Why a Pooled Contrast Can Move
A treatment contrast calculated by pooling participants summarizes not only how treatment performs within subgroups, but also which subgroups contribute to each treatment arm. In an adaptive trial, eligibility rules, enrichment, or allocation can change the composition of enrolled participants. A pooled contrast may then differ from a contrast defined for a fixed target population, even when every subgroup-specific treatment effect is unchanged.
The diagnostic task is to separate those explanations. A change in the subgroup effects is different from a change in the weights assigned to those effects. And if the subgroup proportions differ between treatment arms, differences in subgroup prognosis can affect the crude pooled contrast even when treatment has no effect in any subgroup.
The weights \(q_g\) define the population whose average effect is being targeted. They might represent a prespecified eligible population or another explicitly justified target distribution. They are not automatically the observed proportions in an adaptive trial. Let \(w_{ag}\) denote the proportion of the pooled participants in arm \(a\) who belong to subgroup \(g\), so \(\sum_g w_{ag}=1\) for each arm. The crude pooled contrast is \(\Delta_{\mathrm{pool}}=\sum_g w_{1g}\mu_{1g}-\sum_g w_{0g}\mu_{0g}\).
These expressions concern means on an additive scale. For outcomes where larger values are worse, a negative contrast may indicate benefit; throughout the numerical examples, larger values are taken to be better. We use population means to explain the decomposition. In an analysis, sample subgroup means and observed subgroup proportions give the corresponding estimated quantities.
An Exact Decomposition
Because \(\mu_{1g}=\mu_{0g}+\tau_g\), the treatment-arm mean can be written as a control mean plus a subgroup effect. Substituting this into the pooled contrast gives an exact identity:
The first term, \(\tau_q\), is the effect for the fixed target population. The second term records how the treatment-arm subgroup mix weights the treatment effects differently from the target weights. The third term records the effect of having different subgroup mixes in the two arms, evaluated using control means as the reference. That last term can be nonzero even if every \(\tau_g=0\), whenever subgroups have different control means and their arm proportions differ.
This is an exact accounting identity, not a claim that one term is always a causal bias. Its interpretation depends on the design and on how the weights arose. For example, unequal arm proportions can result from random variation, or from recruitment and adaptation decisions. The decomposition identifies which differences in means and weights contribute to the pooled number; it does not, by itself, establish why those differences occurred.
The identity can also be justified by direct subtraction: adding \(\tau_q\) and the effect-weighting term gives \(\sum_g w_{1g}\tau_g\), since the \(q_g\tau_g\) terms cancel. Adding the arm-composition term then gives \(\sum_g w_{1g}(\tau_g+\mu_{0g})-\sum_gw_{0g}\mu_{0g}\), which is precisely \(\Delta_{\mathrm{pool}}\). No approximation is involved.
The choice of control means as the reference is convenient because it gives the decomposition above, but the allocation of a total difference across components can depend on the reference. For example, rewriting with treatment means as the reference yields a different composition component. The pooled contrast and the fixed target contrast do not depend on that choice; when explaining a particular decomposition, state the reference used.
Worked Diagnoses
Worked Example: A Changed Common Mix With Heterogeneous Effects
Suppose there are two baseline subgroups, A and B. The target gives each weight \(q_A=q_B=0.50\). Control means are 20 points in A and 10 points in B. Treatment effects are 8 points in A and 2 points in B. In a later adaptive cohort, both arms have weights \(r_A=0.80\) and \(r_B=0.20\). Assume the outcome and assessment time are unchanged, and that the subgroup means validly represent the same treatment and control conditions in both cohorts.
The fixed target effect is \(\tau_q=(0.50)(8)+(0.50)(2)=5.00\) points. The treatment means are 28 points in A and 12 points in B. Applying the common observed weights, the pooled treatment mean is \((0.80)(28)+(0.20)(12)=24.8\) points. The pooled control mean is \((0.80)(20)+(0.20)(10)=18.0\) points. Therefore, \(\Delta_{\mathrm{pool}}=24.8-18.0=6.80\) points.
Here the arm-composition term is zero because both arms have the same mix. The effect-weighting term is \((0.80-0.50)(8)+(0.20-0.50)(2)=2.40-0.600=1.80\) points. Thus, the pooled contrast is \(5.00+1.80=6.80\) points. The shift from 5.00 to 6.80 does not mean either subgroup effect changed; the pooled number gives more weight to subgroup A, which has the larger effect.
Worked Example: Different Subgroup Mixes Across Arms
Suppose the target weights are \(q_A=q_B=0.50\). Control means are 50 points in A and 30 points in B; subgroup treatment effects are 10 points in A and 4 points in B. The treatment arm has weights \(w_{1A}=0.70\), \(w_{1B}=0.30\), while control has \(w_{0A}=0.40\), \(w_{0B}=0.60\). Assume consistent outcome measurement and valid within-subgroup arm means.
The target effect is \((0.50)(10)+(0.50)(4)=7.00\) points. Treatment means are 60 points in A and 34 points in B. The pooled treatment mean is \((0.70)(60)+(0.30)(34)=52.2\) points; the pooled control mean is \((0.40)(50)+(0.60)(30)=38.0\) points. Hence \(\Delta_{\mathrm{pool}}=52.2-38.0=14.2\) points.
The effect-weighting term is \((0.70-0.50)(10)+(0.30-0.50)(4)=2.00-0.800=1.20\) points. The arm-composition term, using control means, is \((0.70-0.40)(50)+(0.30-0.60)(30)=15.0-9.00=6.00\) points. The decomposition gives \(7.00+1.20+6.00=14.2\) points, matching the directly calculated contrast. In particular, the 6.00-point term reflects the treatment arm having more participants in the higher-control-mean subgroup. It is not additional within-subgroup treatment benefit.
Worked Example: An Apparent Effect When Subgroup Effects Are Zero
Suppose treatment and control means are both 100 points in subgroup A and both 60 points in subgroup B. Thus \(\tau_A=\tau_B=0\), and the fixed target effect is zero for any target weights. Let the treatment arm have 75% subgroup A and 25% subgroup B, while control has 25% subgroup A and 75% subgroup B. Assume these are the relevant arm-specific weights and the subgroup means are measured comparably.
The pooled treatment mean is \((0.75)(100)+(0.25)(60)=90\) points. The pooled control mean is \((0.25)(100)+(0.75)(60)=70\) points. The pooled contrast is \(90-70=20.0\) points, despite zero treatment effect within either subgroup.
For any fixed target weights, the effect-weighting term is zero because both subgroup effects are zero. The composition term is \((0.75-0.25)(100)+(0.25-0.75)(60)=50.0-30.0=20.0\) points. The pooled difference is entirely due to the different subgroup composition. It should not be interpreted as evidence of a 20-point within-subgroup treatment effect.
Assumptions, Diagnostics, and Alternatives
Before interpreting a decomposition, check that its ingredients are meaningful. Subgroups should be defined using baseline characteristics, be mutually exclusive and exhaustive for the population under consideration, and have the same definitions across arms and cohorts. The outcome, time point, and treatment conditions represented by \(\mu_{ag}\) must also be compatible. If the target weights describe a population, specify that population and ensure the subgroup effects being combined apply to it.
A causal interpretation of the within-subgroup effects requires an appropriate basis for comparing treatment with control. Random assignment within the relevant subgroups can support that comparison, subject to the trial’s conduct and outcome ascertainment. If treatment assignment or inclusion in the analyzed data is confounded by prognostic characteristics within a subgroup, a difference in observed subgroup means need not be a causal treatment effect. Likewise, this algebra does not repair missing outcomes or make incomparable treatment conditions equivalent.
A practical diagnosis starts by displaying, for each period or cohort and arm, the subgroup counts or proportions alongside the subgroup-specific outcome means. Then calculate the target-weighted effect and compare it with the crude pooled contrast. This reveals whether the discrepancy is mainly consistent with heterogeneous effects receiving different weights, unequal arm mixes interacting with prognosis, or both. A numerical decomposition is a description of the contrast; uncertainty in the estimates still matters when making inferential claims.
When the objective is the fixed target effect, a direct alternative to crude pooling is standardization: estimate each \(\tau_g\), then compute \(\sum_g q_g\tau_g\) using the prespecified target weights. Stratified reporting makes the subgroup contributions visible. A regression model with subgroup indicators and treatment-by-subgroup interactions can also estimate subgroup effects, but the target weights still need to be applied to obtain the desired marginal contrast. These approaches answer different questions from a crude contrast if the crude arm weights differ from \(q_g\).
| Quantity | What it weights | Diagnostic use |
|---|---|---|
| Fixed target effect \(\tau_q\) | Subgroup effects by target weights \(q_g\) | Answers the prespecified target-population question |
| Common-mix pooled contrast | Subgroup effects by a shared mix \(r_g\) | Shows how a changed shared mix matters when effects differ |
| Crude arm-specific pooled contrast | Each arm by its own weights \(w_{ag}\) | Can include a prognosis-related composition term |
Common Pitfall and Limits
A common mistake is to see a larger pooled contrast after enrichment and conclude that treatment became more effective. That reasoning ignores the possibility that the enriched subgroup has a larger treatment effect, so its increased weight changes the marginal effect while subgroup effects remain constant. A different mistake is to assume that a shared mix across arms guarantees the target contrast: it removes the arm-composition term, but does not make that mix equal to the target or eliminate effect-weighting differences.
The decomposition is limited by the subgroup definitions and the outcome scale. A weighted average of subgroup effects is a marginal effect on the specified additive mean-difference scale; the corresponding decomposition may not transfer unchanged to a ratio scale. Sparse or empty subgroup-by-arm cells also make some means and effects unstable or unavailable. In those circumstances, the arithmetic identity remains a useful conceptual guide, but the data may not support reliable estimates of all its components.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Two subgroups have effects of 6 and 2 points. The target weights are 0.50 and 0.50, but both arms have a shared mix of 0.75 and 0.25. What is the target effect, the pooled effect, and their difference?
- If all subgroup treatment effects are zero, can a crude pooled contrast be nonzero? State the condition in the decomposition that permits this.
- In the exact identity, what does the term \(\sum_g(w_{1g}-q_g)\tau_g\) diagnose?
- Why does having identical subgroup weights in both arms not, by itself, guarantee that the pooled contrast equals the fixed target effect?
Solutions. (1) The target effect is \((0.50)(6)+(0.50)(2)=4.00\) points. The shared-mix pooled effect is \((0.75)(6)+(0.25)(2)=5.00\) points, which is 1.00 point above target. (2) Yes. The arm-composition term \(\sum_g(w_{1g}-w_{0g})\mu_{0g}\) can be nonzero when subgroup weights differ across arms and control means differ across subgroups. (3) It measures the difference between weighting subgroup effects by the treatment-arm mix and weighting them by the fixed target mix. (4) Identical arm weights eliminate the arm-composition term, but if the shared mix differs from the target and effects vary by subgroup, the effect-weighting term can remain nonzero.