Aggregating Benefit-Risk Across Patient Subgroups

Why Aggregate Across Patient Subgroups?

A treatment can have different expected utilities in different patient subgroups. A treatment may be preferred for one subgroup and its comparator for another. A population-level decision therefore requires more than listing those subgroup results: it requires specifying which population is being considered and how much each subgroup contributes to that population.

The weights are not a technical afterthought. If one subgroup has a larger share of the target population, its subgroup-specific expected utility contributes more to the overall result. Consequently, a treatment preference for one target population need not hold for a population with a different subgroup composition.

The calculation in this tutorial assumes that subgroup-specific expected utilities have already been defined on a common utility scale. The task here is to combine those results transparently, not to choose the utility scale or to infer treatment effects from study data.

Definition and Aggregation Rule

Definition: Let \(S\) denote a patient's subgroup, and let \(a\) denote a treatment. The subgroup-specific expected utility \(EU_s(a)\) is the expected utility of treatment \(a\) among patients in subgroup \(s\). For a target population partitioned into \(K\) subgroups, let \(w_s\) be the proportion of that population in subgroup \(s\). The population-level expected utility is \(\sum_{s=1}^{K} w_s EU_s(a)\), where \(w_s\geq 0\) and \(\sum_{s=1}^{K}w_s=1\).

The weights describe the target population. They might come from a defined eligible population, a planned treatment setting, or a clearly specified scenario. They are not automatically the proportions enrolled in a trial. Trial proportions are appropriate only if they represent the population for which the decision is intended.

For two treatments, \(A\) and \(B\), define the subgroup-specific expected-utility difference as \(\Delta_s=EU_s(A)-EU_s(B)\). The population-level difference is the weighted average of these subgroup differences. A positive difference favors \(A\), a negative difference favors \(B\), and zero means that the two treatments have equal expected utility under the specified weights and utility model.

EU_{\mathcal P}(a)=\sum_{s=1}^{K}w_s EU_s(a), \qquad \Delta_{\mathcal P} =EU_{\mathcal P}(A)-EU_{\mathcal P}(B) =\sum_{s=1}^{K}w_s\Delta_s.

This rule requires an exhaustive and mutually exclusive subgroup partition: each person in the target population belongs to exactly one of the listed subgroups. The weights must be nonnegative and sum to one. The subgroup utilities must also be comparable: treatment utility should use the same outcome definitions, time horizon, and utility scale in every subgroup. If any of these conditions fails, the weighted sum does not represent the intended population-level expected utility.

Why the Weighted Sum Represents the Target Population

The aggregation rule follows from conditioning on subgroup membership. Let \(U_a\) be the utility outcome for a patient receiving treatment \(a\). A target-population patient belongs to subgroup \(s\) with probability \(w_s\). Within that subgroup, the mean utility is \(EU_s(a)\). Averaging over the possible subgroups gives the overall expected utility:

E[U_a] =\sum_{s=1}^{K}E[U_a\mid S=s]P(S=s) =\sum_{s=1}^{K}EU_s(a)w_s.

Subtracting the corresponding expression for treatment \(B\) from the expression for treatment \(A\) gives the weighted-difference formula. Thus aggregation can be done either by weighting each treatment's subgroup expected utilities and then subtracting, or by first subtracting within each subgroup and then weighting the differences. Both approaches agree when the same weights are used for both treatments.

This is an expectation for a population with the specified composition. It is not necessarily the expected utility for every patient, nor does a favorable population average imply that every subgroup favors the same treatment.

Worked Example: Combining Two Subgroup Results

Suppose a hypothetical target population is 60% subgroup 1 and 40% subgroup 2. On a common utility scale, treatment \(A\) has subgroup expected utilities of 0.72 and 0.38; treatment \(B\) has subgroup expected utilities of 0.62 and 0.50. Calculate the population-level expected utilities and compare the treatments.

The weights are \(w_1=0.60\) and \(w_2=0.40\), and they sum to one. For treatment \(A\):

EU_{\mathcal P}(A) =(0.60)(0.72)+(0.40)(0.38) =0.432+0.152 =0.584.

For treatment \(B\):

EU_{\mathcal P}(B) =(0.60)(0.62)+(0.40)(0.50) =0.372+0.200 =0.572.

The population-level difference is \(0.584-0.572=0.012\) utility units in favor of \(A\). The subgroup differences are \(0.72-0.62=0.10\) in subgroup 1 and \(0.38-0.50=-0.12\) in subgroup 2. Weighting them gives \((0.60)(0.10)+(0.40)(-0.12)=0.060-0.048=0.012\), the same result. This conclusion applies to the specified 60%/40% population; it does not establish a preference that must hold under other subgroup compositions.

Subgroup Composition Can Change the Preference

When subgroup-specific differences have opposite signs, the overall difference depends directly on the weights. For two subgroups, write \(w\) for the share in subgroup 1 and \(1-w\) for the share in subgroup 2. Then:

\Delta_{\mathcal P}=w\Delta_1+(1-w)\Delta_2.

If \(\Delta_1>0\) but \(\Delta_2<0\), increasing the share of subgroup 1 pulls the population-level difference toward treatment \(A\); increasing the share of subgroup 2 pulls it toward treatment \(B\). This is a consequence of averaging subgroup-specific values, not a change in either subgroup's utility result.

Worked Example: A Preference Reversal Under Different Weights

In a hypothetical assessment, treatment \(A\) has expected utility 0.80 in subgroup 1 and 0.40 in subgroup 2. Treatment \(B\) has expected utility 0.60 in subgroup 1 and 0.50 in subgroup 2. Compare the treatments in two target populations: one with 30% in subgroup 1 and one with 70% in subgroup 1.

The subgroup differences are \(\Delta_1=0.80-0.60=0.20\) and \(\Delta_2=0.40-0.50=-0.10\). In the first population, \(w_1=0.30\) and \(w_2=0.70\):

\Delta_{\mathcal P} =(0.30)(0.20)+(0.70)(-0.10) =0.060-0.070 =-0.010.

Treatment \(B\) has greater population-level expected utility by 0.010 utility units for this composition. In the second population, \(w_1=0.70\) and \(w_2=0.30\):

\Delta_{\mathcal P} =(0.70)(0.20)+(0.30)(-0.10) =0.140-0.030 =0.110.

Treatment \(A\) has greater population-level expected utility by 0.110 utility units for the second composition. Neither calculation changes the subgroup-specific results; only the target population weights differ. The example shows why an aggregate preference should always be reported alongside its population weights.

Starting From Outcome Distributions

Sometimes the available inputs are outcome probabilities rather than subgroup expected utilities. In that case, calculate expected utility within each subgroup and treatment first, then apply the target-population weights. The same utility values must be used across subgroups for this aggregation to compare like with like.

Worked Example: From Subgroup Outcome Probabilities to an Overall Comparison

Consider two hypothetical subgroups and two treatments. Outcomes are favorable, neutral, or a severe harm, with utility values 1, 0, and \(-2\), respectively. In subgroup 1, treatment \(A\) has probabilities 0.70, 0.25, and 0.05; treatment \(B\) has probabilities 0.60, 0.38, and 0.02. In subgroup 2, treatment \(A\) has probabilities 0.45, 0.50, and 0.05; treatment \(B\) has probabilities 0.50, 0.47, and 0.03. Each set of probabilities sums to one. Suppose the target population is 40% subgroup 1 and 60% subgroup 2.

For each treatment and subgroup, expected utility is the sum of each outcome's utility multiplied by its probability. In subgroup 1:

EU_1(A)=(0.70)(1)+(0.25)(0)+(0.05)(-2)=0.60, \qquad EU_1(B)=(0.60)(1)+(0.38)(0)+(0.02)(-2)=0.56.

In subgroup 2:

EU_2(A)=(0.45)(1)+(0.50)(0)+(0.05)(-2)=0.35, \qquad EU_2(B)=(0.50)(1)+(0.47)(0)+(0.03)(-2)=0.44.

Now weight each treatment's subgroup results using \(w_1=0.40\) and \(w_2=0.60\):

EU_{\mathcal P}(A)=(0.40)(0.60)+(0.60)(0.35)=0.24+0.21=0.45,

EU_{\mathcal P}(B)=(0.40)(0.56)+(0.60)(0.44)=0.224+0.264=0.488.

The population-level difference is \(0.45-0.488=-0.038\) utility units, favoring \(B\). This result depends both on the stated outcome probabilities and on the common utility values and population weights. It is not a comparison of the raw probability of a favorable outcome alone.

Practical Checks, Alternatives, and Limitations

Before reporting an aggregate, check that the subgroup weights correspond to the intended decision population and that they sum to one. Confirm that every target-population member is represented by exactly one subgroup, and that subgroup expected utilities use compatible outcome definitions and utility scales. Recompute the aggregate from the subgroup-specific differences as a simple arithmetic check; it should match the difference between the two weighted treatment utilities.

A common alternative is to report subgroup results separately rather than combine them. This is informative when decisions are intended to be subgroup-specific or when subgroup outcomes differ materially. A pooled overall estimate can also be suitable if it truly represents the target population and is derived in a way consistent with the subgroup definitions. However, a pooled estimate based on a study's enrollment mix may answer a different question from one based on the intended treatment population. Explicit standardization with target-population weights makes that distinction visible.

The calculation has limits. It summarizes average utility for the specified population and can conceal opposing subgroup preferences. It also treats the supplied subgroup utilities and weights as suitable inputs; if either is uncertain or poorly supported, the aggregate can be unstable. Coarse subgroup categories may conceal meaningful differences within a subgroup, and a weighted average does not establish that treatment effects are causal. The aggregate should therefore be read together with its subgroup results, assumptions, and target-population definition.

Wrong reasoning: “Treatment \(A\) was preferred in the trial overall, so it must be preferred in the population where it will be used.”
Correction: An overall result reflects the weights and analysis used to produce it. Recalculate with weights representing the intended population, particularly when subgroup differences point in opposite directions.
Wrong reasoning: “The larger subgroup-specific utility should receive more weight because its utility is numerically larger.”
Correction: Weights are population proportions, not utility magnitudes. Use the subgroup's share of the target population, not the size of its utility score.

Key takeaway: A population-level expected utility is a weighted average of subgroup-specific expected utilities. State the target population and weights explicitly, and retain subgroup results so readers can see whether the overall preference conceals differences between groups.

Check Your Understanding

Try each problem, then compare your work with the solutions below.

A target population is 25% subgroup 1 and 75% subgroup 2. Treatment \(A\) has subgroup expected utilities 0.8 and 0.4; treatment \(B\) has 0.6 and 0.5. Calculate both population-level expected utilities and the difference favoring \(A\) or \(B\).
Why should trial enrollment proportions not automatically be used as weights for a treatment decision in a different population?
Suppose subgroup differences favor treatment \(A\) by 0.15 in subgroup 1 and favor \(B\) by 0.05 in subgroup 2. What is the population-level difference if subgroup 1 makes up 20% of the target population?
Name two checks required before interpreting a weighted subgroup aggregate.

Solutions. 1. \(EU_{\mathcal P}(A)=(0.25)(0.8)+(0.75)(0.4)=0.20+0.30=0.50\). \(EU_{\mathcal P}(B)=(0.25)(0.6)+(0.75)(0.5)=0.15+0.375=0.525\). The difference is \(0.50-0.525=-0.025\), so \(B\) is favored by 0.025 utility units. 2. Enrollment proportions describe the trial sample and may not match the intended treatment population; using them can answer a different population-level question. 3. The subgroup differences are \(0.15\) and \(-0.05\), so \(\Delta_{\mathcal P}=(0.20)(0.15)+(0.80)(-0.05)=0.030-0.040=-0.010\), favoring \(B\) by 0.010 utility units. 4. For example, verify that weights are nonnegative and sum to one, that subgroups form an exhaustive mutually exclusive partition, and that subgroup utilities use compatible scales and outcome definitions.