Day 32: SUTVA & Interference · Math for Causal Inference

Theory

The stable unit treatment value assumption

Writing $Y_i(1)$ and $Y_i(0)$ quietly assumed two things, together called SUTVA (Rubin 1980):

No interference: unit $i$ 's outcome depends only on its own treatment, not on who else is treated.
No hidden versions (consistency): there is only one version of each treatment level, so $Y_i = Y_i(T_i)$ .

Without (1), unit $i$ has a potential outcome for every assignment vector $\mathbf t = (t_1, \dots, t_n)$ : $2^n$ of them. Nothing can be learned without some structure.

Where interference comes from

Mechanism	Example
Contagion	vaccination protects the unvaccinated (herd immunity); deworming cuts reinfection
Information	farmers copy treated neighbours' techniques
Competition / displacement	job-search assistance helps treated jobseekers take jobs controls would have got
Prices (general equilibrium)	cash transfers raise local prices, affecting non-recipients

Partial interference and exposure mappings

A common, testable structure: units interact within groups (villages, schools, markets) but not across them, and only through the share of their group that is treated. Then

Y_{ig} = Y_{ig}(t_{ig},\, s_g), \qquad s_g = \text{saturation (share treated in group } g).

Effects at saturation $s$ (averages over units):

Direct effect: $\text{DE}(s) = E[Y(1, s) - Y(0, s)]$ (own treatment, holding others' fixed).
Spillover (indirect) effect: $\text{SE}(s) = E[Y(0, s) - Y(0, 0)]$ (on the untreated, of others being treated).
Total effect: $E[Y(1, s) - Y(0, 0)] = \text{DE}(s) + \text{SE}(s)$ .

Designs

Cluster randomisation (Day 40) puts the whole group in one arm, so spillovers stay inside: estimates the total policy effect.
Saturation (two-stage) designs first randomise groups to saturations (0%, 50%, 100%…), then individuals within groups (Hudgens & Halloran 2008; Baird et al. 2018). Comparing untreated units in treated groups with pure controls identifies spillovers.

Part 1 · Warm-up

Exercise 1 · How many potential outcomes?

A village has 3 farmers. Without SUTVA, how many potential outcomes does each farmer have? With SUTVA?

Show solution

Step 1. Without no-interference, each farmer's yield can depend on all three treatments: $2^3 = 8$ assignment vectors.

Step 2. With SUTVA, only own treatment matters: 2.

Answer: 8 versus 2. With 1,000 farmers it would be $2^{1000}$ versus 2, which is why structure is essential.

Exercise 2 · Spot the violation

For each case, say whether SUTVA fails and which part: (a) a measles vaccine; (b) a training programme run by two NGOs of different quality; (c) fertiliser applied to a farmer's own field.

Show solution

Step 1. (a) Vaccinating others lowers my infection risk: interference.

Step 2. (b) "Training" is really two treatments: hidden versions.

Step 3. (c) Fertiliser on my plot affects my yield only (unless it runs off into neighbours' fields): SUTVA is plausible.

Exercise 3 · An exposure mapping

In a village of 10 households, 4 are treated, including household $i$ . What are $i$ 's own treatment and the share of other households treated?

Show solution

Step 1. Own treatment $t_i = 1$ .

Step 2. Other treated households: $4 - 1 = 3$ of the other 9: share $3/9 = 1/3$ .

Answer: exposure $(1, 1/3)$ .

Exercise 4 · Direct effect

At 50% saturation, treated households average 120 (consumption, $/month) and untreated households in the same villages average 100. What is the direct effect at $s = 0.5$ ?

Show solution

Step 1. $\text{DE}(0.5) = E[Y(1, 0.5)] - E[Y(0, 0.5)]$ , both measured in the same villages.

Step 2. $120 - 100 = 20$ .

Answer: $20/month (assuming individuals were randomised within villages).

Exercise 5 · Spillover effect

Untreated households in pure-control villages average 92. With Exercise 4, compute the spillover and total effects.

Show solution

Step 1. Spillover $= E[Y(0, 0.5)] - E[Y(0, 0)] = 100 - 92 = 8$ .

Step 2. Total $= E[Y(1, 0.5)] - E[Y(0, 0)] = 120 - 92 = 28 = 20 + 8$ .

Answer: spillover $8, total $28.

Part 2 · Proofs

Exercise 6 · SUTVA collapses the outcomes

Show that under no interference $Y_i(\mathbf t) = Y_i(t_i)$ , and that consistency then gives $Y_i = T_iY_i(1) + (1 - T_i)Y_i(0)$ .

Show solution

Step 1. No interference says $Y_i(\mathbf t) = Y_i(\mathbf t')$ whenever $t_i = t'_i$ , so $Y_i(\mathbf t)$ depends on $\mathbf t$ only through $t_i$ : write it $Y_i(t_i)$ .

Step 2. Consistency says the observed outcome equals the potential outcome of the treatment received: $Y_i = Y_i(T_i)$ .

Step 3. If $T_i = 1$ , $Y_i = Y_i(1)$ ; if $T_i = 0$ , $Y_i = Y_i(0)$ . Both cases are $T_iY_i(1) + (1 - T_i)Y_i(0)$ . $\blacksquare$

Exercise 7 · Within-group randomisation identifies the direct effect

Assume partial interference with exposure $(t, s)$ . In groups assigned saturation $s$ , individuals are treated completely at random. Show $E[Y \mid T = 1, s] - E[Y \mid T = 0, s] = \text{DE}(s)$ .

Show solution

Step 1. In a saturation- $s$ group, every unit has exposure $(T_i, s)$ , so $Y_i = Y_i(T_i, s)$ .

Step 2. Within those groups, $T_i$ is randomised, so it is independent of $(Y_i(1, s), Y_i(0, s))$ .

Step 3. Hence $E[Y \mid T = 1, s] = E[Y(1, s)]$ and $E[Y \mid T = 0, s] = E[Y(0, s)]$ (Day 6, Exercise 14).

Step 4. The difference is $\text{DE}(s)$ . $\blacksquare$

Exercise 8 · Total = direct + spillover

Prove $E[Y(1, s) - Y(0, 0)] = \text{DE}(s) + \text{SE}(s)$ .

Show solution

Step 1. Add and subtract $Y(0, s)$ : $Y(1, s) - Y(0, 0) = [Y(1, s) - Y(0, s)] + [Y(0, s) - Y(0, 0)]$ .

Step 2. Take expectations (linearity): $\text{DE}(s) + \text{SE}(s)$ . $\blacksquare$

Exercise 9 · Contaminated controls

Individuals are randomised within every village at saturation $s$ , and the analyst treats the within-village difference in means as "the effect". Show it equals the total effect minus the spillover on controls.

Show solution

Step 1. By Exercise 7 the comparison estimates $E[Y(1, s)] - E[Y(0, s)]$ .

Step 2. Rewrite using Exercise 8: $E[Y(1, s)] - E[Y(0, s)] = \underbrace{E[Y(1, s) - Y(0, 0)]}_{\text{total}} - \underbrace{E[Y(0, s) - Y(0, 0)]}_{\text{spillover on controls}}$ .

Answer: positive spillovers (herd immunity) make the within-village comparison understate the policy's total benefit; negative spillovers (displacement) make it overstate it.

Exercise 10 · Hidden versions average out

A "treatment" has two versions, $a$ and $b$ , delivered with probabilities $\pi$ and $1 - \pi$ independently of potential outcomes. Show the difference in means estimates $\pi\,\text{ATE}_a + (1 - \pi)\,\text{ATE}_b$ .

Show solution

Step 1. A treated unit's outcome is $Y_i(a)$ with probability $\pi$ and $Y_i(b)$ otherwise, so $E[Y \mid T = 1] = \pi E[Y(a)] + (1 - \pi)E[Y(b)]$ .

Step 2. $E[Y \mid T = 0] = E[Y(0)]$ by randomisation.

Step 3. Difference: $\pi(E[Y(a)] - E[Y(0)]) + (1 - \pi)(E[Y(b)] - E[Y(0)])$ . $\blacksquare$

The estimate is a mixture: it describes this blend of versions, not either version alone.

Part 3 · Applied

Exercise 11 · Deworming and reinfection

In a school deworming trial, infection rates are: treated pupils in treatment schools 0.20; untreated pupils in treatment schools 0.35; pupils in control schools 0.50. Compute the direct, spillover and total effects, and what a within-school comparison would report.

Show solution

Step 1. Direct effect (within treatment schools): $0.20 - 0.35 = -0.15$ .

Step 2. Spillover on untreated pupils: $0.35 - 0.50 = -0.15$ .

Step 3. Total effect for treated pupils: $0.20 - 0.50 = -0.30$ .

Step 4. A within-school comparison reports the direct effect, −0.15.

Answer: the within-school comparison captures only half the benefit; deworming also protects untreated classmates. This was a central finding of Miguel & Kremer (2004).

Exercise 12 · Job-search assistance and displacement

Job-finding rates within 6 months: assisted jobseekers 0.40; unassisted jobseekers in labour markets where half were assisted 0.28; jobseekers in untreated markets 0.32. Compute the direct and spillover effects, and the net effect on the average jobseeker in a 50%-assisted market.

Show solution

Step 1. Direct effect: $0.40 - 0.28 = 0.12$ .

Step 2. Spillover on the unassisted: $0.28 - 0.32 = -0.04$ (they lose jobs to assisted competitors).

Step 3. Average job-finding in a 50%-assisted market: $0.5(0.40) + 0.5(0.28) = 0.34$ .

Step 4. Net effect versus untreated markets: $0.34 - 0.32 = 0.02$ .

Answer: the direct effect (12 points) vastly overstates the 2-point net gain: most of the "effect" is reshuffling who gets the jobs (as in Crépon et al., 2013).

Exercise 13 · Herd immunity

When 80% of a town is vaccinated, the infection risk of unvaccinated residents falls from 0.10 to 0.04. The vaccine reduces a vaccinated person's risk to 0.01 at that coverage. Compute the indirect and direct effects at 80% coverage.

Show solution

Step 1. Indirect (spillover) effect on the unvaccinated: $0.04 - 0.10 = -0.06$ .

Step 2. Direct effect at 80% coverage: $0.01 - 0.04 = -0.03$ .

Step 3. Total effect for a vaccinated person versus no programme: $0.01 - 0.10 = -0.09 = -0.03 - 0.06$ .

Answer: at high coverage, two-thirds of a vaccinated person's protection comes from others being vaccinated. A trial randomising individuals at low coverage would miss this.

Exercise 14 · Cash transfers and local prices

A large cash transfer to half the households in each village raises local food prices by 5%. Which SUTVA component fails, who is affected, and how should the experiment be designed?

Show solution

Step 1. No interference fails: non-recipients face higher prices, a (negative) spillover through markets.

Step 2. Recipients' measured gains are also net of the price rise.

Step 3. Design: randomise at the market (village or cluster-of-villages) level, ideally with varying saturations, and measure prices directly.

Answer: a within-village comparison mis-states welfare effects; market-level randomisation captures the general-equilibrium effect (as in Egger et al., 2022).

Exercise 15 · Two training providers

A national training programme is delivered by provider A (effect +$3,000) to 40% of trainees and provider B (effect +$1,000) to 60%, assigned independently of workers' potential earnings. What does the evaluation estimate, and what would scaling up through provider B alone deliver?

Show solution

Step 1. By Exercise 10: $0.4(3{,}000) + 0.6(1{,}000) = 1{,}200 + 600 = 1{,}800$ .

Step 2. A scale-up through B only would deliver $1,000 per trainee.

Answer: the evaluation reports $1,800, which overstates the scale-up effect by 80%. Record which version each unit received; "the treatment" must match the policy being considered.

Verify in Python

import numpy as np

rng = np.random.default_rng(32)
G, m = 400, 20                                  # villages, households per village
sat = rng.choice([0.0, 0.5], G)                 # half the villages untreated, half 50% treated
DE, SPILL = 20.0, 8.0                           # direct effect and spillover at 50% saturation
rows = []
for g in range(G):
    T = np.zeros(m, bool)
    T[rng.choice(m, int(sat[g] * m), replace=False)] = True
    y = 92 + DE * T + SPILL * (sat[g] > 0) + rng.normal(0, 15, m)   # everyone in a treated village gets the spillover
    rows += [(sat[g], t, v) for t, v in zip(T, y)]
s, t, y = map(np.array, zip(*rows))
direct = y[(s == 0.5) & t].mean() - y[(s == 0.5) & ~t].mean()
spill = y[(s == 0.5) & ~t].mean() - y[s == 0].mean()
total = y[(s == 0.5) & t].mean() - y[s == 0].mean()
print(f"direct {direct:.1f} (truth 20), spillover {spill:.1f} (truth 8), total {total:.1f} (truth 28)")

# Job-search displacement (Exercise 12)
print("net effect in a 50% market:", round(0.5 * 0.40 + 0.5 * 0.28 - 0.32, 3))

Key takeaways

SUTVA = no interference + no hidden versions; without it, "the" effect is not even defined.
With spillovers, distinguish direct, spillover and total effects; within-group comparisons capture only the direct effect.
Saturation designs and cluster randomisation turn spillovers from a nuisance into an estimand.

Day 32. SUTVA & Interference

Theory

The stable unit treatment value assumption

Where interference comes from

Partial interference and exposure mappings

Designs

Part 1 · Warm-up

Exercise 1 · How many potential outcomes?

Exercise 2 · Spot the violation

Exercise 3 · An exposure mapping

Exercise 4 · Direct effect

Exercise 5 · Spillover effect

Part 2 · Proofs

Exercise 6 · SUTVA collapses the outcomes

Exercise 7 · Within-group randomisation identifies the direct effect

Exercise 8 · Total = direct + spillover

Exercise 9 · Contaminated controls

Exercise 10 · Hidden versions average out

Part 3 · Applied

Exercise 11 · Deworming and reinfection

Exercise 12 · Job-search assistance and displacement

Exercise 13 · Herd immunity

Exercise 14 · Cash transfers and local prices

Exercise 15 · Two training providers

Verify in Python

Key takeaways

Simulation Lab: Spillovers & Saturation