_i(1)=-I_i\). By contrast, spillover(first path) minus spillover(second path) is \(S_i(1)-S_i(0)=I_i\). Keeping the path order explicit prevents the sign reversal and shows why the two component changes cancel.