Estimands Under Different Treatment Conditions

Two Ways to Define a Treatment Condition

A treatment contrast is meaningful only after its treatment conditions are clear. In a randomized trial, one condition can be defined by which treatment participants are assigned to. Another can be defined by which treatment strategy they actually receive. These conditions need not coincide: participants may stop, switch, reduce a dose, or receive a different treatment. As a result, an assignment-based contrast and a received-treatment contrast can have different targets, even when they are calculated from the same trial.

The distinction is not simply a choice between two calculation methods. It concerns which outcomes are being compared. An assignment-based question asks about outcomes under assignment to each randomized condition, with intercurrent events handled as specified. A treatment-received question asks about outcomes under clearly defined conditions of receiving treatment. In particular, sorting participants by what they actually received does not, by itself, preserve the protection provided by randomization.

For a mean outcome, write \(Y^{R=1}\) and \(Y^{R=0}\) for the potential outcomes under assignment to A and B. What those potential outcomes include depends on how the estimand handles events after assignment. Under a treatment-policy approach, for example, outcomes after discontinuation or rescue treatment remain part of the outcome under the assigned condition. A different specified strategy could define the assignment-based target differently.

For treatment-received conditions, \(Y^{g_1}\) and \(Y^{g_0}\) denote potential outcomes under well-defined strategies \(g_1\) and \(g_0\). A strategy must say enough to distinguish the conditions: a dose or dose range, timing, duration, and relevant rules for switching or stopping may matter. “Received treatment A” can be too imprecise if it combines materially different exposure histories.

Assignment-Based Contrasts

Randomization makes assignment a basis for a comparison: before treatment begins, assignment is independent of participants’ potential outcomes, subject to the trial’s randomization scheme. If outcomes are ascertained for the randomized groups, the difference between their mean outcomes estimates the mean contrast between the assignment conditions. The condition remains assignment, not adherence to the assigned regimen.

For a mean difference under a treatment-policy approach, the target can be written as

Here the expectation is over the target population. If someone assigned to A stops A, the outcome after stopping is still part of the outcome associated with assignment to A under this approach. Randomization does not, on its own, resolve missing outcome data: the outcome comparison still requires complete outcome ascertainment or defensible assumptions and methods for missingness.

Received-Treatment Contrasts and Causal Interpretation

The observed mean difference between participants who received A and those who received B is an as-treated contrast. Receipt occurs after randomization and can depend on prognosis: a participant’s health, response, side effects, or preferences may affect whether treatment is started or continued. Consequently, the received groups can differ in outcome-related characteristics even though the original assignment groups were randomized.

A causal contrast between treatment-received strategies has a different requirement. The strategies must be well defined, and the data and assumptions must support inference about outcomes if members of the target population followed each strategy. In a simple baseline-treatment setting, a common identification argument requires consistency, conditional exchangeability given measured baseline covariates \(L\), and positivity. Consistency means that the observed outcome equals the potential outcome for the strategy actually followed. Conditional exchangeability means that, within levels of \(L\), the received strategy is independent of the potential outcomes being compared. Positivity means that each strategy has a nonzero probability at each covariate pattern in the target population.

Under these assumptions, a mean under strategy \(g\) can be identified by standardizing conditional outcome means over the target population:

The formula requires that \(L\) adequately captures the relevant common causes of receipt and outcome, and that the observed receipt categories correspond to the strategies being compared. For treatment that changes over time, past treatment and evolving prognosis can complicate these requirements; the simple formula should not be applied as if those complications were absent.

A Short Derivation: Assignment Means Mix Receipt Histories

The relation between the two kinds of observed summaries can be seen by partitioning each assignment group according to receipt. Let \(D\) take values \(d\), and let \(\bar{Y}_{r,d}\) be the mean outcome among participants assigned to \(r\) who received \(d\). If all outcomes are observed, the assignment-group mean is a weighted average of these receipt-specific means:

The weights are the proportions with each receipt history within assignment group \(r\), and they sum to 1. This identity follows by writing the total outcome in the assignment group as the sum of outcomes across its receipt subgroups, then dividing by the group size. The earlier switching example gives \(0.72(12)+0.28(6)=10.32\) for assignment to A and \(0.08(12)+0.92(6)=6.48\) for assignment to B.

This identity is a decomposition of observed means, not a causal formula for treatment received. The assignment contrast compares two mixtures of receipt histories generated under the trial’s assignment conditions. A causal receipt contrast instead compares outcomes under specified receipt strategies in the same target population. The mixtures can differ because assignment affects receipt, and observed receipt groups can differ because prognosis affects receipt.

Choosing and Checking the Treatment Condition

The main alternative approaches answer distinct questions; neither is universally preferable. The assignment-based contrast retains the randomized comparison and targets the consequences of assignment under its specified handling of intercurrent events. An as-treated comparison describes observed groups by receipt but can be confounded. A causal received-strategy contrast can directly target adherence to, or receipt of, a specified regimen, but requires stronger identification assumptions and suitable data.

Several checks help reveal when a treatment condition is not doing its intended job:

• Check what “received” means. A group defined by “ever received A” could include one dose, sustained dosing, and switching. Those histories may not represent one coherent strategy.
• Check the timing of prognosis and receipt. If worsening health affects later treatment, a simple comparison of people who did and did not receive treatment can be especially misleading.
• Check overlap. If some covariate groups almost never receive one of the strategies, the data provide little direct information about that strategy for those groups.
• Check that outcome handling matches the question. Omitting outcomes after discontinuation changes what is being summarized; it does not automatically create a contrast under sustained treatment.
• Check the target population. Assignment and receipt contrasts should be described for the population whose outcomes are being summarized, rather than silently changing populations between conditions.

A common mistake is to say, “The trial was randomized, so the participants who actually received A and B are randomized groups.” They are not: randomization assigned \(R\), while \(D\) is determined after assignment and may depend on prognosis. The correction is to preserve the assignment-based comparison when the target concerns randomized assignment, or to define a causal received-treatment strategy and justify the assumptions required to identify its contrast. An observed as-treated difference alone does neither.

These approaches also have limits. An assignment contrast may be diluted when many participants do not follow their assigned regimen, but that does not make it an estimate of the effect of receiving treatment. A received-strategy contrast can be difficult to identify if relevant causes of receipt and outcome are unmeasured, if strategies are not sufficiently specified, or if some strategy is rare for important patient profiles. The estimand should therefore make clear whether “treatment condition” refers to assignment or receipt before a numerical contrast is interpreted.