Why the Average Depends on the Question
When treatment effects differ across patient subgroups, there is no single subgroup effect that automatically describes the whole target population. An overall effect is a weighted average, and the weights must match the population the comparison is intended to describe. Separately, the outcomes observed under a trial’s allocation pattern depend on which patients are assigned to each treatment. These are related quantities, but they answer different questions.
The earlier material on treatment-effect heterogeneity distinguishes subgroup-specific effects from patient benefit. Here the focus is how those subgroup effects combine into a population-average comparison, and why different allocation patterns can produce different observed arm means even when the subgroup effects themselves have not changed.
This average uses the subgroup proportions in the target population, not necessarily the proportions among patients assigned to treatment or control in a particular trial. The target must therefore be stated. For example, an average for all eligible patients can differ from an average for a population enriched for a subgroup with a larger treatment effect.
Assumptions for Interpreting the Average
The potential-outcome definition is meaningful when each patient has a well-defined outcome under each treatment, the observed outcome equals the potential outcome for the treatment actually assigned (consistency), and one patient’s assignment does not change another patient’s outcome (no interference). To estimate subgroup effects from randomized trial data, treatment assignment must also be randomized within the relevant subgroups, or otherwise be appropriately controlled for; each subgroup must have a positive chance of receiving each treatment. The examples below assume these conditions hold.
Under these assumptions, a subgroup treatment contrast estimates \(\tau_k\). Combining the subgroup contrasts using specified target weights estimates \(\tau_{\text{target}}\). In contrast, simply subtracting the observed control mean from the observed treatment mean need not estimate that target average if the arms have different subgroup compositions.
Worked Example: Weighting Subgroup Effects for a Target Population
Suppose a hypothetical target population consists of 70% patients in subgroup 1 and 30% in subgroup 2. The treatment increases the mean outcome by 2 units in subgroup 1 and 10 units in subgroup 2. Assume the units and outcome definition are the same in both subgroups, and the subgroup effects are valid for this target population.
The target average treatment effect uses the population proportions as weights:
The average treatment effect is 4.4 outcome units in this target population. The unweighted average, \((2+10)/2=6\), would describe a population with equal proportions in the two subgroups, not the stated target population. It is not a different estimate of the same weighted average; it uses different weights and therefore defines a different average.
Allocation Patterns and the Mix in Each Arm
Let \(A=1\) indicate assignment to treatment and \(A=0\) assignment to control. For a simple allocation scheme, write \(r_k=P(A=1\mid S=k)\). The fraction of the target population assigned to treatment is \(\bar r=\sum_k\pi_k r_k\). Among those assigned to treatment, subgroup \(k\) has population share \(w_{1k}=\pi_k r_k/\bar r\). Among those assigned to control, its share is \(w_{0k}=\pi_k(1-r_k)/(1-\bar r)\). These expressions require both arms to have positive allocation probability.
If treatment assignment is randomized with the same probability in every subgroup, \(r_k=r\), then \(w_{1k}=w_{0k}=\pi_k\). In expectation, both arms represent the same subgroup mix, so their population mean contrast corresponds to the target average treatment effect. In an actual finite sample, the realized subgroup proportions can still differ by chance.
If assignment probabilities differ across subgroups, the expected arm means instead average subgroup outcomes using different weights. Define \(\mu_{0k}=E[Y(0)\mid S=k]\), so \(E[Y(1)\mid S=k]=\mu_{0k}+\tau_k\). The difference between the treatment-arm and control-arm means under the allocation scheme is
The first term averages subgroup effects using the treatment-arm mix. The second reflects differences in baseline mean outcomes due to the two arms having different subgroup mixes. Consequently, this arm contrast need not equal \(\sum_k\pi_k\tau_k\). The discrepancy is a mismatch between the crude comparison and the stated target estimand; it does not, by itself, show that any subgroup effect changed.
Worked Example: Unequal Subgroup Allocation Changes a Crude Contrast
Consider a hypothetical population with 80% in subgroup 1 and 20% in subgroup 2. Control mean outcomes are 10 and 30 units, respectively; treatment effects are 4 and 12 units. Thus treatment mean outcomes are 14 and 42 units. Assume random assignment within each subgroup with treatment probabilities \(r_1=0.25\) and \(r_2=0.75\), and assume the population proportions and means stated above apply to the trial.
The target average treatment effect is
The treatment-arm population share assigned from subgroup 1 is \(0.80(0.25)=0.20\), and from subgroup 2 it is \(0.20(0.75)=0.15\). The total assigned fraction is \(0.35\), so the treatment-arm subgroup shares are \(0.20/0.35=0.5714\) and \(0.15/0.35=0.4286\). Its mean outcome is
For control, the assigned fractions are \(0.80(0.75)=0.60\) and \(0.20(0.25)=0.05\), totaling \(0.65\). The control-arm mean is
The crude arm contrast is \(26.00-11.54=14.46\) units, not the target average of 5.6 units. Much of the difference reflects the larger share of subgroup 2 in the treatment arm, where the control mean is higher as well as the treatment effect. The subgroup effects can instead be combined using the stated target weights to recover the 5.6-unit target comparison.
Standardizing the Comparison
A direct way to target a specified population is to estimate the treatment effect within each subgroup and then combine those estimates using that population’s subgroup proportions. If \(\hat{\tau}_k\) estimates \(\tau_k\), the standardized estimate is
The weights should be fixed from the target definition, rather than chosen to match whichever arm has the larger sample. When the trial’s subgroup proportions differ from the target proportions, the same principle applies: use the target weights, provided the subgroup effects can be estimated from the trial and the target subgroups are represented.
This procedure is distinct from a crude contrast of the two arm means. It also differs from reporting each subgroup effect separately: standardization gives a single summary for a stated population while retaining the subgroup-specific effects as its ingredients. The summary can hide important variation, so it should not replace subgroup results when those differences matter to the question.
Worked Example: Standardizing Subgroup Results to a New Target
A hypothetical trial estimates effects of 3 units in subgroup A and 9 units in subgroup B. Its enrolled patients are split equally between the subgroups, but the intended target population is 75% subgroup A and 25% subgroup B. Assume randomization within both subgroups, adequate representation in both treatment arms, and that the subgroup effects apply to the target population.
The trial’s equal subgroup mix is not the target mix. Standardizing to the target gives
An unweighted average of 6 units would summarize an equal-subgroup population. The standardized 4.50-unit estimate answers the stated target-population question, subject to uncertainty in the estimated subgroup effects and target proportions.
Derivation and Practical Checks
The weighting rule follows directly by partitioning the target population into mutually exclusive subgroups. Let \(S\) take one of the values \(1,\ldots,K\). The law of total expectation gives
Thus the population average is not an extra assumption about how effects combine: it follows from averaging the individual treatment contrast over the target population. The substantive choices are which population defines the weights and whether the subgroup effects are relevant to that population.
A practical comparison should make the following checks explicit:
- Check the target weights. State the population represented by each \(\pi_k\), and verify that the weights are nonnegative and sum to 1.
- Check subgroup representation. If a target subgroup has no patients receiving one treatment, its treatment effect cannot be estimated directly from that trial without additional assumptions.
- Compare arm composition. Report subgroup proportions by assigned treatment. Large differences make a crude contrast especially hard to interpret as a target-weighted effect.
- Check outcome definitions and units. Effects being averaged must refer to a common outcome scale with comparable meaning.
- Keep uncertainty in view. A weighted estimate is still estimated from data. Uncertainty in subgroup estimates and, when estimated, target weights affects uncertainty in the overall comparison.
For an allocation scheme whose probabilities change in response to accumulated trial information, a single fixed \(r_k\) may not describe the assignment process. The target average \(\sum_k\pi_k\tau_k\) remains defined by the target population, but interpreting a crude arm contrast requires accounting for the actual allocation process and its relation to patient characteristics. The simple fixed-probability calculations above should not be applied mechanically to a changing scheme.
Comparing Common Summaries
| Summary | Weights or comparison | Question answered | Main caution |
|---|---|---|---|
| Subgroup-specific effects | One contrast within each subgroup | How does the treatment effect differ by subgroup? | May be imprecise, especially in small subgroups |
| Target-standardized effect | Target proportions \(\pi_k\) applied to subgroup effects | What is the average effect in the stated target population? | Requires suitable subgroup estimates and relevant target weights |
| Crude arm contrast | Observed treatment-arm mean minus observed control-arm mean | How do the means of the two assigned groups differ? | Can mix treatment effects with differences in subgroup composition |
| Mean outcome under an allocation policy | Outcomes averaged over assignment probabilities and patient mix | What mean outcome is expected under that assignment policy? | Is not, by itself, a treatment-versus-control effect |
Common Mistake and Limitations
Wrong reasoning: “The treatment arm has more patients in the subgroup with the larger effect, so its mean difference is the overall treatment effect.” This overlooks that the control arm may have a different subgroup mix and that baseline mean outcomes may differ between subgroups. A crude contrast can combine both kinds of differences.
Correction: First specify the target population and its subgroup proportions. Then estimate subgroup effects under valid within-subgroup comparisons and combine them using those target weights. Report the crude arm contrast only with a clear description of what it compares.
Standardization does not solve every problem. If treatment effects vary within the defined subgroups, a coarse subgroup classification can leave meaningful heterogeneity hidden. If a target subgroup is absent or lacks treatment overlap, its effect is not identified from the trial data alone. And a population average can conceal effects that differ in direction across subgroups. These limitations call for careful subgroup definition and transparent reporting, not for changing the weights until a preferred answer appears.
Worked Example: Mean Outcome Under an Allocation Policy
Use the population from the earlier allocation calculation: subgroup proportions are 0.80 and 0.20; control means are 10 and 30 units; treatment effects are 4 and 12 units. Compare a policy assigning treatment with probability 0.50 in both subgroups to one assigning with probabilities 0.25 and 0.75, respectively. Assume the stated means and probabilities apply to the population.
Under the equal-probability policy, the expected outcome within subgroup 1 is \((0.50)(14)+(0.50)(10)=12\) units. Within subgroup 2 it is \((0.50)(42)+(0.50)(30)=36\) units. Averaging over the population gives
Under the second policy, the expected outcome in subgroup 1 is \((0.25)(14)+(0.75)(10)=11\) units. In subgroup 2 it is \((0.75)(42)+(0.25)(30)=39\) units. The population mean under this policy is
The policies therefore have different expected outcomes for the trial population: 16.80 versus 16.60 units. These are policy-specific mean outcomes, not estimates of the target treatment effect, which remains 5.6 units for the stated population. A policy mean averages treatment and control outcomes according to assignment probabilities; the treatment effect compares the two potential outcomes for the target population.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A target population is 60% subgroup A and 40% subgroup B. Their treatment effects are 1 and 7 units. What is the target average treatment effect?
- Why can an unweighted average of subgroup effects fail to describe the target population?
- In one sentence, distinguish a target-standardized treatment effect from a mean outcome under an allocation policy.
- What allocation feature makes subgroup composition differ between treatment and control arms?
- If a target subgroup has no patients assigned to control, what is the key limitation for estimating its treatment effect from that trial?
Solutions. (1) \((0.60)(1)+(0.40)(7)=0.60+2.80=3.40\) units. (2) An unweighted average assigns equal weights to subgroups, which describes an equal-subgroup population unless the target proportions are equal. (3) A standardized effect averages within-subgroup treatment contrasts using target-population weights, whereas a policy mean averages outcomes across assigned treatments using an allocation rule. (4) Subgroup-specific treatment probabilities that differ across subgroups can create different subgroup mixtures in the two arms. (5) The subgroup lacks treatment overlap, so its control mean and treatment effect cannot be estimated directly from that trial without additional assumptions.