Changing Own Treatment While Holding Peer Exposure Fixed
When one person’s treatment can affect another person’s outcome, the usual notation \(Y_i(z_i)\) may omit an important part of the causal setting: the treatments received by the other units. An outcome can depend both on unit \(i\)’s own treatment and on the treatment environment created by its peers. A direct effect isolates the contrast associated with changing the first of these while fixing the second.
“Direct” here names a particular comparison, not a claim that the effect operates through a particular biological or social mechanism. The effect is defined relative to a chosen mapping of peers’ treatment assignments into a manageable exposure summary. Its interpretation therefore depends on that mapping.
The order of the contrast matters: treatment minus control. Thus, if larger outcomes are favorable, a positive direct effect means that own treatment increases the outcome on average at the specified peer exposure. If smaller outcomes are favorable, the same positive number instead indicates an increase in an undesirable outcome.
The exposure mapping \(g_i\) summarizes the treatment assignments of the peers relevant to unit \(i\). For the notation \(Y_i(z_i,g_i)\) to represent outcomes under interference, the mapping must be adequate: assignments that give unit \(i\) the same own treatment and the same mapped peer exposure must imply the same potential outcome for that unit. Formally, if two full treatment assignments \(z\) and \(z'\) satisfy \(z_i=z'_i\) and \(g_i(z_{-i})=g_i(z'_{-i})\), then the exposure-mapping condition is $$ Y_i(z)=Y_i(z'). $$ Here \(z_{-i}\) denotes the treatment assignments of units other than \(i\). This condition does not rule out all interference; it says that any peer influence relevant to the outcome is captured by the chosen summary \(g_i\).
How to Interpret the Fixed-Exposure Contrast
In \(\tau_i(g)\), the two potential outcomes refer to two treatment states for the same unit, with peers held at the same mapped exposure. If \(g\) is a peer-treatment proportion, for example, the contrast compares own treatment states at that same proportion. It does not compare a treated unit surrounded by highly treated peers with a control unit surrounded by mostly untreated peers.
A fixed exposure can represent many different peer assignments. Under a saturation mapping, the same proportion might be produced by different identities of treated peers. The exposure-mapping condition treats those assignments as equivalent for the potential outcome. If which particular peers are treated matters beyond the proportion, saturation alone is not an adequate mapping, and the resulting direct effect may not describe the causal contrast the analyst intends.
The average \(\tau(g)\) averages individual contrasts over the specified target units, giving each unit equal weight in this definition. Other target populations or weights can be used, but they must be stated: changing the units or their weights can change the average effect. The fixed peer exposure \(g\) is also part of the estimand. Direct effects at different exposure levels need not be equal.
Worked Example: An Individual Effect at Half Peer Saturation
Suppose a hypothetical clinic assigns a binary treatment to each patient. For a particular patient, the mapped peer exposure is the proportion of the patient’s four peers assigned treatment. At peer saturation \(g=0.5\), suppose the patient’s potential outcomes are:
| Own treatment \(z_i\) | Peer exposure \(g\) | Potential outcome: symptom-free days in 30 days |
|---|---|---|
| 0 | 0.5 | 18 |
| 1 | 0.5 | 22 |
The individual direct effect at \(g=0.5\) is $$ \tau_i(0.5)=Y_i(1,0.5)-Y_i(0,0.5)=22-18=4\text{ days}. $$ For this hypothetical patient, changing own treatment from control to treatment while holding peer saturation at 50% increases the potential number of symptom-free days by 4. This is an individual causal contrast, not an estimate from an observed comparison: only one of the two potential outcomes could be observed for this patient under a single assignment.
Peer Saturation and Average Direct Effects
A common exposure mapping is peer saturation, the proportion of a unit’s peers assigned treatment. If unit \(i\) has \(m_i\) peers, one such mapping is $$ g_i(z_{-i})=\frac{1}{m_i}\sum_{j\ne i,\ j\in\mathcal{P}_i}z_j, $$ where \(\mathcal{P}_i\) is the set of peers defined for unit \(i\). For binary peer treatments, \(g_i\) ranges from 0 to 1 in increments of \(1/m_i\). A direct effect at a specified saturation compares own treatment states at the same peer-treatment proportion. The mapping must be chosen to fit the substantive setting; it is not automatically the correct summary just because it is convenient to calculate.
Worked Example: Averaging Direct Effects at a Fixed Saturation
In a hypothetical group program, each participant has two peers. The exposure mapping is the proportion of those two peers assigned treatment, so \(g=0.5\) means exactly one of the two peers is treated. Suppose four participants’ potential outcomes, measured as a symptom score for which lower is better, are as follows at \(g=0.5\):
| Participant | \(Y_i(0,0.5)\) | \(Y_i(1,0.5)\) | \(\tau_i(0.5)\) |
|---|---|---|---|
| A | 14 | 11 | 11 − 14 = −3 |
| B | 10 | 9 | 9 − 10 = −1 |
| C | 12 | 12 | 12 − 12 = 0 |
| D | 8 | 10 | 10 − 8 = 2 |
The equal-weight average direct effect for these four participants is $$ \tau(0.5)=\frac{(-3)+(-1)+0+2}{4}=\frac{-2}{4}=-0.5\text{ score points}. $$ Equivalently, averaging the treatment potential outcomes gives \((11+9+12+10)/4=10.5\), and averaging the control potential outcomes gives \((14+10+12+8)/4=11\). Their difference is \(10.5-11=-0.5\) score points. In this hypothetical population, own treatment lowers the mean symptom score by 0.5 points at 50% peer saturation. Individual effects differ, including one positive effect, so the average does not imply that treatment lowers every participant’s score.
The equality between “average the individual differences” and “difference the averages” follows directly when the same units and weights appear in both terms. For equal weights, $$ \frac{1}{N}\sum_{i=1}^N\{Y_i(1,g)-Y_i(0,g)\} =\frac{1}{N}\sum_{i=1}^NY_i(1,g)-\frac{1}{N}\sum_{i=1}^NY_i(0,g). $$ The equality uses the distributive property of summation. It would not justify subtracting averages over different sets of units: that would change the target contrast.
Worked Example: A Binary Outcome at Fixed Peer Saturation
Consider a hypothetical finite population of five units, each with two peers. At \(g=0.5\), the binary outcome is whether the unit meets a specified recovery criterion by day 30. A value of 1 indicates recovery and 0 indicates no recovery. Suppose the potential outcomes at this fixed peer saturation are:
| Unit | \(Y_i(0,0.5)\) | \(Y_i(1,0.5)\) | Direct effect |
|---|---|---|---|
| 1 | 0 | 1 | 1 |
| 2 | 1 | 1 | 0 |
| 3 | 0 | 0 | 0 |
| 4 | 0 | 1 | 1 |
| 5 | 1 | 0 | −1 |
The average direct effect is $$ \tau(0.5)=\frac{1+0+0+1+(-1)}{5}=\frac{1}{5}=0.20. $$ The mean recovery outcome under own treatment is \((1+1+0+1+0)/5=0.60\), while under own control it is \((0+1+0+0+1)/5=0.40\). The risk difference is therefore \(0.60-0.40=0.20\), or 20 percentage points. For this hypothetical population, own treatment increases the recovery proportion by 20 percentage points at 50% peer saturation. One unit has a negative individual effect, which is compatible with a positive population average.
What the Definition Does—and Does Not—Establish
The potential-outcome contrast defines the causal quantity of interest. It does not, by itself, say how to identify or estimate that quantity from observed data. For each unit, only the outcome under the treatment and peer exposure actually received is observed; the other potential outcomes remain counterfactual. A comparison of observed outcomes can be interpreted as the defined direct effect only if the assignment process and available information justify that interpretation.
Even with a suitable assignment process, the exposure mapping matters. If peers’ identities, treatment timing, or another aspect of their assignments affects outcomes beyond the mapped saturation, then \(Y_i(z_i,g_i)\) is not sufficient to represent the potential outcomes. In that case the fixed-saturation contrast is not fully specified by the chosen exposure summary. Defining the mapping is therefore a substantive modeling decision, not just a notation choice.
A direct effect at fixed peer exposure also does not answer how outcomes change when peer exposure changes while own treatment stays fixed. That is a distinct causal contrast. Keeping the two questions separate prevents a change in peers’ treatment from being mistaken for a change in own treatment.
Common Mistake: Comparing Different Peer Environments
A tempting but incorrect calculation is to subtract the average outcome for treated units in one peer environment from the average outcome for control units in another and call the result a direct effect. For example, comparing treated units with \(g=0.75\) to control units with \(g=0.25\) changes both own treatment and peer exposure. That difference does not hold peer exposure fixed, so it does not match \(Y_i(1,g)-Y_i(0,g)\) for any single \(g\).
The correction is to name the exposure level first, then define both potential outcomes at that same level, and use the same target units or weights on both sides of the contrast. If the exposure mapping fails to capture relevant peer-assignment differences, refining the mapping is necessary before interpreting the resulting contrast as the intended direct effect.
Check Your Understanding
Answer each question by identifying the treatment contrast and the peer exposure being held fixed.
- For a binary outcome, \(Y_i(1,0.25)=1\) and \(Y_i(0,0.25)=0\). What is the individual direct effect at \(g=0.25\), and what does its sign mean?
- Why must the two potential outcomes in a direct-effect contrast use the same peer exposure?
- Four units have direct effects at a specified \(g\) of \(2,-1,0,\) and \(3\) outcome units. What is their equal-weight average direct effect?
- Under a peer-saturation mapping with three peers, what peer-treatment proportions are possible?
- Explain why comparing treated units at \(g=0.75\) with control units at \(g=0.25\) does not isolate a direct effect at fixed peer exposure.