Why allocation targets can depend on profile
When response probabilities differ across baseline covariate profiles, a single allocation ratio need not be equally efficient for every profile. A participant’s profile may identify a group in which the two treatment arms have different response variances. If the design objective is to estimate the treatment difference within each profile as precisely as possible, those variances determine a profile-specific allocation target.
This target is not a rule that says which treatment is better for an individual, and it does not by itself specify a complete adaptive randomization procedure. It is a benchmark allocation fraction derived from a stated precision objective and a response model. The introductory material on covariate-adjusted response-adaptive allocation provides the broader setting; here the focus is the conditional calculation.
Deriving the conditional target
Fix a profile \(x\), and suppose there are \(N(x)\) participants with that profile. Write \(n_1(x)\) and \(n_0(x)\) for the numbers assigned to arms 1 and 0, so \(n_1(x)+n_0(x)=N(x)\). In this profile, the estimand is the difference in response probabilities \(p_1(x)-p_0(x)\). Its direct estimator is \(\hat p_1(x)-\hat p_0(x)\), where \(\hat p_a(x)\) is the observed response proportion in arm \(a\) among participants with profile \(x\).
For the following fixed-count calculation, assume that responses are Bernoulli with the stated probabilities, that each arm has a positive sample size, and that outcomes are independent across all participants. In particular, the two arm-specific response proportions then have zero covariance. Their variances are \(p_a(x)\{1-p_a(x)\}/n_a(x)\), so the variance of their difference is the sum of those two quantities. The independence assumption matters: independence only among participants within each arm does not, on its own, guarantee zero covariance between the arm-specific proportions.
To find the target, minimize this variance over the allocation of a fixed \(N(x)\). Temporarily treat the arm counts as positive real numbers, put \(n_0(x)=N(x)-n_1(x)\), and differentiate with respect to \(n_1(x)\). At an interior minimum, the derivative is zero:
Rearranging gives \(n_1(x)/n_0(x)=\sqrt{v_1(x)/v_0(x)}\). Converting this ratio to fractions that sum to one yields the Neyman allocation target for this profile:
The square roots arise because the objective contains each response variance divided by its arm’s sample size. Thus the arm with the larger conditional response variance receives a larger target fraction. This is a statement about precision for the within-profile difference, not a statement about which arm has the higher response probability.
Worked calculations
Worked Example: A Target From Two Conditional Response Probabilities
Suppose, hypothetically, that for profile \(x\) the response probabilities are \(p_1(x)=0.70\) and \(p_0(x)=0.50\). Assume independent outcomes across participants and consider \(N(x)=200\) participants with this profile. Calculate the precision-based target and the approximate arm counts.
The Bernoulli variances are \(v_1(x)=0.70(1-0.70)=0.2100\) and \(v_0(x)=0.50(1-0.50)=0.2500\). The target for arm 1 is therefore
The target counts are approximately \(200(0.4782)=95.64\) in arm 1 and \(200(0.5218)=104.36\) in arm 0. Integer counts must be chosen in practice; for example, 96 and 104 are close to these targets. This is a small precision gain over equal allocation, not a prediction that arm 0 is a superior treatment.
Using the unrounded target fractions, the model-based variance is approximately \(0.2100/95.64+0.2500/104.36=0.004591\), giving a standard error of \(\sqrt{0.004591}=0.06776\) for the difference in response proportions. Equal allocation at 100 per arm gives variance \(0.2100/100+0.2500/100=0.004600\), slightly larger. The small gain reflects the relatively similar conditional variances.
Worked Example: Calculating a Target From a Binary Response Model
Consider a hypothetical logistic response model for a binary baseline profile indicator \(x\in\{0,1\}\):
For a participant with profile \(x=1\), the linear predictor is \(0.4\) in arm 0 and \(1.0\) in arm 1. Applying the inverse-logit function gives \(p_0(1)=1/(1+e^{-0.4})=0.5987\) and \(p_1(1)=1/(1+e^{-1.0})=0.7311\). For this calculation, treat the model probabilities as specified and assume the independent-outcome, fixed-count conditions described above.
The conditional variances are \(v_0(1)=0.5987(1-0.5987)\approx0.2403\) and \(v_1(1)=0.7311(1-0.7311)\approx0.1966\). Hence
For 80 participants with profile \(x=1\), the corresponding target counts are approximately \(80(0.4749)=37.99\) in arm 1 and \(80(0.5251)=42.01\) in arm 0, or about 38 and 42. Although the model predicts a higher response probability in arm 1, the precision target gives slightly more participants to arm 0 because its conditional Bernoulli variance is larger. A response-maximizing allocation would answer a different question.
Worked Example: Targets Can Differ Across Profiles
Suppose a hypothetical model gives \(p_1(x_A)=0.90\) and \(p_0(x_A)=0.60\) for profile \(x_A\), while for profile \(x_B\) it gives \(p_1(x_B)=0.60\) and \(p_0(x_B)=0.40\). Assume independent outcomes and apply the same within-profile precision objective to each profile.
For \(x_A\), the variances are \(v_1(x_A)=0.90(0.10)=0.0900\) and \(v_0(x_A)=0.60(0.40)=0.2400\). Thus \(q_1(x_A)=0.3000/(0.3000+\sqrt{0.2400})=0.3000/(0.3000+0.4899)=0.3798\). For \(x_B\), both variances equal \(0.60(0.40)=0.2400\) and \(0.40(0.60)=0.2400\), so \(q_1(x_B)=q_0(x_B)=0.5000\).
With 100 participants in each profile, the approximate targets are 38 in arm 1 and 62 in arm 0 for \(x_A\), and 50 in each arm for \(x_B\). The contrast between these targets is not contradictory: the optimization is performed separately within each profile, using that profile’s two response variances.
Assumptions, diagnostics, and failure modes
A calculated target is only as useful as its assumptions and inputs. In particular, verify that the response is binary and that \(p_a(x)\) refers to the response probability for the same clearly defined profile under arm \(a\). A fitted outcome model may supply these probabilities, but a model-based target is not automatically reliable merely because the formula is correct. Examine whether predicted probabilities are plausible for the profiles being used, and whether there is enough information to support profile-specific predictions.
The variance derivation also has a specific scope. It is a fixed-count benchmark for the difference between two arm-specific sample proportions within a profile. If outcomes are correlated across participants, or the two sample proportions have nonzero covariance, the variance of the difference instead includes a covariance term:
Ignoring a nonzero covariance can therefore invalidate the optimization. In an adaptive trial, arm counts may themselves depend on earlier outcomes. The fixed-count calculation should not be mistaken for a complete variance analysis of the resulting adaptive estimator; the allocation process and its dependence on observed outcomes may also matter.
Other warning signs include targets that change sharply across nearby profiles, probabilities near 0 or 1 based on very little data, and predicted targets that are practically impossible to maintain because a profile is rare. These do not prove the target is wrong, but they call for checking model support, profile definitions, and whether the chosen precision objective is appropriate. If a response variance is zero, the interior derivation does not apply: the square-root formula may put all allocation in the other arm, or become undefined if both variances are zero. Such boundary targets are not a reason to assume that an arm needs no observations for all scientific or operational purposes.
How this target differs from alternatives
| Allocation approach | What it targets | How it differs here |
|---|---|---|
| Equal allocation | Half of participants to each arm | Matches the precision target only when the two conditional response variances are equal. |
| Precision-based conditional target | Minimizes the variance of the within-profile difference for a fixed profile-specific total | Uses both conditional response variances and can differ by profile. |
| Allocation aimed at more observed responses | Assigns more participants to the arm expected to produce more responses | Optimizes a different objective; the arm with more responses need not have the larger response variance. |
| One common allocation ratio | Uses the same target across all profiles | May be simpler, but does not generally minimize the variance separately within every profile. |
The conditional target also has a limitation that is easy to overlook: optimizing each profile separately does not, by itself, prove that the resulting collection of targets minimizes the variance of every possible overall treatment-effect estimator. An overall estimand may combine profiles with different weights, and the total numbers of participants in those profiles may be unequal. The target derived here is specifically for profile-specific contrasts under the stated fixed-count objective.
A common mistake
Wrong reasoning: “Arm 1 has the larger response probability, so the precision target should assign more participants to arm 1.” That confuses a response-maximization objective with a variance-minimization objective. For a binary response, precision depends on \(p_a(x)\{1-p_a(x)\}\), not on \(p_a(x)\) alone. The correct calculation first finds both conditional variances and then allocates in proportion to their square roots. Also, the variance sum used in the derivation requires zero covariance between the two arm-specific proportions; independence within each arm alone is not enough to establish that condition.
Check Your Understanding
Try each question, then compare your reasoning with the brief solutions.
- For a profile with \(p_1(x)=0.40\) and \(p_0(x)=0.60\), calculate \(q_1(x)\) and \(q_0(x)\). What is the target for 100 participants?
- If the two conditional response variances are equal and positive, what allocation target minimizes the variance under the stated fixed-count objective?
- Why is it not valid to omit the covariance term merely because outcomes are independent within each arm?
- Suppose \(p_1(x)=0.95\) and \(p_0(x)=0.50\). Which arm receives the larger precision target, and why?
Solutions. (1) Both variances are \(0.40(0.60)=0.2400\). Their square roots are equal, so \(q_1(x)=q_0(x)=0.5000\), or 50 participants per arm. (2) Equal allocation, because equal positive variances give equal square roots in the target formula. (3) Within-arm independence does not establish that the two arm-specific sample proportions have zero covariance. The variance of their difference includes \(-2\operatorname{Cov}\{\hat p_1(x),\hat p_0(x)\}\) unless zero covariance is separately justified; independence across all participants is a sufficient condition in the stated setup. (4) Arm 0 receives the larger target. Its variance is \(0.50(0.50)=0.2500\), while arm 1’s is \(0.95(0.05)=0.0475\); the target is proportional to the square root of variance, not the response probability.