Ethical Targets With Minimum Information Per Arm

Why Put a Floor Under Every Arm?

If the only objective is to maximize expected patient utility, and one arm has a higher expected utility per participant than every other arm, the welfare-maximizing allocation sends everyone to that arm. That answer may leave too few participants in other arms to estimate their outcomes usefully or compare them with the preferred arm. A minimum sample-size requirement prevents this complete exclusion: every arm receives at least a prespecified number of participants.

The requirement creates a direct trade-off. Participants allocated to a lower-utility arm contribute to the information available about that arm, but, under the assumed utility values, they reduce expected total utility relative to assigning them to the highest-utility arm. The calculation below makes that trade-off explicit. A sample-size floor is a design constraint; by itself, it is not a guarantee of adequate precision or power.

The utilities \(u_j\) must be on a common scale and refer to the same patient-level objective. For example, they could be expected numbers of beneficiaries per participant, or expected net utility points per participant. The optimization does not determine which utility measure is ethically appropriate; it takes the specified values as inputs. The earlier material on patient benefit, treatment-related harm, and treatment costs supplies examples of how such a measure can be defined.

The basic model assumes that the total \(N\) is fixed; each participant’s expected utility depends on the assigned arm and does not change with the allocation of other participants; and utilities are correctly specified for the population and treatment context being considered. These assumptions make expected total utility additive. If outcomes or utilities depend on calendar time, interference, or the numbers already assigned, the fixed-value calculation need not describe expected utility accurately.

Feasibility and the Constrained Optimum

Before optimizing, check feasibility. Since every arm must receive its minimum, the total number required by the floors cannot exceed the total sample size. For integer sample sizes, the allocation is feasible exactly when \(\sum_j m_j \le N\). If this condition fails, no allocation can meet all the requirements with the stated \(N\); at least one of the constraints or the total sample size must change.

When the constraints are feasible, first assign each arm its minimum. The remaining participants can then be allocated to increase expected utility. The following result describes that step for a linear expected-utility objective.

To justify the result, write each allocation as \(n_j=m_j+x_j\), where \(x_j\ge 0\) is the number assigned beyond the minimum. The remaining assignments satisfy \(\sum_j x_j=R\). Expected total utility is therefore the utility earned by the minimum allocations plus the utility from these remaining assignments:

Because \(u_j\le u_{\max}\) for every arm, and \(x_j\ge 0\), the second sum cannot exceed \(R u_{\max}\):

This upper bound is attained by assigning all remaining participants to arms whose utility equals \(u_{\max}\). If utilities tie for the maximum, any distribution of the remainder among those arms attains the same value. This also shows why the minimum constraints alter the unconstrained welfare target: they reserve \(m_j\) participants for every arm before the welfare-maximizing allocation of the remainder.

Worked Examples

Using the Result Carefully

In two arms, the result has a particularly simple form. If Arm B has higher utility, the constrained target is \(n_A=m_A\) and \(n_B=N-m_A\). If Arm A has higher utility, it is \(n_A=N-m_B\) and \(n_B=m_B\). If the utilities are equal, every feasible allocation has the same expected utility. These formulas show that the nonpreferred arm’s minimum—not its own minimum—limits how many participants can be assigned to the higher-utility arm.

For a fixed total, a minimum count can support outcome estimation in every arm, but it is only a coarse proxy for information. The precision of an estimate also depends on outcome variability, the outcome distribution, the estimator, and the inferential goal. For example, a count that is adequate for a common binary outcome may be inadequate for a rare outcome. Conversely, a count floor alone does not establish a power level or a target standard error. The present optimization answers the welfare question subject to the stated count floors; it does not solve those separate design questions.

It is useful to distinguish this constrained welfare target from common alternatives. Unconstrained welfare maximization allocates everyone to an arm with the highest specified utility. Equal allocation assigns the same number to each arm when feasible, regardless of the utility differences. A precision-oriented allocation instead uses an explicit statistical precision objective or constraint; a minimum count can be part of such a design, but it is not equivalent to optimizing precision. The welfare target derived here uses the specified utilities and minimum counts, and does not claim to be optimal for every objective at once.

A frequent mistake is to start from the unconstrained target and simply add the minimum counts to it. For example, if all participants would go to Arm B without constraints, assigning all \(N\) to B and then adding \(m_A\) participants to A creates a total larger than \(N\). The correction is to reserve the required minimum for every arm first, then allocate only the remaining \(R=N-\sum_jm_j\) participants to the highest-utility arm or arms.

Another mistake is to treat a minimum count as proof that the comparison is informative enough. The constraint guarantees only \(n_j\ge m_j\). It does not guarantee a desired precision, and it cannot compensate for incorrect utility assumptions. If the minimums sum to more than \(N\), the target is infeasible; if utilities are tied, the welfare objective does not choose among the tied allocations; and if the expected utilities are misspecified, the calculated optimum is optimal only for the values supplied.

In practice, allocations are integer counts. The theorem is exact for integer \(n_j\) and integer \(m_j\): after meeting the floors, assign the remaining whole participants to the arm or arms with the highest utility. If a target is expressed as a proportion rather than a count, converting it to integer enrollments requires a rounding rule that preserves the total and all minimums. Rounding should not silently reduce any arm below its required count.