Total and Overall Effects Under Interference

Why Effects Must Specify the Assignment Being Compared

When one person’s treatment can affect another person’s outcome, changing treatment for one unit while holding everyone else’s treatment fixed answers a different question from changing treatment for the whole group. The first kind of contrast isolates a particular change in assignment. A total effect, as used here, compares outcomes when all units receive treatment with outcomes when none do. An overall effect compares average outcomes under two possibly randomized assignment policies, allowing treatment to vary across units under each policy.

These are causal estimands: quantities defined using potential outcomes under specified interventions. Their definitions do not by themselves guarantee that observed data identify them. To interpret either contrast causally, treatment assignments must correspond to well-defined interventions, and the potential outcomes must specify the relevant effects of the full assignment. The two-unit calculations below treat the potential outcomes as given; they do not assume that the outcomes could be recovered from observational data without further conditions.

Definition: With two units, let $Z_1,Z_2\in\{0,1\}$ denote their treatment assignments. Let $Y_i(z_1,z_2)$ be unit $i$'s potential outcome under the joint assignment $(z_1,z_2)$. Thus, the potential outcome can depend on both the unit’s own assignment and the other unit’s assignment. The notation assumes that these two assignments capture all treatment exposure relevant to the outcome in this example.

A potential outcome is indexed by the whole assignment vector, not just by the unit’s own treatment. For example, $Y_1(1,0)$ is unit 1’s outcome when unit 1 is treated and unit 2 is untreated; $Y_1(1,1)$ is unit 1’s outcome when both are treated. These need not be equal. Potential outcomes are fixed quantities for the units in the population, even though only the outcome corresponding to the assignment that actually occurs is observed for each unit.

Total Effects: Changing Own and Peer Treatment Together

For a two-unit group, the all-treated assignment is $(1,1)$, and the all-control assignment is $(0,0)$. The unit-level total effect for unit $i$ is the contrast between that unit’s potential outcomes under these two joint assignments. The average total effect across the two units averages those unit-level contrasts.

Definition: The unit-level total effect for unit $i$ is $Y_i(1,1)-Y_i(0,0)$. For a population of $N$ units, the average total effect comparing all treated with all untreated is $$ \tau_T=\frac{1}{N}\sum_{i=1}^{N}\left\{Y_i(\mathbf{1})-Y_i(\mathbf{0})\right\}, $$ where $\mathbf{1}$ denotes the assignment with every unit treated and $\mathbf{0}$ the assignment with every unit untreated. For two units, these assignments are $(1,1)$ and $(0,0)$.

The contrast changes every unit’s assignment together. It therefore includes whatever changes in outcomes follow from changing own treatment and the treatment received by peers under those two group assignments. It is not a comparison that holds peer treatment fixed. The word “total” describes the scope of the assignment contrast; it does not mean that the contrast is automatically a sum of separately identified components.

Worked Example: All-Treated Versus All-Control Assignment

Suppose the outcome is a hypothetical symptom score, measured in points, for two people. Lower scores indicate fewer symptoms. The potential outcomes are:

Assignment $(z_1,z_2)$	Unit 1 outcome $Y_1(z_1,z_2)$	Unit 2 outcome $Y_2(z_1,z_2)$
(0, 0)	2	1
(1, 0)	5	3
(0, 1)	4	4
(1, 1)	8	7

For unit 1, the total effect is $Y_1(1,1)-Y_1(0,0)=8-2=6$ symptom-score points. For unit 2, it is $Y_2(1,1)-Y_2(0,0)=7-1=6$ points. Therefore, the average total effect is

\tau_T=\frac{(8-2)+(7-1)}{2} =\frac{6+6}{2} =6\text{ points}.

Under this hypothetical set of potential outcomes, assigning both people to treatment rather than assigning neither to treatment increases the average symptom score by 6 points. This is a contrast between two group assignments; it does not say what would happen if only one person’s treatment changed.

Overall Effects: Averaging Over Assignment Policies

An assignment policy specifies a probability distribution over joint assignments. It may, for example, treat both units, treat neither, or assign treatment randomly. Let $\pi(z_1,z_2)$ be the probability that policy $\pi$ assigns the pair $(z_1,z_2)$. The probabilities across the four possible assignments are nonnegative and sum to 1. Under this policy, each unit’s expected potential outcome is an average over the assignments the policy can produce.

Definition: For a policy $\pi$, define unit $i$'s policy-averaged outcome as $$ \mu_i(\pi)=\sum_{z_1=0}^{1}\sum_{z_2=0}^{1}\pi(z_1,z_2)Y_i(z_1,z_2). $$ The overall mean outcome under $\pi$ is $\mu(\pi)=\frac{1}{N}\sum_{i=1}^{N}\mu_i(\pi)$. The overall effect comparing policies $\pi$ and $\pi'$ is $$ \tau_O(\pi,\pi')=\mu(\pi)-\mu(\pi'). $$

This definition averages first over the assignment distribution within each unit, then across units. Equivalently, it averages the population mean outcome across the joint assignments that each policy can produce. A deterministic policy is a special case: it assigns probability 1 to one joint assignment and 0 to all others. Thus, the total effect comparing all treated with all untreated is also an overall effect comparing those two deterministic policies. The term “overall effect” is useful when the contrast is between more general assignment policies, not only those two extremes.

Worked Example: Comparing Two Independent Assignment Policies

Use the hypothetical symptom-score potential outcomes in the table above. Under policy $\pi_{0.50}$, each person is independently assigned treatment with probability 0.50. Under policy $\pi_{0.25}$, each is independently assigned treatment with probability 0.25. Independence means the probability of a joint assignment is the product of its two individual assignment probabilities.

Under $\pi_{0.50}$, each of the four assignments has probability $0.25$. Unit 1’s policy-averaged outcome is

\mu_1(\pi_{0.50}) =0.25(2)+0.25(5)+0.25(4)+0.25(8) =4.75\text{ points}.

Unit 2’s policy-averaged outcome is

\mu_2(\pi_{0.50}) =0.25(1)+0.25(3)+0.25(4)+0.25(7) =3.75\text{ points}.

Averaging across the two units gives $\mu(\pi_{0.50})=(4.75+3.75)/2=4.25$ points. Under $\pi_{0.25}$, the assignment probabilities are $0.5625$ for $(0,0)$, $0.1875$ for each of $(1,0)$ and $(0,1)$, and $0.0625$ for $(1,1)$. They sum to 1. The corresponding outcomes are

\begin{aligned} \mu_1(\pi_{0.25}) &=0.5625(2)+0.1875(5)+0.1875(4)+0.0625(8)\\ &=3.3125\text{ points},\\ \mu_2(\pi_{0.25}) &=0.5625(1)+0.1875(3)+0.1875(4)+0.0625(7)\\ &=2.3125\text{ points}. \end{aligned}

Therefore, $\mu(\pi_{0.25})=(3.3125+2.3125)/2=2.8125$ points, and the overall effect is

\tau_O(\pi_{0.50},\pi_{0.25}) =4.25-2.8125 =1.4375\text{ points}.

For these hypothetical outcomes, the policy assigning treatment independently with probability 0.50 produces an average outcome 1.4375 points higher than the policy assigning it independently with probability 0.25. This policy contrast is not the same as the all-treated versus all-control total effect of 6 points: the policies average over several joint assignments rather than selecting only the two extreme assignments.

Why the Joint Assignment Distribution Matters

A policy is a distribution over joint assignments. Knowing only the probability that each person is treated may not specify that distribution: assignments can be dependent. With interference, that distinction can change policy-averaged outcomes because the outcome under one person’s treatment can depend on the other person’s assignment.

Worked Example: Same Treatment Marginals, Different Policies

Consider two hypothetical policies using the same symptom-score potential outcomes. Policy A assigns $(0,0)$ with probability 0.50 and $(1,1)$ with probability 0.50. Policy B assigns $(1,0)$ with probability 0.50 and $(0,1)$ with probability 0.50. Under both policies, each person has a 0.50 probability of treatment. However, Policy A always assigns both people the same status, while Policy B always assigns exactly one person to treatment.

Under Policy A, the policy-averaged outcomes are

\begin{aligned} \mu_1(A)&=0.50(2)+0.50(8)=5\text{ points},\\ \mu_2(A)&=0.50(1)+0.50(7)=4\text{ points},\\ \mu(A)&=(5+4)/2=4.5\text{ points}. \end{aligned}

Under Policy B, they are

\begin{aligned} \mu_1(B)&=0.50(5)+0.50(4)=4.5\text{ points},\\ \mu_2(B)&=0.50(3)+0.50(4)=3.5\text{ points},\\ \mu(B)&=(4.5+3.5)/2=4.0\text{ points}. \end{aligned}

Thus, $\tau_O(A,B)=4.5-4.0=0.5$ symptom-score points, even though the two policies have the same treatment probability for each person. The difference arises from how each policy distributes probability across joint assignments, which matters for these potential outcomes.

A Short Derivation of the Policy Contrast

The overall effect can be written directly as a weighted contrast across assignments. Start with the average policy outcome, substitute the definition of each unit’s policy-averaged outcome, and subtract the corresponding expression for the comparison policy:

\begin{aligned} \tau_O(\pi,\pi') &=\frac{1}{N}\sum_{i=1}^{N} \left[ \sum_{\mathbf z}\pi(\mathbf z)Y_i(\mathbf z) - \sum_{\mathbf z}\pi'(\mathbf z)Y_i(\mathbf z) \right]\\ &=\frac{1}{N}\sum_{i=1}^{N} \sum_{\mathbf z} \left\{\pi(\mathbf z)-\pi'(\mathbf z)\right\}Y_i(\mathbf z), \end{aligned}

The second line follows because both sums range over the same finite set of joint assignments, so their terms can be combined assignment by assignment. This form shows precisely what the contrast does: each assignment’s average potential outcome is weighted by the difference between its probability under the two policies. Assignments given the same probability by both policies contribute zero to the contrast. This is an algebraic identity for the defined estimand, not an identification result about observed outcomes.

Interpretation and a Common Pitfall

Both effects depend on specifying the assignment contrast before interpreting the number. A total effect comparing $(1,1)$ with $(0,0)$ changes own and peer treatment together. An overall effect compares the average outcomes induced by two stated distributions over assignments. The units of either effect are the units of the outcome; in the examples, they are symptom-score points.

Common mistake: “Both policies treat each person with probability 0.50, so they must have the same overall effect.” That reasoning uses only the individual treatment probabilities and overlooks the joint assignment distribution. As the worked example shows, one policy can concentrate on $(0,0)$ and $(1,1)$, while another concentrates on $(1,0)$ and $(0,1)$. With interference, potential outcomes can differ across those assignments, so the policy averages need not agree.

A related interpretive caution is to avoid treating a defined causal contrast as automatically estimable. To learn an overall effect from data requires a design or assumptions that connect observed outcomes to the potential outcomes under the assignments used by both policies. The definitions here specify the target comparison; they do not supply that connection.

Key takeaway: A total effect contrasts the all-treated and all-untreated joint assignments. An overall effect contrasts expected outcomes under two assignment distributions. Under interference, both the assignments being compared and, for policies, their joint probabilities are part of the causal question.

Check Your Understanding

Use the definitions and hypothetical potential outcomes in this tutorial to answer the following questions.

In the two-unit setting, write the potential-outcome contrast defining unit 2’s total effect.
If a policy assigns $(0,0)$ with probability 1, what is the policy-averaged outcome for unit 1 in the example?
Explain why the all-treated versus all-control total effect is a special case of an overall effect.
Two policies give each unit the same probability of treatment but differ in dependence between assignments. Why might their overall mean outcomes differ?
Does defining $\tau_O(\pi,\pi')$ alone establish that it can be estimated from observational data? Explain briefly.