Deriving an Adaptation-Invariant Treatment Effect

Why Derive the Target Before Looking at the Adaptive Sample?

An adaptive trial may change its enrollment criteria, recruit more participants with a particular baseline profile, or alter the probability of assignment to each treatment. As a result, the mix of participants observed in later stages can differ from the mix observed earlier. A marginal treatment contrast computed in each stage may then change even when the treatment effects within baseline profiles have not changed. The central question here is how to define a treatment-effect target that does not move merely because the trial’s enrollment or randomization mix changes.

The solution is to define the target using potential outcomes and a baseline covariate distribution fixed in advance. The target is an average of conditional treatment effects, with weights determined by that reference distribution—not by the participants who happen to enroll in a particular stage or treatment arm. This definition does not by itself guarantee that trial data identify the target. Identification also requires appropriate randomization, support, and, for an external reference population, a transportability assumption.

Definition: Let $X$ denote baseline covariates, $Y(1)$ the outcome that would be observed under the specified treatment, and $Y(0)$ the outcome that would be observed under the specified control condition. For a fixed target population with baseline covariate distribution $F^*$, define $m_a^*(x)=E[Y(a)\mid X=x,\text{target}]$, for $a\in\{0,1\}$. The adaptation-invariant marginal treatment effect is $$ \Delta^*=\int \{m_1^*(x)-m_0^*(x)\}\,dF^*(x). $$ For discrete covariates with target-profile weights $w_j^*$, this is $\Delta^*=\sum_j w_j^*\{m_1^*(x_j)-m_0^*(x_j)\}$, where $w_j^*\geq 0$ and $\sum_j w_j^*=1$.

The target distribution $F^*$ must describe a population and be specified independently of the changing stage-specific enrollment mix. It could represent a prespecified eligible population or another clearly defined reference population. The treatment and control conditions, outcome, follow-up period, and handling of events such as rescue treatment must also be specified: changing any of these can change the potential outcomes themselves, rather than merely changing the weights used to average them.

Deriving Invariance to Enrollment and Allocation

Suppose stage $s$ enrolls participants with baseline distribution $F_s$. If the conditional mean treatment effect is the same across stages, write it as $\tau(x)=m_1(x)-m_0(x)$. The stage-specific marginal effect is then an average over the stage’s own covariate distribution, while the fixed target effect averages the same conditional contrast over $F^*$:

\Delta_s=\int \tau(x)\,dF_s(x), \qquad \Delta^*=\int \tau(x)\,dF^*(x).

These quantities need not be equal. But the fixed target expression itself contains no stage-specific enrollment distribution and no treatment-allocation probability. Thus, if the conditional contrast and the specified target distribution remain fixed, changes to enrollment mix or randomization probabilities do not alter $\Delta^*$. This is a statement about the definition of the estimand; it is not a claim that every estimator computed from every adaptive sample will estimate it without bias.

The derivation is an application of the law of total expectation. Let $Y^*$ denote the potential outcome in the fixed target population under a specified treatment condition. Averaging first within baseline profiles and then across the target population gives $E[Y^*(a)\mid\text{target}]=\int E[Y(a)\mid X=x,\text{target}]\,dF^*(x)$. Subtracting the control mean from the treatment mean gives:

\begin{aligned} E[Y^*(1)-Y^*(0)\mid\text{target}] &=E[Y^*(1)\mid\text{target}]-E[Y^*(0)\mid\text{target}]\\ &=\int m_1^*(x)\,dF^*(x)-\int m_0^*(x)\,dF^*(x)\\ &=\int\{m_1^*(x)-m_0^*(x)\}\,dF^*(x) =\Delta^*. \end{aligned}

This identity explains the construction: the fixed target effect is the marginal mean difference for the target population, expressed as a weighted average of conditional effects. It does not say the conditional effects must be constant. If effects differ across profiles, the target weights determine how those differences contribute to the marginal effect.

Assumptions for Interpreting Trial Data as the Fixed-Target Effect

Potential outcomes define the causal contrast, but observed data reveal only the outcome under the treatment actually assigned. The following assumptions clarify when trial observations can inform $\Delta^*$.

Well-defined treatment conditions and consistency. Each treatment condition is sufficiently specified, and a participant’s observed outcome equals the potential outcome for the condition received. If dose, follow-up, outcome definition, or intercurrent-event strategy changes, the treatment condition or outcome may have changed; invariance cannot be presumed.
Randomization within the trial. Among enrolled participants, assignment is independent of potential outcomes conditional on the baseline covariates and any randomization history that governs assignment. In a sequentially randomized design, this condition applies at the time each participant is randomized. Randomization supports treatment comparisons for the enrolled population; it does not alone make that population representative of an external target.
Treatment positivity. For target profiles represented by trial participants, each treatment condition needed for the contrast has a positive probability of assignment under the design. If an arm is never assigned for a profile, its conditional mean cannot be learned from randomized outcomes in that profile without additional assumptions.
Target support. The trial must contain relevant information for the baseline profiles receiving positive weight under $F^*$. If the target includes profiles absent from the trial, observed trial data do not identify their conditional treatment effects without extrapolation or other external information.
Conditional transportability when the target is external. Given the covariates used for standardization, the potential-outcome conditional means among trial participants must equal those in the reference population: $E[Y(a)\mid X=x,\text{trial}]=E[Y(a)\mid X=x,\text{target}]$, for each relevant $a$ and supported $x$. Randomization does not establish this assumption. It requires substantive justification that the measured covariates adequately account for differences relevant to conditional outcomes.
Stability across adaptations. The conditional potential-outcome means being used must retain the same meaning across stages. For example, an adaptation must not silently change the treatment version or follow-up definition. If effects genuinely change over time or with context, the target must specify how those versions or contexts are to be combined.

Under randomization, consistency, and treatment positivity, trial data can identify conditional mean outcomes for the trial population at supported profiles. To interpret those means as $m_a^*(x)$ for an external target, conditional transportability is an additional requirement. Alternatively, one can define $m_a(x)$ explicitly as the trial response function and restrict the target interpretation to a specified population for which that response function is relevant. In either case, the target distribution and the conditional response functions must be distinguished.

Worked Example: Standardizing Two Profile-Specific Effects

Suppose a hypothetical trial defines two baseline profiles: biomarker-positive and biomarker-negative. Its prespecified target population has 60% positive and 40% negative participants. Assume, for this illustration, that the conditional mean outcomes transport to that target population and that the treatment and outcome definitions are stable. The conditional mean outcomes, in points, are:

Baseline profile	Control mean	Treatment mean	Conditional treatment effect
Positive	50	58	8 points
Negative	40	42	2 points

The target weights are $w_+^*=0.60$ and $w_-^*=0.40$. Therefore the fixed-target treatment effect is:

\Delta^*=0.60(58-50)+0.40(42-40) =0.60(8)+0.40(2) =4.8+0.8 =5.6\text{ points}.

Equivalently, the standardized treatment mean is $0.60(58)+0.40(42)=51.6$ points and the standardized control mean is $0.60(50)+0.40(40)=46.0$ points; their difference is $51.6-46.0=5.6$ points. This is the mean treatment-minus-control contrast for the specified target distribution, under the stated transportability and stability assumptions.

What Changes When the Trial’s Composition Changes?

The following example holds the conditional effects fixed but changes the baseline mix. It illustrates why a raw marginal contrast can differ across stages even though the fixed-target estimand does not.

Worked Example: Two Enrollment Mixes, One Fixed Target

Consider the same hypothetical profile-specific means: positive participants have control and treatment means of 50 and 58 points, while negative participants have means of 40 and 42 points. Keep the target weights fixed at 60% positive and 40% negative. Suppose stage A has a composition of 80% positive and 20% negative, while stage B has a composition of 20% positive and 80% negative.

For a population with the stage A composition, the marginal treatment mean is $0.80(58)+0.20(42)=54.8$ points and the marginal control mean is $0.80(50)+0.20(40)=48.0$ points. The composition-specific contrast is $54.8-48.0=6.8$ points.

For stage B, the marginal treatment mean is $0.20(58)+0.80(42)=45.2$ points and the marginal control mean is $0.20(50)+0.80(40)=42.0$ points. The composition-specific contrast is $45.2-42.0=3.2$ points.

Neither stage-specific contrast is the prespecified fixed-target contrast of 5.6 points. Reweighting the conditional effects to the target gives $0.60(8)+0.40(2)=5.6$ points in either stage, provided both profiles are supported and the conditional effects remain relevant. These calculations use population compositions and conditional means; they do not treat random realized treatment-arm counts as fixed or claim that a ratio formed from expected counts is the expected value of a random arm mean.

Allocation probabilities can change as well. If randomization probabilities vary by profile, an unadjusted treatment-arm mean and an unadjusted control-arm mean can reflect different profile mixes. Randomization within profiles can still identify profile-specific treatment contrasts, but the arm-level marginal comparison need not use the target weights. Changing allocation probabilities affects the information available in profiles and may affect precision; it does not enter the definition of $\Delta^*$.

Worked Example: A Hypothetical External Reference Population

Suppose an adaptive trial observes two baseline profiles, A and B. Its enrolled sample is 30% profile A and 70% profile B, while an external reference population has prespecified weights of 70% and 30%. Assume the conditional treatment effects are 6 points in profile A and 1 point in profile B. Also assume the trial and reference population have the same conditional potential-outcome means given these profiles, both profiles are represented in the trial, and both treatments have positive assignment probability within each profile.

The trial-composition average is $0.30(6)+0.70(1)=1.8+0.7=2.5$ points. The fixed external-target effect is instead:

\Delta^*=0.70(6)+0.30(1)=4.2+0.3=4.5\text{ points}.

The 4.5-point value is the treatment effect averaged over the specified reference distribution, not a claim that the trial’s participants had that composition. If conditional transportability were not justified, the trial’s randomized comparisons would identify trial-conditional means, but the external-target interpretation of 4.5 points would not follow from the calculation alone.

Diagnostics, Alternatives, and Limits

A useful audit of an adaptation-invariant target asks whether the ingredients of its definition have remained fixed. Record the target population and its weights, the baseline covariates used to define profiles, the treatment versions, the outcome and assessment time, and the conditional response functions to be transported or estimated. Then check whether each positively weighted target profile is represented and whether assignment to both conditions is possible there. This audit helps identify violations; it cannot prove transportability or stability from the randomized data alone.

Three common summaries answer different questions:

Approach	What is averaged?	Main limitation for this target
Unadjusted marginal comparison	Outcomes in the observed treatment and control groups	Those groups may have different profile mixes, especially when allocation probabilities vary by profile.
Stage-specific marginal effect	Conditional effects averaged using that stage’s enrollment distribution	The estimand changes when the stage distribution changes.
Fixed-target standardization	Conditional effects averaged using the prespecified target distribution	Requires adequate profile support and valid conditional effects for the target population.

A fixed target is not automatically more scientifically appropriate than a stage-specific target. The choice depends on the question. If the question concerns the participants actually enrolled in a particular stage, that stage’s distribution may be relevant. If the aim is to compare treatment effects across adaptations on a common population basis, a fixed target makes the comparison’s weighting rule explicit.

A frequent mistake is to say, “Randomization was used, so the trial identifies the treatment effect in the external target population.” Randomization addresses treatment assignment among trial participants; it does not remove differences between trial participants and an external reference population. The correction is to state the conditional transportability assumption, justify it using the design and substantive knowledge, and verify support. Another mistake is to interpret a changing marginal contrast as proof that the conditional treatment effect changed. A change may instead arise because the mix of baseline profiles changed; the fixed-target calculation separates that weighting change from the conditional contrasts.

Finally, invariance has limits. If adaptation changes the treatment dose or version, the outcome definition, follow-up duration, or the conditional response to treatment, then the potential outcomes or conditional effects may differ across stages. Reusing the same weights cannot make different treatment questions identical. The target remains adaptation-invariant only with respect to changes in enrollment and allocation that leave the specified target distribution and relevant conditional potential-outcome contrasts unchanged.

Key takeaway: Define the target effect as a weighted average of conditional potential-outcome contrasts over a fixed baseline distribution. Changes in who enrolls or how assignment is allocated do not change that definition; support, randomization, stable treatment conditions, and any required transportability assumptions determine whether observed trial data can identify it.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

A fixed target places 25% weight on profile A and 75% on profile B. The conditional treatment effects are 10 and 2 points. What is the fixed-target effect?
Why does randomization within a trial not, by itself, identify conditional potential-outcome means in an external reference population?
A trial changes allocation probabilities but retains the same treatment conditions, outcome, and target weights. Which part of the estimand’s definition changes?
A target assigns positive weight to a profile that is never enrolled in the trial. What assumption problem arises?
What must be reconsidered if an adaptation changes the treatment dose?

Solutions. (1) $\Delta^*=0.25(10)+0.75(2)=2.5+1.5=4.0$ points. (2) Randomization supports comparisons among trial participants; transporting conditional means to an external population additionally requires conditional transportability given the covariates and adequate support. (3) None of the target weights or conditional contrasts in the estimand’s definition changes solely because allocation probabilities change, although estimation and precision may be affected. (4) There is a support failure: the trial supplies no randomized outcomes for that profile, so its conditional effect is not identified from these trial data without additional information or assumptions. (5) The treatment condition and its potential outcomes may have changed, so the conditional effects may no longer be the same; the target’s treatment contrast must be specified and its invariance reassessed.