Fixed Allocation As The Benchmark

Why a Fixed-Allocation Benchmark Is Needed

Response-adaptive randomization changes assignment probabilities as trial responses accumulate. To understand what that change contributes, it helps to compare the adaptive design with a clearly specified design whose allocation rule does not use those responses. The fixed-allocation design is a benchmark: it provides a reference for describing how the trial would be organized under a pre-specified allocation plan.

The benchmark is not automatically an equal-allocation design, nor does naming one settle every question about which trial features are comparable. Allocation is only one part of a trial. A meaningful comparison must also specify such features as the planned sample size, outcome definition and measurement, estimand, and analysis. If these differ, a resulting contrast cannot be attributed solely to the allocation rule.

For a two-treatment fixed-probability design, an allocation ratio of 1:1 means that each participant has probability \(1/2\) of assignment to either treatment. A ratio of 2:1 could instead mean probabilities \(2/3\) and \(1/3\). Neither ratio requires exact realized counts when assignments are made independently with those probabilities. By contrast, a randomized schedule designed to fill exactly 60 places in one arm and 30 in another guarantees those totals, while keeping the schedule independent of responses.

For comparison, let \(A_i\) denote the treatment assigned to participant \(i\), and let \(\mathcal H_{i-1}\) denote the assignment and response history available before that assignment. A response-adaptive design allows the probability of the next assignment to depend on the observed history, such as \(P(A_i=j\mid\mathcal H_{i-1})\). A fixed-probability design instead uses \(P(A_i=j\mid\mathcal H_{i-1})=\pi_j\) at each assignment. Other fixed schedules can depend on pre-specified design features, such as a participant’s stratum, but not on trial responses.

What Must Be Held Constant

A comparison intended to isolate the role of allocation should describe both designs using the same trial framework. The following features are central.

• Sample size and stopping plan. Specify the same planned number of participants and the same rules for ending enrollment or follow-up. If one design stops earlier because it uses a different rule, then allocation is not the only design difference.
• Outcome and measurement. Define the same outcome, time point, measurement procedure, and handling of missing or unavailable measurements. “Response” must refer to the same event or quantity in both designs.
• Estimand. State the treatment effect the trial is intended to estimate, including the treatment contrast, outcome summary, and relevant population. For example, the target might be the difference in 30-day response probabilities between two treatments in the eligible trial population.
• Analysis. Use the same planned statistical model or estimator, covariate strategy, and inferential procedure. The analysis must be appropriate for the stated estimand and design. If the analysis differs, a difference in reported results may reflect that choice as well as allocation.

Holding these features constant does not mean the two designs produce identical data. Different allocation rules can produce different arm counts and different participants assigned to each treatment. Those are possible consequences of the allocation rule itself. The point is to prevent unrelated changes—such as switching from a 30-day outcome to a 90-day outcome—from being mistaken for allocation effects.

The comparison also relies on a coherent common trial setting: the same eligibility criteria, treatments, enrollment process, follow-up procedures, and outcome ascertainment should apply. The assumed treatment-outcome relationship should be the same when describing the two designs. Otherwise, the designs are being compared under different scenarios rather than under a common set of trial conditions.

Matching the Outcome, Estimand, and Analysis

An estimand makes the target of the comparison explicit. For example, with a binary outcome \(Y\) measured at a specified time, one possible two-treatment estimand is the difference in response probabilities, \(p_A-p_B\). It is not interchangeable with a difference in mean outcomes or an odds ratio. The endpoint, time point, population, and treatment contrast all matter to its interpretation.

A planned analysis is the procedure for estimating or testing that target. A benchmark comparison should name it rather than merely saying “analyze the results.” The estimator and model need to correspond to the estimand. Identical analyses cannot make different estimands equivalent; conversely, a common estimand does not by itself ensure that two analyses are comparable if the planned procedures differ.

A Short Justification for the Fixed-Probability Count

The expected-count calculation above follows directly from indicator variables. For each participant, \(I_i\) is one for assignment to A and zero otherwise. Under independent assignment with the same probability \(\pi_A\), \(E(I_i)=\pi_A\), and the variance of a zero-one variable is \(\pi_A(1-\pi_A)\). Summing the indicators gives the total count. Independence makes the covariances between distinct indicators zero, yielding:

This result assumes a fixed number \(n\) of assignments and independent assignment indicators with a common probability \(\pi_A\). It does not apply unchanged to every fixed schedule. In particular, if a schedule guarantees an exact number of assignments, the count has no assignment variation. The distinction matters when describing the benchmark: an allocation ratio alone does not tell the reader whether arm totals are merely expected or are constrained by design.

Diagnostics, Alternatives, and Limitations

Before interpreting a design comparison, check that the benchmark is operationally clear. Ask whether the allocation probabilities are stated for every treatment, whether the plan is based on probabilities or guaranteed counts, and whether the stopping rule and planned sample size are explicit. Then verify that the endpoint, time point, estimand, and analysis match across designs. If any item is ambiguous, the comparison may not have a unique interpretation.

The main alternative to a fixed allocation plan is response-adaptive allocation: future assignments can depend on observed outcomes. A fixed benchmark is often easier to describe because its allocation probabilities are set in advance. An adaptive rule may be more complicated to specify because the probability for a new participant can depend on the history of the trial. Neither label alone determines which design is preferable. Their aims and consequences depend on the full design and the evaluation criterion.

A fixed allocation need not be equal. Unequal fixed probabilities may be selected in advance for practical or scientific reasons. Likewise, a fixed exact-count schedule and independent fixed-probability assignments are not identical mechanisms, even if they share the same target ratio. The benchmark should name which one is intended rather than treating “fixed randomization” as a single universal procedure.

Finally, holding trial features constant makes a comparison interpretable; it does not guarantee that every assumption is satisfied or that all sources of variation disappear. A change in who enrolls, how outcomes are observed, or how missing outcomes are handled can complicate interpretation even when the written protocol appears similar. A careful comparison states its common trial conditions and reports design differences explicitly.

Common Mistake

Wrong reasoning: “The fixed design has a 2:1 allocation ratio, so with 90 participants it must assign exactly 60 to one treatment and 30 to the other.” This confuses target probabilities with guaranteed counts.

Correction: If each assignment is made independently with probabilities \(2/3\) and \(1/3\), the expected counts are 60 and 30, but realized counts can differ. Exact totals require a schedule that enforces them. State which mechanism defines the benchmark.