Why a Treatment Label May Not Define an Intervention
Consistency links an observed outcome to the potential outcome for the treatment actually received. That link requires the treatment to be specified clearly enough that its potential outcome has a definite meaning. A label such as “treated” or “received the drug” may conceal different doses, formulations, routes, durations, or delivery settings. If those versions can affect the outcome, the label alone does not identify one intervention.
This is a distinct problem from confounding or lack of treatment support. Even when the label is randomized and both label values occur, an observed outcome among those assigned the label may not represent the outcome under the particular treatment version a researcher intends to study.
A target such as \(E[Y^{1,v_1}]-E[Y^0]\) asks about a policy that gives everyone version \(v_1\) rather than control. By contrast, \(E[Y\mid A=1]\) describes outcomes under the versions that happened to be delivered to the labelled treated group. These quantities need not agree. If a study instead targets a policy that deliberately assigns versions according to a stated rule, that rule must also be part of the treatment definition.
Worked Examples: What the Label Conceals
Worked Example: Two Versions of the Same Therapy
Suppose “therapy” is recorded as \(A=1\), but it is delivered either as an oral formulation \(v_1\) or an injected formulation \(v_2\). For a particular patient, suppose \(Y^{1,v_1}=1\), \(Y^{1,v_2}=0\), and \(Y^0=0\), where \(Y=1\) denotes recovery.
If the patient receives the oral version, version-specific consistency gives \(Y=Y^{1,v_1}=1\). If the same patient receives the injection, it gives \(Y=Y^{1,v_2}=0\). In both cases the recorded label is \(A=1\), but the observed outcome is not the same potential outcome under a single, version-free intervention.
Thus, writing \(Y=Y^1\) for every patient with \(A=1\) would silently assert that the versions are interchangeable for the outcome. The label itself does not establish that assertion.
Worked Example: The Treated Mean Is a Mixture
In a treated group of 100 people, suppose 60 receive \(v_1\) and 40 receive \(v_2\). Among those receiving \(v_1\), 30 recover; among those receiving \(v_2\), 8 recover. The observed recovery proportions are \(30/60=0.50\) and \(8/40=0.20\), respectively.
The recovery proportion under the versions actually delivered is \((30+8)/100=0.38\). Equivalently, it is the weighted average \(0.60(0.50)+0.40(0.20)=0.30+0.08=0.38\).
This calculation describes the observed mixture: 60% \(v_1\) and 40% \(v_2\). It does not give the recovery proportion if everyone received \(v_1\), which would concern \(E[Y^{1,v_1}]\), nor does it give the effect of a version-independent treatment label. Those targets require additional information or assumptions.
Worked Example: Randomized Label, Unidentified Version Effect
Consider a population with two equally common types, \(U=0\) and \(U=1\). Treatment label \(A\) is randomized independently of type. Controls have outcome \(Y^0=0\). Among people with \(A=1\), the delivered version is \(v_1\) for type \(U=1\) and \(v_2\) for type \(U=0\). The version is not recorded.
In both of the following causal models, set \(Y^{1,v_1}=1\) for type \(U=1\) and \(Y^{1,v_2}=0\) for type \(U=0\). Therefore, among labelled treated people, half have outcome 1 and half have outcome 0. The observed treated mean is \(0.5\) in either model; controls all have outcome 0.
In Model I, set the unobserved potential outcome \(Y^{1,v_1}=1\) for type \(U=0\) as well. The population mean under everyone receiving \(v_1\) is \(E[Y^{1,v_1}]=0.5(1)+0.5(1)=1\), so its contrast with control is \(1-0=1\).
In Model II, instead set \(Y^{1,v_1}=0\) for type \(U=0\). Then \(E[Y^{1,v_1}]=0.5(1)+0.5(0)=0.5\), and the contrast with control is \(0.5-0=0.5\).
The observed data have the same distribution in both models: randomized \(A\), control outcomes all zero, and labelled treated outcomes split equally between zero and one. Yet the effect of assigning version \(v_1\) differs. Therefore that effect is not identified from these observed data, even though the treatment label was randomized and both label values had positive probability.
Why the Usual Identification Link Breaks
For a well-defined treatment, consistency supplies the equality between an observed outcome under treatment and the corresponding potential outcome. With multiple outcome-relevant versions under one label, the equality is instead version-specific. The observed treated mean can be expressed as a mixture over versions, but the mixture is not automatically the mean under any one version.
Proof. Because the possible versions form a finite partition of the event \(A=1\), the law of total expectation gives \(E[Y\mid A=1]=\sum_v E[Y\mid A=1,V=v]P(V=v\mid A=1)\). On the event \(A=1,V=v\), version-specific consistency gives \(Y=Y^{1,v}\). Substituting this equality into each conditional expectation yields the stated identity. The terms are conditional means for people who actually received each version, not necessarily population means under assignment of that version. No equality to \(E[Y^{1,v^\ast}]\) follows without further assumptions. \(\square\)
A useful sufficient condition for a label-level potential outcome is version irrelevance: for every pair of versions \(v,w\) included under the label, \(Y^{1,v}=Y^{1,w}\) almost surely. Then define \(Y^1\) to be this common random variable. If \(A=1\), consistency gives \(Y=Y^{1,V}=Y^1\), so the observed outcome does represent the label-level potential outcome.
Proof. Choose any admitted version \(v_0\) and define \(Y^1=Y^{1,v_0}\). By the assumed equality, \(Y^{1,V}=Y^{1,v_0}=Y^1\) for every admitted version, almost surely. When \(A=1\), version-specific consistency gives \(Y=Y^{1,V}\). Combining the equalities gives \(Y=Y^1\). \(\square\)
The condition is sufficient, not necessary for every possible target. A treatment label could also be made precise by defining a policy that assigns versions according to a specified distribution or rule. That policy has its own potential outcome. What is not justified is treating the observed, possibly undocumented allocation of versions as though it were automatically the intended policy.
Assumptions, Diagnostics, and Alternatives
Exchangeability and positivity address other links in causal identification. Exchangeability concerns whether treatment assignment is independent of relevant potential outcomes, possibly conditional on measured history. Positivity concerns whether treatment options occur in the relevant groups. Neither condition specifies what a treatment label means. In the randomized counterexample, label assignment is independent of type, and both labels occur, yet the target version effect remains unidentified because the version is concealed and version-specific outcomes are not observed for everyone.
Useful diagnostics begin with the intervention itself. Ask whether the label uniquely determines dose, formulation, route, timing, duration, and delivery procedure. Then check whether those details were recorded, whether their distribution differs across sites or patient groups, and whether outcomes plausibly vary by version. Large differences across versions are evidence against casually treating them as interchangeable; absence of measured differences does not prove that their potential outcomes are identical.
| Approach | What it targets | Key requirement |
|---|---|---|
| Refine the treatment definition | The effect of a specified version, such as \(v_1\) versus control | Version information and assumptions sufficient to identify that comparison |
| Define a policy over versions | The effect of a stated rule for assigning versions | The policy, including its version distribution or assignment rule, is explicit |
| Pool versions as one label | A common label-level treatment effect | Versions are outcome-equivalent for the target, or the pooling has a justified interpretation |
A common mistake is to argue, “Treatment was randomized, so the treated mean estimates the outcome under treatment.” Randomization can balance the groups assigned to the labels; it cannot turn several different interventions into one. The correction is to state the target version or version policy, then establish the assumptions needed for that target. If version details were not observed, the data may support only a mixture description, not a version-specific causal effect.
In longitudinal studies the same issue can arise at any decision time: a recorded treatment \(A_t\) may conceal versions \(V_t\), and a regime described only by the sequence of labels may fail to specify what is delivered. The remedy is not merely more elaborate adjustment for measured histories. The intervention and its versions must first be defined sufficiently clearly for the relevant potential outcomes to be meaningful.
Check Your Understanding
For each question, identify the relevant target and explain what the observed data do or do not establish.
- A trial randomizes a label “high-dose treatment,” but clinics use two different formulations. Does randomization alone identify the effect of formulation \(v_1\) versus control?
- Suppose the observed treated group is 70% version \(v_1\) and 30% version \(v_2\), with version-specific observed means 0.4 and 0.8. What is the observed treated mean?
- State a sufficient condition under which a single label-level potential outcome \(Y^1\) can represent all versions under label 1.
- Two causal models produce the same observed distribution but different values of \(E[Y^{1,v_1}]\). What does this imply about identification from those observed data?
Solutions. (1) No. Randomization concerns assignment of the label; it does not identify the effect of \(v_1\) if version-specific outcomes are not distinguished and additional assumptions or information are absent. (2) The observed mean is \(0.70(0.4)+0.30(0.8)=0.28+0.24=0.52\). This is the mean under the observed version mixture, not necessarily under either version alone. (3) A sufficient condition is \(Y^{1,v}=Y^{1,w}\) almost surely for every pair of versions admitted under label 1. Version-specific consistency then gives the same observed outcome for every delivered version. (4) The target is not identified: the observed distribution alone cannot distinguish models that imply different target values.