Expected Beneficiaries Under Competing Targets

From Allocation Targets to Expected Successes

An allocation target determines what fraction of participants are assigned to each arm. A separate question is what that target implies for the expected number of successful outcomes. These quantities are related, but the target that produces the most successes is not automatically the target that best serves every trial objective. Equal allocation may be familiar and easy to implement; Neyman allocation is aimed at precision; an ethical target may reflect patient benefit and constraints. Calculating expected successes makes one part of their trade-off explicit.

Let $N$ be the total number of participants, $q$ the fraction assigned to treatment, $p_T$ the probability of success on treatment, and $p_C$ the probability of success on control. The corresponding target counts are $Nq$ and $N(1-q)$, before any integer rounding.

Definition: Under a target treatment share $q$, the expected number of successful outcomes is $$ E(S\mid q)=Nq p_T+N(1-q)p_C =N\{p_C+q(p_T-p_C)\}, $$ where $S$ is the total number of successes. The expected number is an average over repetitions of the trial under the stated response probabilities; it need not be an integer and does not predict the exact number observed in one trial.

For example, if each assigned participant has the same response probability within their arm, the outcome is recorded for everyone, and the probabilities do not change with allocation, then the expected successes in an arm equal its participant count times its response probability. Adding those two arm-specific expectations gives the expression above. Independence of participant outcomes is a common assumption for further variance calculations, but is not needed for this expectation calculation.

Assumptions and What the Calculation Represents

The calculation is useful only when its inputs match the trial being considered. Treat $p_T$ and $p_C$ as the response probabilities for the specified population, outcome definition, and follow-up period. Assume that assignment precedes outcome measurement, each participant contributes one binary success-or-failure outcome, and the response probability in an arm is stable across participants and does not itself change because the allocation share changes. Also assume complete outcome ascertainment, or that the stated response rates already account for the intended missing-outcome handling.

For a fixed target with specified arm counts, linearity of expectation gives the formula directly. In a response-adaptive design, realized arm counts may be random. If assignment is made before the current participant’s outcome and the arm-specific response probabilities remain fixed, the conditional expected outcome for that participant is the response probability of the assigned arm. Summing over participants gives $E(S)=p_T E(N_T)+p_C E(N_C)$, where $N_T$ and $N_C$ are the final arm counts. Thus the same calculation applies to expected arm counts, though the final counts need not equal their target values exactly.

This is not a model for changing response rates over time, interference between participants, or outcomes whose probabilities depend on how many people have already been assigned to each arm. If those features matter, a single pair $p_T,p_C$ and a single target share may not describe expected trial outcomes adequately. Likewise, expected successes are not a measure of treatment harm, cost, or the value of information for estimating a treatment effect.

Worked Comparisons

Worked Example: Equal, Neyman, and Ethical Targets

Suppose a hypothetical trial plans $N=100$ participants. The assumed success probabilities are $p_T=0.70$ and $p_C=0.50$. Compare equal allocation, Neyman allocation for estimating the difference in binary response proportions, and an ethically justified target of $q_E=0.60$. Assume equal per-participant costs. The ethical target is supplied by a separate utility assessment; this calculation evaluates its expected successes rather than deriving that target.

Under equal allocation, $q=0.50$, so the target is 50 participants per arm. The expected successes are $$ E(S\mid q=0.50)=50(0.70)+50(0.50)=35+25=60. $$

For binary outcomes, the standard deviations are $\sqrt{p_T(1-p_T)}=\sqrt{0.21}$ and $\sqrt{p_C(1-p_C)}=\sqrt{0.25}$. Under equal costs, the Neyman treatment share is the treatment-arm standard deviation divided by the sum of the two standard deviations: $$ q_N=\frac{\sqrt{0.21}}{\sqrt{0.21}+\sqrt{0.25}} =\frac{0.4582576}{0.4582576+0.5000000} \approx 0.4782196. $$

The unrounded target counts are approximately $100(0.4782196)=47.82196$ treatment participants and $52.17804$ control participants. The expected successes are $$ E(S\mid q_N)=100\{0.50+0.4782196(0.70-0.50)\} =50+9.564392 \approx 59.56. $$ If the target is rounded to 48 treatment and 52 control participants, the corresponding expectation is $48(0.70)+52(0.50)=59.60$. This small difference is due to rounding the target counts.

For the ethical target $q_E=0.60$, the target counts are 60 and 40. Its expected successes are $$ E(S\mid q_E)=60(0.70)+40(0.50)=42+20=62. $$

With these response rates, the ethically supplied target yields the largest expected success count among the three, followed by equal allocation and then Neyman allocation. This comparison does not show that the ethical target is always best overall: it shows its expected success count under these assumptions. Neyman allocation is designed for precision, not for maximizing successes.

Worked Example: When the Treatment Response Rate Is Lower

Consider another hypothetical trial with $N=100$, $p_T=0.45$, and $p_C=0.60$. Suppose a utility analysis has selected an ethical target of $q_E=0.35$, for example because treatment has relevant burdens in addition to its response rate. Compare expected successes under this target, equal allocation, and Neyman allocation.

Equal allocation gives $$ E(S\mid q=0.50)=50(0.45)+50(0.60)=22.5+30=52.5. $$ For Neyman allocation, the arm standard deviations are $\sqrt{0.45(0.55)}=\sqrt{0.2475}\approx0.4974937$ and $\sqrt{0.60(0.40)}=\sqrt{0.24}\approx0.4898979$. Therefore $$ q_N=\frac{0.4974937}{0.4974937+0.4898979}\approx0.503846. $$ The corresponding expected success count is $$ E(S\mid q_N)=100\{0.60+0.503846(0.45-0.60)\} =60-7.55769 \approx52.44. $$

The ethical target assigns 35 participants to treatment and 65 to control. Its expected successes are $$ E(S\mid q_E)=35(0.45)+65(0.60)=15.75+39=54.75. $$

Here, increasing the treatment share lowers expected successes because $p_T-p_C=-0.15$. The supplied ethical target has the largest expected success count of these three, partly because it allocates more participants to the higher-response control arm. Its ethical justification, however, must come from the utility assessment, not from this count alone.

Why the Targets Differ: A Short Derivation

The expected-success formula is linear in the treatment share. For any two targets $q_1$ and $q_2$, subtract their expected counts:

\begin{aligned} E(S\mid q_1)-E(S\mid q_2) &=N\{p_C+q_1(p_T-p_C)\} -N\{p_C+q_2(p_T-p_C)\}\\ &=N(q_1-q_2)(p_T-p_C). \end{aligned}

This identity gives the direction and size of the expected-count difference. If $p_T>p_C$, a target with a larger treatment share has more expected successes; if $p_T<p_C$, it has fewer. If the response probabilities are equal, all targets have the same expected success count, even though their precision or ethical properties may differ. The result concerns expected counts under fixed response probabilities; it does not say that the observed count in every realized trial will follow the same ordering.

Neyman allocation chooses arm shares to reduce the variance of an estimator of the treatment-control difference under its assumptions. For binary responses and equal costs, that allocation depends on the arm standard deviations, not simply on which arm has the higher response probability. The share can therefore differ from both one-half and the share that maximizes expected successes. An ethical target instead reflects a stated ethical objective, such as patient utility, together with any imposed constraints. The expected-success calculation can describe one consequence of that target, but it does not replace its ethical rationale.

Worked Example: Checking the Direction Before Calculating

A hypothetical trial plans $N=200$ participants, with $p_T=0.80$ and $p_C=0.60$. Compare equal allocation, Neyman allocation, and an ethical target of $q_E=0.65$. Because treatment has the higher stated success probability, a larger treatment share should have a larger expected success count.

Equal allocation gives $q=0.50$, so $$ E(S\mid q=0.50)=200\{0.60+0.50(0.20)\}=140. $$ For Neyman allocation, the standard deviations are $\sqrt{0.80(0.20)}=0.4000000$ and $\sqrt{0.60(0.40)}\approx0.4898979$. Thus $$ q_N=\frac{0.4000000}{0.4000000+0.4898979}\approx0.4494897, $$ and $$ E(S\mid q_N)=200\{0.60+0.4494897(0.20)\} =120+17.97959 \approx137.98. $$

The ethically selected target gives $200(0.65)=130$ treatment participants and 70 control participants: $$ E(S\mid q_E)=130(0.80)+70(0.60)=104+42=146. $$ Neyman allocation assigns a smaller treatment share than equal allocation in this example, so its expected success count is also smaller. The ethical target assigns a larger share than either and consequently has the largest expected count among these three. If Neyman target counts are rounded to 90 treatment and 110 control, the rounded-count expectation is $90(0.80)+110(0.60)=138$, consistent with the unrounded calculation.

Interpreting and Diagnosing a Comparison

A reliable comparison begins by stating the response rates and the target shares, not just the resulting counts. Check that the same definition of “success” and the same follow-up window apply to both arms. Confirm that the response rates refer to the population relevant to the proposed trial, and state whether the allocations are exact counts, nominal shares, or expected shares. For a fixed total, verify that treatment and control shares sum to one and that the arm counts sum to $N$.

Several quick checks can catch errors. The expected total must lie between zero and $N$. If $p_T=p_C=p$, every allocation should give $Np$. If $p_T>p_C$, increasing $q$ should not decrease the expected count; if it does, check the arm labels or subtraction. A calculated count that is inconsistent with these checks often signals that a response rate was paired with the wrong arm, a share was used as a count, or a rounded share was mixed with unrounded target counts.

For a binary outcome, Neyman allocation uses the response standard deviations $\sqrt{p(1-p)}$, not the response rates themselves. A common error is to put $p_T$ and $p_C$ directly into the Neyman ratio. Another is to claim that Neyman allocation must yield the most successes: it optimizes an estimation-precision criterion under its stated assumptions. Equal allocation is a convenient benchmark and may be chosen for other design reasons, but it is not guaranteed to maximize either precision or expected successes.

Target	What it is based on	What expected-success calculation tells you
Equal allocation	Same planned share in each arm	A benchmark count under the stated response rates
Neyman allocation	Arm outcome variability and the precision objective, under relevant cost assumptions	The expected success count associated with a precision-oriented target
Ethically constrained allocation	A specified utility objective and ethical or operational constraints	One outcome-yield consequence of the ethically selected target, not its full justification

Limitations and Common Pitfall

Expected successes are an average, not a forecast with certainty. Two trials using the same target can observe different totals because outcomes vary. The formula also does not compare uncertainty in estimated treatment effects, the distribution of benefit across participants, treatment-related harms, costs, or the consequences of receiving control. Those may be central to choosing an allocation target even when expected success counts are available.

A common mistake is: “The target with the most expected successes is therefore the correct allocation.” The correction is that expected successes answer a narrower question: how many successful outcomes are expected under specified response probabilities and a specified allocation? A design choice must also reflect its objective and constraints. In particular, a precision-oriented Neyman target and an ethical target answer different questions; neither should be judged solely by its expected success count.

Key takeaway: Under stable arm-specific response probabilities, expected successes equal $N\{p_C+q(p_T-p_C)\}$. The difference between two targets’ expected counts is $N(q_1-q_2)(p_T-p_C)$. Use this to compare outcome yield, while keeping precision and ethical utility as distinct design considerations.

Check Your Understanding

Use the formula and show the target share and response rates in each calculation.

A trial plans 80 participants, with $p_T=0.65$ and $p_C=0.55$. Find expected successes under equal allocation and under a treatment share of $q=0.70$.
For $N=120$, $p_T=0.40$, and $p_C=0.40$, compare the expected counts under $q=0.25$ and $q=0.75$. Explain why they agree.
For $N=100$, $p_T=0.50$, and $p_C=0.70$, which target has more expected successes: $q_1=0.30$ or $q_2=0.60$? Calculate the difference using the two-target identity.
Why does a Neyman target not necessarily maximize expected successes?

Solutions. (1) Equal allocation gives $40(0.65)+40(0.55)=26+22=48$. At $q=0.70$, the expected count is $80\{0.55+0.70(0.10)\}=80(0.62)=49.6$. (2) Both targets give $120(0.40)=48$, since the response rates are equal and the formula does not depend on $q$ in that case. (3) The difference for $q_1=0.30$ versus $q_2=0.60$ is $100(0.30-0.60)(0.50-0.70)=6$. Thus $q_1$ has six more expected successes; directly, the counts are 64 and 58. (4) Neyman allocation is based on precision and arm variability, whereas expected successes depend linearly on the treatment share and the response-rate difference. These are distinct objectives.