Treatment Versions and Well-Defined Interventions

Why Treatment Labels Can Be Too Broad

A treatment variable often records a category such as “treated” or “received the drug.” But that label may hide what was actually done: the dose, timing, duration, route, or method of delivery. If those differences can change an outcome, “treated” does not by itself describe one intervention with one potential outcome. The difficulty is not merely imprecise data recording. It is that the causal question may not yet specify what action is being compared.

The earlier material on consistency for longitudinal treatment strategies expresses the basic requirement: for a person whose observed treatment history matches a specified strategy, the observed outcome equals the potential outcome under that strategy. When a strategy is recorded only as a broad treatment label, we must ask whether its hidden versions can legitimately be treated as the same intervention.

The phrase “for an outcome” matters. Two delivery methods might produce the same value of one outcome but different values of another. Whether versions can be grouped is therefore a substantive, outcome-specific judgment, not a rule that can be settled from the treatment variable’s name alone.

Consistency When Versions Are Recorded

Suppose one decision is observed as a pair \((A,V)\). The version-specific consistency condition is \(Y=Y^{A,V}\): the observed outcome equals the potential outcome under the treatment level and version actually received. If a longitudinal history records both levels and versions, the corresponding condition is \(Y=Y^{\bar A,\bar V}\). This says what consistency means for fully specified observed treatment; it does not assert that the data identify any potential-outcome mean.

For a broad level \(a\) with possible versions in a set \(\mathcal V_a\), a single potential outcome \(Y^a\) is coherent only when the version-specific potential outcomes agree, at least almost surely, across the versions that the broad level is intended to include. If they do agree, write their common value as \(Y^a\). If they do not, the level-specific label alone leaves multiple possible outcomes and does not define one version-free potential outcome.

To justify the result, first suppose the version-specific outcomes agree. Choose one version \(v_0\in\mathcal V_a\) and set \(Y^a=Y^{a,v_0}\). For every \(v\in\mathcal V_a\), the assumed equality gives \(Y^{a,v}=Y^{a,v_0}=Y^a\) almost surely. In the other direction, if \(Y^{a,v}=Y^a\) and \(Y^{a,v'}=Y^a\) almost surely, then \(Y^{a,v}=Y^{a,v'}\) almost surely by transitivity of almost-sure equality. Thus a version-free outcome is not an additional observed quantity: it is a coherent summary only when the version-specific outcomes support it. \(\square\)

Three Ways a Broad Label Can Hide an Intervention

Means Under a Policy That Selects Versions

A broad treatment level need not be discarded merely because its versions have different effects. Instead, the intervention can explicitly select among versions. Let \(G\) denote a random version drawn according to probabilities \(g(v)\), where \(\sum_{v\in\mathcal V_a}g(v)=1\), and define \(Y^{a,G}=Y^{a,V}\) when \(G=V\). Assume the version draw is independent of the potential-outcome vector \(\{Y^{a,v}:v\in\mathcal V_a\}\). This independence describes the policy’s randomization; it is not a claim about how versions were assigned in observational data.

For a finite set, partition according to the selected version. Independence gives \(P(G=v\mid \{Y^{a,u}\}_u)=g(v)\). Consequently, \(E[Y^{a,G}]=\sum_v E[\mathbf 1(G=v)Y^{a,v}]\). For each term, independence implies \(E[\mathbf 1(G=v)Y^{a,v}]=g(v)E[Y^{a,v}]\). The finite sum yields the stated identity; finiteness of each absolute expectation ensures each term is integrable.

For a countable set, the same partition gives a sum of expectations. The condition \(\sum_v g(v)E[|Y^{a,v}|]<\infty\) implies \(E[\sum_v\mathbf 1(G=v)|Y^{a,v}|]=\sum_v g(v)E[|Y^{a,v}|]<\infty\), by Tonelli’s theorem for nonnegative terms. Hence the selected outcome is integrable, and the sum of signed expectations can be interchanged with expectation by absolute integrability. This proves the identity and ensures its value is finite. Without this condition, finite expectations for each individual version do not by themselves guarantee a finite policy mean. \(\square\)

A specified version policy is an alternative to demanding that all versions have identical effects. The two approaches answer different questions: version-free consistency treats the versions as equivalent for the outcome, whereas a policy mean averages over versions with stated probabilities. A fixed version is another option, as is refining the treatment variable so that substantively distinct versions are separate levels.

Diagnostics, Common Mistakes, and Limits

When reviewing a treatment definition, list the features that can vary within each recorded level. For a medication, check dose, schedule, duration, and route. For a procedure, check timing, provider, and relevant implementation details. Ask whether these differences are part of the intended intervention, whether the potential outcome is plausibly unchanged across them, or whether a rule for selecting versions should be specified. In data, examine whether versions are actually recorded and whether treatment labels combine visibly different protocols. These checks diagnose ambiguity; they do not prove that unrecorded versions have no effect.

• Common mistake: assuming the treatment name is the intervention. The reasoning “everyone received the drug, so consistency gives one \(Y^1\)” fails if “received the drug” combines doses with different potential outcomes. The correction is to specify a version or policy, or defend the claim that the included versions have the same potential outcome.
• Common mistake: confusing version ambiguity with confounding. Defining a dose policy does not establish that people receiving that dose are comparable to people not receiving it. A precise intervention and assumptions for causal identification are separate requirements.
• Diagnostic warning: versions are unmeasured. If dose or timing is absent from the data, it may be impossible to check whether the recorded category combines versions. A broad label can remain ambiguous even when the data contain no visible evidence of differences.

The main alternatives each have limits. Refining categories clarifies the intervention only when the relevant version information is measured and the resulting categories are meaningful. Specifying a stochastic policy makes version selection explicit, but its mean still depends on the chosen probabilities. Treating all versions as equivalent can simplify the estimand, but requires substantive justification and may fail for one outcome even if it is plausible for another. None of these choices, by itself, supplies the assumptions needed to identify the causal effect from observed data.