Allocation Changes and Marginal Treatment Effects

Allocation Is Not the Target Population

An adaptive trial can change the probability that an arriving participant is assigned to each treatment arm. That changes the distribution of observed assignments, and it may change the mix of participants observed in each arm. It does not, by itself, change a marginal treatment effect defined for a fixed population. Keeping these ideas separate is essential: an allocation ratio is a feature of the randomization design, while a marginal effect is a contrast between treatment outcomes averaged over a specified population.

The distinction matters because a pooled difference between observed arm means need not equal the effect for the population of interest. If allocation probabilities vary with baseline characteristics or trial stage, the two arms can contain different mixes of participants. Their pooled means then average outcomes using different weights. A fixed-population effect instead uses the same stated population weights for both treatment conditions.

Definition: Let $Y(1)$ and $Y(0)$ denote the outcomes a participant would have under treatment conditions 1 and 0, respectively, with the outcome and intercurrent-event strategy specified in the estimand. Let $X$ denote baseline characteristics, and let $F^*$ be the fixed target-population distribution of $X$. Define $m_a(x)=E[Y(a)\mid X=x]$, for $a\in\{0,1\}$. The fixed-population marginal treatment effect is $$ \Delta^*=\int \{m_1(x)-m_0(x)\}\,dF^*(x). $$ For a discrete set of strata $k$, this is $\Delta^*=\sum_k w_k\{m_1(k)-m_0(k)\}$, where $w_k$ are the target-population proportions and $\sum_k w_k=1$.

The allocation probability does not appear in this definition. In contrast, it does determine how often each treatment is assigned, and therefore affects the information available in each arm and the distribution of observed assignments. The target weights $w_k$ describe whose outcomes are being averaged; they are not the proportion assigned to treatment 1 or treatment 0.

Conditions for an Allocation Change to Leave the Effect Fixed

The claim that an allocation change does not alter $\Delta^*$ is conditional, not automatic. The target population $F^*$, the two treatment conditions, the outcome definition, and the handling strategy for intercurrent events must remain the same. The treatment conditions must be sufficiently consistent across participants and stages for $Y(1)$ and $Y(0)$ to retain the same meaning. If an allocation change is accompanied by a changed treatment version, different eligibility, or a changed target distribution, the original estimand may no longer describe the trial's target.

The usual potential-outcomes interpretation also assumes consistency (the observed outcome equals the potential outcome for the treatment actually received) and no interference (one participant's assignment does not change another participant's outcome). For estimating the effect from randomized data, assignment must be randomized according to the design, conditional on the information used to adapt allocation. The assignment probability for each treatment must be positive in the target groups being estimated. These conditions allow treatment comparisons to be made within relevant strata; they do not guarantee that a raw pooled contrast uses the desired target weights.

A short algebraic argument shows why a change in allocation probability is not itself a change in the effect. For any fixed $F^*$, linearity of integration gives:

\Delta^* =\int m_1(x)\,dF^*(x)-\int m_0(x)\,dF^*(x).

The allocation probability is absent from both terms. Altering that probability can change the random variables recording who receives which treatment, but it does not change either potential-outcome mean under the stated fixed treatment conditions and population. An estimate can still change, because a different allocation may produce different observed data and different precision.

Worked Examples

Worked Example: Standardizing to a Fixed Population

Suppose a hypothetical trial targets a population with equal proportions in two baseline-risk strata: lower-risk and higher-risk. The hypothetical mean outcomes, measured in points, are 12 under treatment and 10 under control in the lower-risk stratum, and 28 under treatment and 20 under control in the higher-risk stratum. Calculate the marginal effect for the fixed target population.

The stratum-specific contrasts are $12-10=2$ points and $28-20=8$ points. The target weights are both 0.50, so the target-population contrast is:

\Delta^*=0.50(2)+0.50(8)=1+4=5\text{ points}.

Equivalently, the standardized treatment mean is $0.50(12)+0.50(28)=20$ points, and the standardized control mean is $0.50(10)+0.50(20)=15$ points; their difference is $20-15=5$ points. This result uses the target population's equal stratum weights, not the proportions assigned to either arm.

Worked Example: Changing Allocation Across Trial Stages

Consider a hypothetical trial in which the mean outcome improves over calendar time for reasons unrelated to treatment. In stage 1, the treatment mean is 14 points among 20 participants and the control mean is 10 points among 20. In stage 2, after allocation changes, the treatment mean is 24 points among 20 participants and the control mean is 20 points among 10. The treatment contrast is 4 points in each stage. Calculate the pooled contrast and compare it with the effect for the enrolled population.

The pooled treatment mean is:

\bar{Y}_1=\frac{20(14)+20(24)}{20+20} =\frac{280+480}{40}=19\text{ points}.

The pooled control mean is:

\bar{Y}_0=\frac{20(10)+10(20)}{20+10} =\frac{200+200}{30}=13.333\text{ points}.

Thus the pooled contrast is $19-13.333=5.667$ points, even though the contrast is 4 points in each stage. The enrolled population consists of 40 participants in stage 1 and 30 in stage 2, so its stage weights are $40/70=4/7$ and $30/70=3/7$. Standardizing both treatment conditions to those same weights gives:

\mu_1^*=\frac{4}{7}(14)+\frac{3}{7}(24)=\frac{128}{7}, \qquad \mu_0^*=\frac{4}{7}(10)+\frac{3}{7}(20)=\frac{100}{7}, \qquad \Delta^*=\frac{28}{7}=4\text{ points}.

The discrepancy arises because the pooled treatment and control means use different stage weights: 20/40 and 20/40 for treatment, but 20/30 and 10/30 for control. The allocation change and the stage-related outcome shift combine to give the arms different compositions. The standardized contrast uses one common, explicitly defined population distribution.

Worked Example: Allocation Probabilities and an Inverse-Probability Mean

For a small hypothetical fixed population of four participants, suppose the potential treatment outcomes are 8, 10, 12, and 14 points. Each participant is independently assigned to treatment with probability $q=0.25$. For this example, consider the inverse-probability estimate of the population treatment mean, $\hat{\mu}_1=(1/4)\sum_i I(A_i=1)Y_i/q$, where $I(A_i=1)$ equals 1 if participant $i$ is assigned treatment and 0 otherwise. Show that its expectation is the fixed population mean.

The fixed population mean under treatment is $(8+10+12+14)/4=11$ points. For each participant, the assignment indicator has expectation $E[I(A_i=1)]=q=0.25$. Therefore:

E[\hat{\mu}_1] =\frac{1}{4}\sum_{i=1}^4\frac{E[I(A_i=1)]Y_i(1)}{q} =\frac{1}{4}(8+10+12+14) =11\text{ points}.

If the treatment probability were changed to another positive value, the same cancellation would hold: $E[I(A_i=1)]/q=q/q=1$. The estimate's expected value targets the same fixed population mean, although its realized value and precision can differ with the allocation. This calculation relies on the assignment probability being correctly specified and positive; it is not a claim that a particular trial estimate must equal 11.

Why the Weighting Argument Works—and When It Fails

More generally, let $H_i$ be the information available immediately before participant $i$'s assignment, and let $\pi_i=P(A_i=1\mid H_i,X_i)$ be the known randomization probability for treatment. Under randomized assignment conditional on that history, and with $\pi_i>0$, the assignment-weighted contribution has conditional expectation:

E\left[\frac{I(A_i=1)Y_i(1)}{\pi_i}\,\middle|\,H_i,X_i,Y_i(1)\right] =\frac{\pi_iY_i(1)}{\pi_i}=Y_i(1).

For control, the corresponding expression replaces $I(A_i=1)$ and $\pi_i$ with $I(A_i=0)$ and $1-\pi_i$, requiring $1-\pi_i>0$. Averaging these contributions targets outcomes under the assigned treatment conditions rather than the changing frequency of assignment. To target a population distribution different from the enrolled sample, additional standardization or appropriate population weights are needed. In practice, allocation-weighted and standardized estimates also require suitable variance and inference methods; the expectation calculation alone does not provide them.

The positivity condition is substantive. If a treatment has zero assignment probability in a group, the trial has no randomized observations for that treatment in that group, so the contrast there cannot be recovered from the trial by weighting alone. Very small probabilities can also make weights large and estimates unstable. The method can preserve the estimand's definition without guaranteeing a precise or reliable estimate.

Diagnostics, Alternatives, and Limitations

When allocation changes, first check whether the target has actually remained fixed. Compare the planned target weights with the enrolled population and confirm that treatment conditions, outcome definition, and intercurrent-event strategy are unchanged. Then examine arm composition by the variables that influenced allocation and by stage or calendar time. Large differences between the arm-specific distributions are a warning that a pooled arm-mean difference may not represent the stated target effect.

The main alternatives answer different practical needs. A pooled contrast is simple and can be reasonable when both arms represent the target population under the design, but it can use incompatible weights when allocation varies across groups or time. Direct standardization estimates treatment-specific means within groups and averages each using the same fixed target weights. Inverse-probability weighting reweights observed assignments according to their randomization probabilities; it can accommodate known changing probabilities but can be unstable when probabilities are small. These approaches do not change the estimand merely by being different estimators, but they have different assumptions and precision.

Approach	What it averages	Main caution
Pooled arm means	Each arm's observed participants, using that arm's own composition	Different arm compositions can make the contrast differ from the fixed-target effect
Direct standardization	Stratum-specific means with common, fixed target weights	Requires outcome information in each target stratum under both conditions
Inverse-probability weighting	Observed outcomes reweighted by assignment probabilities	Requires correct probabilities and positivity; extreme weights can reduce precision

Common warning signs include missing treatment observations in a target stratum, changing eligibility or recruitment patterns, a target distribution that was never stated, and pooling stages with different outcome levels but different allocation ratios. A further limitation is that allocation changes can affect recruitment, retention, or which participants are available for randomization. If this changes the population represented by the trial, the target distribution may have changed as well. A fixed estimand is preserved only if the population and treatment conditions it specifies remain fixed, not merely because the randomization ratio changed.

Common Mistake

Pitfall: “The trial assigned more participants to treatment later, so the marginal treatment effect is now the treatment mean minus the control mean in the combined data.” This reasoning confuses assignment frequencies with target-population weights and assumes the pooled arms have the same composition. The correction is to state the target distribution, estimate both treatment means for that same distribution, and then take their difference. A changed allocation may alter the estimate, its precision, or the arm composition; it does not alone redefine a marginal effect for a fixed population.

Check Your Understanding

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

A target population is 25% in stratum A and 75% in stratum B. The treatment effects are 2 and 6 points, respectively. What is the fixed-population marginal effect?
Why can a pooled arm contrast differ from the marginal effect when allocation probabilities vary by baseline stratum?
If a treatment assignment probability changes from 0.50 to 0.25 but remains positive, does the definition of a fixed-population marginal effect change? What may change?
What problem arises if the probability of assigning one treatment is zero in a stratum that has positive weight in the target population?

Solutions. 1. The target-weighted effect is $0.25(2)+0.75(6)=0.5+4.5=5$ points. 2. The pooled treatment and control means can average outcomes using different stratum proportions, whereas the marginal effect uses the same target weights for both conditions. 3. No: the fixed target distribution and potential outcomes define the effect. The observed assignment distribution, realized estimate, and precision may change. 4. The trial has no randomized information for that treatment in that target stratum, so the stratum-specific contrast is not identified by the trial through randomization weighting alone.