Consistency and Treatment Versions

Why Consistency Requires a Specified Intervention

A causal contrast compares outcomes under interventions, not merely outcomes among people with different recorded labels. For that comparison to have a clear meaning, each intervention must specify what would be done. If “treatment” refers to several materially different regimens, or “prevent the competing event” could be accomplished in several ways, there may be no single counterfactual outcome attached to the broad label.

This issue is especially important for competing risks. A treatment or prevention strategy can affect both the event of interest and the competing event. Consequently, two ways of delivering something described as the same treatment, or two ways of preventing a competing event, can produce different counterfactual event histories.

The notation \(a\) must refer to an intervention sufficiently specific to determine the counterfactual history. The same requirement applies to an intervention that changes or prevents a competing event. Consistency is a link between the factual outcome and the counterfactual outcome under the actual intervention received. It does not say that people receiving different interventions would have had the same outcomes, nor does it establish that intervention groups are comparable.

As usual, individual counterfactual outcomes also presume that one person’s intervention does not change another person’s outcome, unless interference is explicitly built into the intervention and outcome definitions. Here the focus is on defining the intervention and its versions. Exchangeability, positivity, and adequate outcome observation are separate requirements for causal estimation.

When a Treatment Has Multiple Versions

Suppose a binary label \(A=1\) includes several versions \(V\), such as different doses, routes, schedules, or co-interventions. Let \(Y_k^{a,v}(\tau)\) denote the counterfactual cause-\(k\) outcome under treatment label \(a\) delivered as version \(v\). The version-specific consistency statement is that a person actually receiving \((A,V)=(a,v)\) has \(Y_k(\tau)=Y_k^{a,v}(\tau)\).

That statement does not generally justify replacing every \(Y_k^{a,v}(\tau)\) with one common \(Y_k^a(\tau)\). Such a single potential outcome is well-defined only when the treatment label identifies a unique intervention, or when versions can justifiably be treated as equivalent for the outcome and time horizon in question. Merely assigning the same name to different regimens does not make their counterfactual outcomes equal.

A useful way to make the estimand explicit is to define a policy that chooses versions. Let \(q(v)\) be the probability that the policy assigns version \(v\), with \(\sum_v q(v)=1\). The risk under that policy is the average of the version-specific risks using those policy probabilities:

This is one well-defined meaning for a broad treatment label: the label is implemented by a declared version policy \(q\). Change \(q\), and the intervention may change. In observational data, the versions received by people with \(A=a\) are often not assigned according to a fixed policy independent of their characteristics. Thus, the observed mixture of versions is not automatically the risk under a desired policy, and consistency by itself does not remove confounding or selection.

Consistency for Competing-Event Prevention

A hypothetical intervention that prevents a competing event also needs an operational definition. Let \(g\) denote a specified prevention strategy and \(Y_k^{a,g}(\tau)\) the target-event outcome under treatment \(a\) combined with that strategy. For example, \(g\) might specify a particular preventive drug, eligibility rules, start time, dose, monitoring, and what happens if adverse effects occur. Consistency links the factual outcome to \(Y_k^{a,g}(\tau)\) only for a person who actually receives and follows that specified strategy.

The phrase “eliminate the competing event” alone may fail to define \(g\). One method might prevent the competing event while also affecting the target event; another might prevent it through a different mechanism. Even if both strategies eliminate the competing event, their counterfactual target-event risks need not agree. A hypothetical elimination intervention therefore cannot be interpreted as a unique real-world intervention unless the intervention is defined or the relevant versions are justified as equivalent.

Derivation: Why the Version Mix Matters

The role of versions follows from partitioning the treated population by the version actually received. For simplicity, take a binary outcome \(Y_k(\tau)\) and suppose version \(V\) is recorded. Version-specific consistency gives \(Y_k(\tau)=Y_k^{a,v}(\tau)\) among people who received \((A,V)=(a,v)\). The law of total probability then yields:

The second line uses consistency within each observed version group. It describes the factual risk among those classified as \(A=a\), with the version proportions and version-specific group composition found in that group. It is not generally the same as \(\sum_v q(v)E[Y_k^{a,v}(\tau)]\): the latter is the risk under a specified policy in the target population. Equality requires additional conditions linking the observed version groups to the target population and policy. This derivation shows why consistency is necessary to connect observed and counterfactual outcomes, but insufficient on its own to identify a policy risk.

Practical Checks, Alternatives, and Limitations

Consistency is principally a design and definition requirement, not a statistical hypothesis that can usually be tested from observed outcomes. For a competing-risk analysis, examine the intervention protocol and ask whether the treatment and any prevention strategy are specified closely enough to define the event history they imply.

• Audit the intervention details. Record dose, route, timing, duration, co-interventions, adherence rules, and clinically relevant changes over time. For a competing-event intervention, state how prevention is achieved and what happens when the strategy cannot be followed.
• Check the event definition. Specify the target cause, competing causes, time origin, and follow-up horizon. A change in the event classification or horizon changes the outcome being defined.
• Look for versions hidden by a broad label. If versions are recorded, compare their protocols and consider whether the scientific question concerns a particular version, a policy mixture, or a justified common effect.
• Do not treat plausibility as proof. Similar names or intended purposes do not establish that versions have identical effects. Conversely, differences in delivery do not automatically prove that the outcome differs; they signal a question that needs substantive justification.

There are several principled responses to multiple versions. One is to define a version-specific contrast. Another is to define a version-allocation policy and target its risk. A third is to defend a common treatment label using a substantive argument that the versions are equivalent for the outcome of interest. For a competing-event intervention, the corresponding choice is to specify a concrete prevention strategy or justify why alternative implementations represent the same intervention. These choices answer different questions and should not be silently substituted for one another.

A common mistake is to say, “Everyone assigned \(A=1\) received treatment, so consistency gives \(Y=Y^1\),” even though \(A=1\) includes materially different regimens. The correction is to write \(Y^{1,v}\) for each relevant version, or explicitly define the policy that selects \(v\). A related mistake is to censor observed competing events and call the result the outcome under prevention. Censoring is a data operation, not itself a specified intervention that prevents the competing event.

Even with an unambiguous intervention and consistency, causal estimation still requires other conditions. In particular, consistency alone does not make intervention groups exchangeable, ensure that every relevant intervention is possible in each covariate group, or resolve incomplete outcome observation. Nor does it show that an imagined prevention strategy could be implemented without changing other aspects of care. A precise intervention makes the causal question interpretable; it does not by itself guarantee that the question is identifiable or practically meaningful.