Why Treatment Labels Need Care
Potential-outcomes analysis describes each unit by the outcome it would have under each treatment condition. The earlier material on potential outcomes uses \(Y_i(1)\) and \(Y_i(0)\) for unit \(i\)’s outcomes under treatment and control. That notation is useful only when “treatment” and “control” identify sufficiently precise conditions for the outcomes to be meaningful. A pill might come in different doses; a tutoring program might be delivered weekly or monthly; one person’s outcome might depend on whether their neighbors also receive the program.
These are not merely wording issues. If the same treatment label covers distinct interventions, or if another unit’s assignment changes a person’s outcome, then the potential outcome cannot generally be represented by the person’s own binary assignment alone. The standard framework addresses these concerns through consistency, no interference, and a requirement that treatment versions be well defined.
Consistency, Versions, and Interference
Let \(Z_i\in\{0,1\}\) denote unit \(i\)’s treatment assignment and \(Y_i\) the observed outcome. Consistency says that the observed outcome is the potential outcome under the condition actually received: \(Y_i=Y_i(Z_i)\). It does not say that treatment was randomly assigned, that treated and control units are comparable, or that a treatment effect has been identified. It is a link between the observed outcome and the potential-outcomes notation.
That link presumes that \(Z_i=1\) refers to a sufficiently specific intervention. If there are several versions of treatment, write \(V_i\) for the version received. A more detailed potential outcome is \(Y_i(z,v)\), and consistency links the observed outcome to the actual assignment and version: \(Y_i=Y_i(Z_i,V_i)\). To use the simpler notation \(Y_i(z)\), outcomes must not depend on which version is used among those grouped under \(z\), or the analysis must define a particular version as the treatment.
Interference is a different issue. If one person’s treatment can affect another person’s outcome, unit \(i\)’s potential outcome may depend on the assignments of other units. With \(Z_{-i}\) denoting the assignments of everyone other than \(i\), write \(Y_i(Z_i,Z_{-i})\). The no-interference condition requires that changing other units’ assignments, while holding \(i\)’s own assignment fixed, does not change \(i\)’s potential outcome.
The two requirements address different sources of ambiguity. No interference concerns who else is treated. The versions requirement concerns what treatment a unit receives. In some presentations, consistency is stated separately from SUTVA; the important point is to state the observed-outcome link and the assumptions that make its notation appropriate.
Worked Example: One Label, Two Medication Doses
Worked Example: One Label, Two Medication Doses
Suppose a hypothetical trial codes receipt of a medication as \(Z_i=1\), but the label includes both a 5-mg dose and a 10-mg dose. For one hypothetical patient, let the outcome be a symptom score after treatment, with lower scores indicating fewer symptoms. Suppose this patient’s potential outcomes are:
Under the 5-mg version, the difference from control for this patient is \(8-9=-1\) symptom-score point. Under the 10-mg version, it is \(5-9=-4\) points. Both versions receive the same binary label \(Z_i=1\), but the patient has two different outcomes under that label. There is therefore no single value of \(Y_i(1)\) determined by the label alone.
The repair is to define the treatment condition more precisely, such as “5 mg,” or to specify a treatment policy that states how the dose is selected. One can then define the relevant potential outcomes for that condition or policy. Merely writing \(Y_i(1)\) does not resolve which dose it denotes.
Worked Example: Spillovers Between Neighbors
Worked Example: Spillovers Between Neighbors
Consider two households in a hypothetical neighborhood program. Let \(Z_1\) and \(Z_2\) indicate whether household 1 and household 2, respectively, receive a home-insulation service. Suppose household 1’s winter energy use, in hundreds of kilowatt-hours, has these potential outcomes:
The first argument is household 1’s own assignment; the second is household 2’s assignment. At a fixed assignment for household 2, household 1’s treatment contrast is:
But household 1’s outcome also changes when household 2’s assignment changes. If household 1 is untreated, the change is \(Y_1(0,1)-Y_1(0,0)=11-7=4\). If household 1 is treated, it is \(Y_1(1,1)-Y_1(1,0)=14-10=4\). Thus household 2’s treatment affects household 1’s outcome by 4 hundreds of kilowatt-hours in this example. No-interference fails, even though household 1’s own treatment contrast happens to equal 3 in both neighbor conditions.
For example, under assignments \((Z_1,Z_2)=(1,0)\), household 1’s outcome is \(Y_1(1,0)=10\). Under assignments \((1,1)\), it is \(Y_1(1,1)=14\). A representation using only household 1’s own assignment would assign both situations the same \(Y_1(1)\), which cannot match both outcomes. The second assignment must be represented, or the analysis must adopt a suitable exposure definition that captures how other assignments matter.
Why the Standard Representation Requires Both Conditions
The logic can be stated directly. With interference and versions allowed, let \(V_i\) denote unit \(i\)’s treatment version and let \(Z_{-i}\) collect other units’ assignments. The observed outcome is linked to the potential outcome under the realized conditions:
To write this as \(Y_i=Y_i(Z_i)\), the outcome must be invariant to changes in the other arguments that the simpler notation omits. In particular, for all relevant versions \(v,v'\) and assignments \(z_{-i},z'_{-i}\), the conditions must imply:
The first equality rules out outcome differences due to hidden versions or other units’ assignments. If either source can change the outcome, the omitted arguments matter and the simpler notation is insufficient. This is why consistency in the usual form depends on both a well-defined treatment condition and no interference.
State what assignment \(Z_i=1\) and \(Z_i=0\) mean in practice, including dose, timing, provider, or other features that define distinct versions.
Ask whether another unit’s assignment could alter unit \(i\)’s outcome. If so, potential outcomes may need to depend on other assignments or on a clearly defined exposure.
Use consistency only after the treatment condition and any relevant exposure have been defined, so the observed outcome is linked to the corresponding potential outcome.
Worked Example: A Program Delivered in Different Ways
Worked Example: A Program Delivered in Different Ways
A hypothetical study records a binary label for an employment-support program, but participating clients receive either weekly coaching or monthly coaching. For illustration, suppose four treated clients have observed job-search activity scores of 12 and 10 under weekly coaching, and 8 and 6 under monthly coaching. Four control clients have scores of 7, 5, 6, and 6.
The mean score among the treated clients is:
The control mean is:
The observed difference in means is \(9-6=3\) activity-score points. This arithmetic is correct, but it does not establish a causal effect of one well-defined program. The treated mean combines two versions; without further definition, it is unclear whether “the program” means weekly coaching, monthly coaching, or a policy that assigns clients to one or the other. The difference also cannot, by itself, establish that treatment caused the difference.
A more precise description could define weekly coaching as the treatment, or define a policy specifying how clients receive the two versions. The potential outcomes would then refer to those specified conditions. If weekly and monthly coaching are intentionally treated as distinct conditions, they should not be silently collapsed into one binary treatment label.
What These Assumptions Do—and Do Not—Establish
SUTVA is a condition on the way treatments and potential outcomes are represented; it is not a guarantee that an experiment is unbiased or that an observed contrast has a causal interpretation. Even when treatment versions are well defined and there is no interference, causal conclusions may require additional design or identification conditions. Conversely, an experiment can be randomly assigned and still have spillovers or ambiguous treatment versions.
These assumptions are often substantive and cannot be verified from the observed outcomes alone. A useful assessment asks how the intervention was implemented, whether units can affect one another, and whether those interactions could change the outcome being studied. Study design, subject-matter knowledge, and detailed measurement of treatment receipt can make the assumptions more plausible or reveal where the standard notation needs refinement.
Common Mistake: Treating the Assignment Code as the Treatment Itself
A common mistake is to reason, “Every treated unit has \(Z_i=1\), so every treated unit has the same treatment and \(Y_i=Y_i(1)\).” The correction is that a code records an assignment category; it does not ensure that the category names a single intervention, nor that other units’ assignments are irrelevant. Check for multiple versions and spillovers before using the standard potential-outcomes representation.
Check Your Understanding
For each question, identify which part of the standard representation is at issue.
- A clinic labels two different medication doses as “treated.” What information is needed before \(Y_i(1)\) has a clear meaning?
- In the neighbor example, which potential outcomes show that no interference fails? State the numerical difference.
- Explain why consistency does not imply that treatment assignment was random.
- A program’s treated clients receive different versions, but the analyst defines treatment as a policy that randomly selects one version according to a specified rule. What must the potential outcome refer to?
- Why does a correctly calculated treated-minus-control mean difference not by itself establish a causal effect?