📊Causal Inference Unit 2 Review

The Stable Unit Treatment Value Assumption (SUTVA) is what makes causal effects well-defined in the potential outcomes framework. Without it, you can't cleanly write down what $Y_i(1)$ and $Y_i(0)$ even mean, because the potential outcomes could shift depending on what happens to other units or how the treatment gets delivered.

SUTVA has two distinct components:

No interference between units — one unit's treatment assignment doesn't affect another unit's outcomes.
No hidden variations in treatment — there's only one version of each treatment level, so "treated" means the same thing for every treated unit.

Both components must hold for the potential outcomes to be stable, well-defined quantities that you can actually estimate.

Potential Outcomes Framework Connection

SUTVA is baked into the notation of the potential outcomes framework itself. When you write $Y_i(t)$ for unit $i$ under treatment $t$ , you're implicitly assuming that this value is a fixed quantity. SUTVA is what justifies that assumption. If other units' treatments could change $Y_i(t)$ , or if treatment $t$ could mean different things for different units, then $Y_i(t)$ isn't a single number — it's a moving target.

No Interference Between Units

This component states that unit $i$ 's potential outcomes depend only on unit $i$ 's own treatment assignment, not on the treatments assigned to units $j, k, l, \ldots$

Formally: $Y_i(t_1, t_2, \ldots, t_n) = Y_i(t_i)$ for all possible treatment vectors. The potential outcome for unit $i$ is a function of $t_i$ alone.

This rules out spillover effects — situations where treating one unit changes outcomes for others. Common violations include:

Peer effects: A job training program changes labor market conditions for non-participants by altering labor supply.
Spatial spillovers: An environmental policy in one region affects pollution levels in neighboring regions.
Contagion: Vaccinating some people in a community reduces disease exposure for unvaccinated people (herd immunity).

No Hidden Variations in Treatment

This component states that there is only one "version" of each treatment level. If unit $i$ and unit $j$ are both assigned $T = 1$ , they receive the same treatment in all relevant respects.

Violations arise when the treatment label masks meaningful differences in what units actually receive:

Dosage differences: Two patients both "receive the drug" but at different doses.
Quality of delivery: Two classrooms both "receive the new curriculum" but with teachers of vastly different skill levels.
Adherence variation: Two participants are both "assigned to exercise" but one exercises 5 days a week and the other exercises 1 day.

When this component fails, $Y_i(1)$ is ambiguous because "treatment = 1" doesn't correspond to a single, well-defined intervention.

Consistency Assumption

Consistency bridges the gap between the theoretical potential outcomes and the data you actually observe. It says: if unit $i$ receives treatment $t$ , then the outcome you observe for unit $i$ equals the potential outcome under that treatment.

Formal Definition

$\text{If } T_i = t, \text{ then } Y_i = Y_i(t)$

where $Y_i$ is the observed outcome, $Y_i(t)$ is the potential outcome under treatment $t$ , and $T_i$ is the treatment actually received.

This might seem obvious, but it's doing real work. Consistency says the potential outcome $Y_i(t)$ is the same outcome you'd observe in the real world when unit $i$ gets treatment $t$ . If the treatment is vaguely defined or has multiple versions, this link breaks down.

Connection to SUTVA

Consistency and SUTVA are tightly linked. SUTVA's "no hidden treatment variations" component is essentially what makes consistency hold. If there are multiple versions of treatment $t$ , then $Y_i(t)$ isn't a single value, and the equation $Y_i = Y_i(t)$ when $T_i = t$ becomes meaningless — which version of $Y_i(t)$ are you referring to?

Some textbooks fold consistency into SUTVA; others treat them as separate assumptions. Either way, the logic is the same: you need well-defined treatments and no interference for the observed data to tell you about potential outcomes.

Consistency vs. Exchangeability

These are different assumptions that address different problems:

Consistency concerns the relationship between observed outcomes and potential outcomes. It asks: does the data I see correspond to the potential outcomes I've defined?
Exchangeability concerns the relationship between treatment assignment and potential outcomes. It asks: are treated and untreated groups comparable, so that differences in observed outcomes reflect causal effects rather than confounding?

You need both for valid causal inference, but they can fail independently. A study could have perfect consistency (well-defined treatments, no interference) but terrible exchangeability (treatment assignment driven by confounders). Or it could have exchangeability through randomization but poor consistency because the treatment varies in implementation.

SUTVA Violations and Their Consequences

What Goes Wrong

When SUTVA or consistency fails, the core machinery of causal inference breaks down in specific ways:

Biased effect estimates: If interference exists, the observed outcome for a control unit may already be influenced by treated units nearby. The treatment effect estimate gets contaminated.
Ambiguous estimands: If treatment has multiple versions, the "average treatment effect" is an average over an undefined mixture of different interventions. You don't know what you're estimating.
Misleading policy conclusions: If spillover effects inflate (or deflate) your estimate, scaling up the intervention could produce very different results than your study predicted.

Potential outcomes framework, Frontiers | How people explain their own and others’ behavior: a theory of lay causal ...

Concrete Examples

Interference example: You randomize villages to receive a microfinance program. But treated villages trade with control villages, and the economic activity spills over. Control village outcomes improve partly because of the treatment in neighboring villages. Your estimated effect underestimates the true direct effect of the program.

Treatment variation example: You study the effect of "surgery vs. no surgery" for a condition, but surgical technique varies widely across hospitals. The causal effect you estimate is a blend of many different surgical procedures, making it hard to know what specific intervention you're recommending.

Strategies for Addressing Violations

Redefining Treatment or Units

Sometimes you can fix a SUTVA violation by changing what counts as a "unit" or a "treatment":

If interference happens within households, analyze at the household level instead of the individual level.
If interference happens within classrooms, randomize at the classroom or school level (cluster randomization).
If treatment varies in intensity, replace a binary treatment indicator with a more granular measure (e.g., hours of tutoring rather than tutored/not tutored).

The goal is to redefine things so that the redefined treatment is truly uniform and the redefined units don't interfere with each other.

Modeling Interference Explicitly

When interference is unavoidable, you can build it into the model rather than assuming it away. This requires specifying the structure of interference:

Partial interference: Units interact within known clusters (e.g., classrooms, villages) but not across clusters. You can define potential outcomes as functions of both a unit's own treatment and the proportion of treated units in its cluster.
Network models: If you know the social network, you can model outcomes as depending on a unit's own treatment and the treatments of its network neighbors.

These approaches require stronger assumptions about how interference works, but they're more honest than pretending it doesn't exist.

Sensitivity Analysis

When you can't rule out violations or model them directly, sensitivity analysis asks: how bad would the violation need to be to change my conclusions?

You vary assumptions about the magnitude of interference or treatment variation and see how the estimated effects shift. If your conclusions hold across a wide range of plausible violations, that's reassuring. If a small amount of interference could flip your result, that's a red flag.

SUTVA and Consistency in Practice

Assessing Plausibility

Before running any analysis, think carefully about whether SUTVA and consistency are reasonable for your setting:

Could units interact? Consider physical proximity, social networks, shared markets, or shared resources.
Is the treatment truly uniform? Consider whether implementation quality, dosage, timing, or adherence varies across units.
Is the treatment well-defined enough? If you can't precisely describe what "receiving treatment" means, consistency is likely shaky.

These assessments should draw on subject-matter knowledge, not just statistical diagnostics.

Role Across Causal Methods

SUTVA and consistency aren't specific to one method. They're required (at least implicitly) by randomized experiments, propensity score methods, instrumental variables, regression discontinuity, and difference-in-differences. No amount of clever identification strategy can rescue you if the potential outcomes themselves are ill-defined.

Limitations and Trade-offs

Addressing violations always involves trade-offs:

Redefining units to a coarser level (e.g., clusters) reduces interference concerns but also reduces your sample size and the precision of your estimates.
Modeling interference explicitly gives you richer results but requires assumptions about the interference structure that may be hard to verify.
Sensitivity analysis doesn't fix the problem; it just tells you how much the problem matters.

Being transparent about which assumptions you're relying on, and how plausible they are, is more valuable than pretending the assumptions are automatically satisfied.

📊Causal Inference Unit 2 Review