Why Hold Own Treatment Fixed?
When one person’s treatment can affect another person’s outcome, a unit’s outcome may depend both on its own treatment and on the treatments received by peers. For example, a patient’s outcome might depend on whether that patient receives a service and on how many nearby patients receive it. To isolate the peer component, compare outcomes under two peer-exposure conditions while keeping the unit’s own treatment unchanged.
This is a spillover effect at fixed own treatment. It answers a focused question: for units assigned to a specified own-treatment level, how would outcomes differ if their mapped peer exposure were changed? The question can be asked separately for units held untreated and units held treated. Those are distinct causal contrasts and need not have the same size or direction.
Potential Outcomes and the Spillover Contrast
Let \(Z_i\in\{0,1\}\) denote unit \(i\)’s own treatment. Let \(G_i\) be a mapped summary of the treatment assignments of that unit’s peers. The mapping might classify peer exposure as low or high, or might be a count or proportion of treated peers. Write \(Y_i(z,g)\) for the outcome unit \(i\) would have if its own treatment were set to \(z\) and its mapped peer exposure were set to \(g\).
The own-treatment value \(z\) stays the same in both potential outcomes. Setting \(z=0\) gives the spillover effect among units held untreated; setting \(z=1\) gives the spillover effect among units held treated. The sign follows the stated order: \(g_1\) minus \(g_0\). A positive effect means the outcome is higher under \(g_1\) than under \(g_0\), for the outcome scale used.
These definitions concern a chosen target population. The target could be a finite set of units or a population represented by an expectation: $$ \tau^{S}(z;g_1,g_0)=E\left[Y_i(z,g_1)-Y_i(z,g_0)\right]. $$ The two forms express the same idea: average the within-unit contrast over the target. If outcomes are measured in days, the effect is measured in days; if the outcome is a binary indicator, the average contrast is a risk difference.
Worked Examples
Worked Example: Spillover Among Units Held Treated
Suppose a hypothetical program has a binary own-treatment indicator and two mapped peer-exposure levels, low and high. For four target units, assume the potential outcomes below are known for illustration. Outcomes are symptom-free days during a follow-up period.
| Unit | \(Y_i(1,\text{low})\) | \(Y_i(1,\text{high})\) | Spillover contrast |
|---|---|---|---|
| A | 8 | 11 | 11 − 8 = 3 |
| B | 6 | 8 | 8 − 6 = 2 |
| C | 10 | 10 | 10 − 10 = 0 |
| D | 7 | 9 | 9 − 7 = 2 |
The target is the four units, held at own treatment \(z=1\), and the contrast is high versus low peer exposure. The average spillover effect is $$ \tau^{S}(1;\text{high},\text{low}) =\frac{3+2+0+2}{4} =\frac{7}{4} =1.75\text{ symptom-free days}. $$
In this hypothetical population, moving from low to high mapped peer exposure increases the average number of symptom-free days by 1.75 among units held treated. This conclusion is specific to that own-treatment level, contrast, outcome, and target population.
Worked Example: Spillover Among Units Held Untreated
For the same four hypothetical units, now consider potential outcomes under own treatment \(z=0\). The outcome is again symptom-free days.
| Unit | \(Y_i(0,\text{low})\) | \(Y_i(0,\text{high})\) | Spillover contrast |
|---|---|---|---|
| A | 5 | 7 | 7 − 5 = 2 |
| B | 4 | 4 | 4 − 4 = 0 |
| C | 6 | 9 | 9 − 6 = 3 |
| D | 3 | 5 | 5 − 3 = 2 |
Holding own treatment at zero, the average high-versus-low spillover effect is $$ \tau^{S}(0;\text{high},\text{low}) =\frac{2+0+3+2}{4} =\frac{7}{4} =1.75\text{ symptom-free days}. $$
The numerical average happens to equal the average in the first example, but the two estimands are different: one compares peer exposure for units held treated, and the other does so for units held untreated. Equality in this particular hypothetical table does not imply the effects must be equal in another population or setting.
Worked Example: Averaging Spillovers Across Target Strata
Suppose a hypothetical study defines low and high peer exposure and divides its target population into baseline low-risk and high-risk strata. The unit’s own treatment is held at \(z=0\) in both strata. Outcomes are hours of missed work during follow-up. The stratum-specific averages are:
| Baseline stratum | Target-population weight | Mean under low exposure | Mean under high exposure |
|---|---|---|---|
| Low risk | 0.60 | 4 hours | 3 hours |
| High risk | 0.40 | 8 hours | 6 hours |
The stratum-specific spillover contrasts are high minus low: \(3-4=-1\) hour in the low-risk stratum and \(6-8=-2\) hours in the high-risk stratum. To average these contrasts over the stated target distribution, multiply each by its target weight and add: $$ \tau^{S}(0;\text{high},\text{low}) =0.60(-1)+0.40(-2) =-0.60-0.80 =-1.40\text{ hours}. $$
For this target population, high rather than low peer exposure reduces average missed work by 1.40 hours among units held untreated. The target weights matter: using a different population mix can produce a different average even if the stratum-specific effects stay unchanged.
Why the Exposure Mapping Matters
The notation \(Y_i(z,g)\) assumes that the mapped exposure \(g\) captures the peer-assignment features relevant to unit \(i\)’s outcome. Formally, assignments that give unit \(i\) the same own treatment \(z\) and the same mapped exposure \(g\) must imply the same potential outcome for that unit. If two peer-assignment patterns receive the same mapped label but can affect the outcome differently, then \(Y_i(z,g)\) is not a sufficiently precise description without refining the mapping.
For example, “two treated peers” may be an inadequate mapping if it matters whether those peers are close relatives or distant neighbors. The numerical count is well-defined, but it may merge substantively different peer conditions. The mapping must therefore be justified for the application; it is not merely a convenient label.
A causal interpretation also requires well-defined versions of own treatment and peer exposure, and consistency: when a unit actually receives own treatment \(z\) and mapped exposure \(g\), its observed outcome equals \(Y_i(z,g)\). The contrast then compares coherent interventions on peer exposure with own treatment held fixed. It does not follow just from observing that units with different peer exposure have different outcomes.
Calculation, Identification, and a Short Justification
When the potential outcomes are available, calculation is direct: form each unit’s contrast at the same \(z\), then average over the target. Linearity of averaging shows why averaging individual contrasts is equivalent to subtracting the two target mean potential outcomes:
The equality follows by distributing the sum across subtraction. It does not require the units to have identical effects: each unit may have a different contrast. It does require the two means to refer to the same target population and the same fixed own-treatment value.
In real data, each unit’s outcome under the exposure actually received is observed, but its outcomes under alternative peer exposures are counterfactual. A raw difference between observed high- and low-exposure groups is therefore not automatically the defined causal effect: the groups may differ in own treatment, baseline characteristics, or other causes of outcomes. Identifying the causal contrast from observed data requires an appropriate design or assumptions that support comparison of the relevant exposure conditions, as well as adequate support for those conditions in the target population. The examples above calculate estimands from hypothetical potential-outcome information; they do not claim that both potential outcomes are observed for the same real unit.
Common Mistakes
- Changing own treatment as well as peer exposure. Comparing \(Y_i(1,g_1)\) with \(Y_i(0,g_0)\) changes two conditions. It is not a spillover contrast at fixed own treatment.
- Reporting one spillover effect for both treatment levels without checking. The contrasts at \(z=0\) and \(z=1\) are separately defined. They may differ because the response to peers can depend on own treatment.
- Reversing the contrast order. High minus low and low minus high have opposite signs. State the order before interpreting the result.
- Treating an observed group difference as a causal effect. A difference between units observed at high and low peer exposure can reflect group composition. A causal interpretation needs a design or assumptions that justify the comparison.
- Using an exposure mapping that hides meaningful differences. If assignments summarized by the same \(g\) can produce different outcomes, the mapped potential outcome is not well-defined as written.
Check Your Understanding
Answer each question using the stated contrast order and outcome scale.
- In \(Y_i(0,\text{high})-Y_i(0,\text{low})\), which treatment condition is held fixed, and which condition changes?
- For two units held treated, the high-minus-low spillover contrasts are \(-2\) and \(4\) minutes. What is their average, and how should it be interpreted?
- Why are the average spillover effects at \(z=0\) and \(z=1\) distinct estimands even when they use the same peer-exposure levels?
- A mapping labels two peer-assignment patterns as “one treated neighbor,” but the patterns can affect outcomes differently. What problem does this create for \(Y_i(z,g)\)?
- Why does a difference in observed outcomes between high- and low-peer-exposure groups not, by itself, establish a causal spillover effect?