Treatment Versions and Well-Defined Interventions in Causal Designs

Why the Treatment Definition Matters

A causal question needs a clear description of what would happen under each treatment condition. A label such as “received the medication” may sound precise, but it might include different doses, routes, or schedules. If those alternatives can lead to different outcomes, the label alone does not identify one intervention. The causal effect then depends on which version is meant, or on how versions are chosen.

Earlier material on potential outcomes represents a unit’s outcome under treatment and under no treatment as \(Y_i(1)\) and \(Y_i(0)\). That notation assumes the treatment condition is sufficiently clear to define one relevant outcome for each unit. The earlier material on consistency connects a unit’s observed outcome to the potential outcome for the treatment actually received; SUTVA also includes a requirement concerning treatment versions. Here the focus is the practical question behind that requirement: what exactly counts as receiving the treatment?

Versions can differ in active ingredient, dose, timing, route, provider, or delivery setting. Differences in labels do not automatically create substantively distinct versions: two descriptions might specify the same intervention in practice. Conversely, the same label can cover different interventions. The key question is whether the differences could change the potential outcome or the interpretation of the intervention.

When a Broad Treatment Label Is Ambiguous

Suppose a study records only whether a patient received a medication, without recording whether it was given orally or by infusion. If the two routes can affect the outcome differently, then “medication” does not specify one treatment condition. For a given patient, \(Y_i(1,\text{oral})\) and \(Y_i(1,\text{infusion})\) may differ. Writing a single \(Y_i(1)\) conceals that distinction.

A sufficient condition for the broad label to have one potential outcome for each unit is that every included version gives that unit the same outcome:

This condition is strong. It is not necessary to assume that every version is literally identical in all respects; what matters for the causal question is whether the differences affect the relevant potential outcomes. If effects could differ, a sound alternative is to define the treatment more narrowly or to specify a policy that determines how versions are assigned.

A version-specific causal effect compares a particular version with a clearly specified comparator. For example, the effect of oral treatment \(v\), relative to no medication, is the population mean contrast \(E[Y_i(1,v)-Y_i(0)]\). The comparator must also be clear: “no medication” might itself cover different usual-care regimens, for instance. A vague comparator can create the same interpretive problem.

Defining Effects for a Policy That Includes Versions

Sometimes the aim is not to estimate the effect of one version. A health system might instead want to evaluate a policy that assigns patients among several versions. In that case, define the policy explicitly. Let \(q_v\) be the probability that the policy assigns version \(v\), with \(q_v \geq 0\) and \(\sum_v q_v=1\). If the version is assigned independently of a unit’s potential outcomes according to these probabilities, the policy’s mean treatment outcome is the weighted mean of the version-specific population means:

The policy effect relative to the specified no-treatment condition is \(E[Y^{q}(1)]-E[Y(0)]\). This is an effect of the policy’s version mix, not an effect of an unspecified treatment label. A different set of assignment probabilities defines a different policy and may therefore define a different effect.

The formula follows from averaging over the version the policy assigns. Conditional on assignment of version \(v\), the mean potential outcome is \(E[Y(1,v)]\); the probability of that assignment is \(q_v\). The law of total expectation therefore gives:

The policy must specify enough to make these probabilities meaningful. If assignments vary across locations or patient groups, the policy may need to state those rules too. The simple weighted formula above applies when the probabilities \(q_v\) describe the policy for the population being considered.

Making the Intervention Operational

An intervention description should let a reader distinguish who receives what from who receives the comparator. For a medication, this may require the drug, dose, route, schedule, and treatment period. For a behavioral intervention, it might specify the program content, frequency, and delivery method. Not every detail is always needed, but details that could change the outcome or the intended target population should not be hidden behind a broad label.

A useful check is to ask whether two people could both be classified as treated while receiving materially different interventions. If so, ask whether the versions are relevant to the causal question. If they are, define version-specific effects, record and analyze the versions, or specify a policy over versions. Merely adding detail to the observed data does not automatically resolve the causal question; the target intervention must also be stated.

Assumptions, Diagnostics, and Alternatives

Defining versions is a conceptual step, not a guarantee that a causal effect can be identified from observed data. The potential outcomes must be meaningful for the intervention being discussed, and observed treatment information must distinguish the relevant conditions well enough to connect each unit’s observed outcome to the corresponding potential outcome. Any identification strategy also requires its own assumptions. Those questions are separate from whether the treatment itself is well-defined.

For a policy over versions, the formula above defines the target contrast. Estimating it from data requires appropriate information about the versions and a design or assumptions that support the required comparisons. In particular, a policy’s version mix should not be confused with the version mix that happened to occur in a study. If the observed mix differs from the target policy, an unadjusted average among treated units need not estimate the policy mean.

Diagnostics begin with the intervention description and the recorded data. Check whether:

• the treatment label combines distinct doses, routes, schedules, providers, or delivery settings;
• the study records which versions each unit actually received;
• the comparator condition is defined with comparable care;
• the target effect names one version or states a policy for allocating versions; and
• the study population includes the versions that the stated target intervention requires.

A missing version field is a warning: if versions may affect outcomes, data grouped only by a broad label may not answer a version-specific question. This does not prove that a particular estimate is biased, but it does mean the target contrast and its interpretation need scrutiny.

The main alternatives are to narrow the treatment definition, retain a broad treatment label only when versions are outcome-equivalent for the question, or define a policy over versions. Narrowing is often easiest to interpret but may target a more specific intervention. Treating versions as equivalent requires substantive justification. A policy effect can match a practical decision but depends on specifying the policy’s version mix. These options answer different causal questions; they are not interchangeable repairs for the same estimand.

Common Mistake

Wrong reasoning: “Everyone in the treated group received the medication, so \(Y(1)\) is the outcome under medication.” This assumes that all versions covered by “medication” define one relevant intervention. If doses or routes can yield different potential outcomes, the assumption has not been justified.

Correction: State the version-specific potential outcomes, such as \(Y(1,v)\), and define the contrast for the version of interest. If the question concerns a mix of versions, specify that mix as a policy and interpret the result as a policy effect. This keeps the causal claim aligned with the intervention actually being studied.