Workshop: Specify a Multiarm Adaptive Rule

From an Adaptive Idea to an Executable Rule

A statement such as “allocation will favor promising treatments” is not a design rule. It does not say when the allocation changes, which data are used, how evidence becomes probabilities, how much randomization remains for each arm, or what evidence is sufficient to remove an arm. Each of those choices affects the trial participants recruited next and must be specified before the platform begins.

This tutorial constructs a rule for a platform with a binary response measured at a prespecified endpoint. The rule is an allocation and arm-removal procedure, not a confirmatory test of treatment efficacy. In particular, an allocation probability is a probability used to randomize the next participant; it is not the posterior probability that a treatment works.

Model and Decision Quantities

Let \(C\) denote the concurrent control and let \(j=1,\ldots,K\) index the currently active experimental arms. For each participant, the endpoint is either response or no response. Assume responses are independent conditional on the assigned arm, with a constant response probability within each arm over the period being modeled. The observations must have the same endpoint definition and follow-up window in all arms. This simple model does not account for calendar-time drift, dependence, or delayed and missing outcomes; those features require additional design provisions.

Write \(x_j\) for the number of responses among \(n_j\) participants with an observed endpoint in arm \(j\), and use \(x_C,n_C\) for control. For each arm, take an independent prior \(p_j\sim\mathrm{Beta}(a_j,b_j)\), where \(p_j\) is that arm’s response probability and \(a_j,b_j>0\). With the binomial likelihood, the posterior is

p_j\mid \text{data}\sim\mathrm{Beta}(a_j+x_j,\ b_j+n_j-x_j).

This update assumes that the model and prior were specified in advance and that the observed outcomes included in the update are available under the endpoint definition. A participant whose endpoint is not yet observed must not be counted as a nonresponse merely because follow-up is incomplete.

Definition: For a prespecified clinically relevant difference \(\delta\geq 0\), define the posterior evidence score for active arm \(j\) as \(q_j=P(p_j>p_C+\delta\mid\text{current data})\). This is a posterior probability of a treatment contrast. The randomization probability \(\pi_j\) is a separate design quantity specifying the chance that the next eligible participant is assigned to arm \(j\).

Under independent Beta posteriors, the evidence score can be computed by integrating over the posterior distributions:

q_j=\int_0^1 f_j(t)\left\{\int_0^{\max(0,t-\delta)} f_C(c)\,dc\right\}dt,

where \(f_j\) and \(f_C\) are the posterior densities for \(p_j\) and \(p_C\). Numerical integration or posterior simulation can evaluate this probability. The value of \(\delta\), including a choice of zero, has a clinical interpretation and should be chosen before examining interim results.

Specify the Update Schedule First

An allocation formula is only reproducible if its data cutoff is reproducible. One suitable specification is to review after each fixed block of \(B\) newly randomized participants, beginning after a startup block, and to use only endpoint outcomes observed by a prespecified data cutoff. The rule should state whether a review is delayed until a minimum fraction of the block has completed follow-up, how pending outcomes are handled, and who verifies the data extract.

For example, a platform might review after every 30 additional randomizations once each active arm has at least 20 endpoint-observed participants. The exact numbers are design choices, not universal recommendations. If the required endpoint information is not available at the scheduled cutoff, the protocol should specify whether the review is postponed or proceeds with the available observations. It should not silently change the definition of an observed outcome between looks.

Rule component: State the review trigger, data cutoff, endpoint-maturity rule, minimum information required for any arm-removal decision, and the procedure for validating the counts before computing probabilities.

Map Evidence Scores to Bounded Allocation Probabilities

A direct, transparent way to adapt is to assign a fixed fraction of randomization to concurrent control and divide the remainder among active experimental arms using evidence-based weights. Choose a control fraction \(\rho_C\), a positive exploration constant \(\epsilon\), and a response parameter \(\gamma>0\). For each active experimental arm set

w_j=(\epsilon+q_j)^\gamma,\qquad r_j=(1-\rho_C)\frac{w_j}{\sum_{h=1}^{K}w_h}.

Here \(r_j\) is an unconstrained target, not the final probability if bounds are applied. Because \(w_j>0\), each active arm retains a positive target. The parameter \(\gamma\) controls how strongly differences in evidence affect the targets: increasing \(\gamma\) generally concentrates more of the experimental allocation on arms with larger \(q_j\). It should not be selected in response to interim results.

Now prespecify a lower bound \(L\) and upper bound \(U\) for each active experimental arm. Require \(L\leq \pi_j\leq U\), \(\sum_j\pi_j=1-\rho_C\), and \(\pi_C=\rho_C\). A simple implementation is constrained proportional redistribution: set any target below \(L\) to \(L\), any target above \(U\) to \(U\), and redistribute the remaining experimental allocation among arms not yet fixed in proportion to their original weights. Repeat until all bounds are satisfied. The chosen values must make the constraints feasible for every possible number of active arms: in particular, \(KL\leq1-\rho_C\leq KU\). If the constraints are infeasible after arm removal, the rule is incomplete and must be revised before the trial.

Worked Example: Calculate and Bound an Allocation Update

Suppose control is protected at \(\rho_C=0.30\), three experimental arms are active, and the evidence scores at a scheduled review are \(q_1=0.80\), \(q_2=0.35\), and \(q_3=0.10\). Use \(\epsilon=0.05\), \(\gamma=2\), and experimental-arm bounds \(L=0.10\), \(U=0.70\).

The weights are \(w_1=(0.05+0.80)^2=0.7225\), \(w_2=(0.05+0.35)^2=0.1600\), and \(w_3=(0.05+0.10)^2=0.0225\). Their sum is \(0.9050\). The unbounded experimental targets are therefore

r_1=0.70(0.7225/0.9050)=0.5588,\quad r_2=0.70(0.1600/0.9050)=0.1238,\quad r_3=0.70(0.0225/0.9050)=0.01740.

Arm 3 is below its lower bound, so set \(\pi_3=0.10\). The remaining experimental probability is \(0.70-0.10=0.60\). Redistribute it between arms 1 and 2 in proportion to their weights, whose sum is \(0.7225+0.1600=0.8825\):

\pi_1=0.60(0.7225/0.8825)=0.4912,\qquad \pi_2=0.60(0.1600/0.8825)=0.1088.

The final probabilities are \(\pi_C=0.30\), \(\pi_1=0.4912\), \(\pi_2=0.1088\), and \(\pi_3=0.10\). They sum to 1, satisfy both experimental-arm bounds, and keep control at 30%. The score \(q_1=0.80\) did not become an 80% chance of assigning a participant to arm 1.

Worked Example: Apply the Formula When No Bound Is Reached

At a later review, suppose only two experimental arms remain, with \(q_1=0.60\) and \(q_2=0.40\). Keep \(\rho_C=0.30\), \(\epsilon=0.05\), and \(\gamma=1\), with \(L=0.10\) and \(U=0.70\).

The weights are \(w_1=0.65\) and \(w_2=0.45\), so their sum is 1.10. Thus

\pi_1=0.70(0.65/1.10)=0.4136,\qquad \pi_2=0.70(0.45/1.10)=0.2864.

Both values lie between 0.10 and 0.70, and \(0.30+0.4136+0.2864=1.0000\). No redistribution is needed. The higher score leads to a higher allocation target, but both arms continue to receive randomization.

Define Arm Removal Separately

The allocation weights answer how to randomize among active arms. They do not, by themselves, define when an arm ceases to be active. A complete protocol gives arm removal its own eligibility conditions and decision threshold. One possible rule is: an experimental arm can be dropped only at a scheduled review, only after at least \(n_{\min}\) endpoint-observed participants in that arm, and only if \(q_j<\eta\) at two consecutive eligible reviews. Here \(\eta\) is a prespecified low posterior probability threshold. After removal, the arm receives no further assignments, its accrued data remain in the record, and it is not reactivated under this rule.

This repeated-look requirement is one design choice to guard against a decision driven by a transient early result; it is not a guarantee of good operating characteristics. The threshold, minimum information, and look schedule should be evaluated together by simulation under plausible response scenarios. If the rule includes an exception for safety or other clinical reasons, that pathway must also be specified rather than folded informally into the efficacy score.

Worked Example: Apply a Repeated-Review Dropping Criterion

Suppose the protocol sets \(\delta=0.05\), \(\eta=0.10\), and \(n_{\min}=40\). Arm A has at least 40 endpoint-observed participants. At its first eligible review, its calculated score is \(q_A=0.08\); at the next scheduled review, it is \(q_A=0.07\). Assume no other removal rule applies.

At the first review, \(0.08<0.10\), but this is only one eligible review, so arm A remains open. At the next review, \(0.07<0.10\) again. The two consecutive-review condition is now met, so the rule drops arm A after this review. Participants already assigned to A remain in the analysis, and future randomization probabilities are recalculated across the remaining active experimental arms while control remains at \(\rho_C=0.30\).

Justification, Checks, and Limitations

The probability construction has two useful properties. First, before bounds are applied, positive weights guarantee that each active experimental arm has a positive target. Second, normalization assigns exactly the experimental allocation remainder. Indeed,

\sum_{j=1}^{K}r_j =(1-\rho_C)\frac{\sum_{j=1}^{K}w_j}{\sum_{h=1}^{K}w_h} =1-\rho_C.

When the bounds are feasible, redistribution preserves that total while bringing each constrained probability into its permitted range. Control remains protected because its probability is specified separately, not included in the evidence-weighted experimental allocation.

A practical review should verify that the active-arm list is correct, the response counts and endpoint-observed denominators reconcile with the trial records, the calculated scores use the prespecified prior and \(\delta\), and the final probabilities sum to 1 within numerical tolerance. It should also check that every probability respects its bound and that any removal condition is evaluated only at an eligible review. If the bounds cannot be satisfied, the software should stop for resolution rather than silently produce invalid probabilities.

This approach is not the only alternative. Fixed equal randomization is simpler and does not change probabilities in response to outcomes, but it does not adapt allocation. A response-adaptive rule such as this one can shift assignment toward arms with more favorable posterior evidence while preserving exploration and control allocation. A near-deterministic rule that assigns almost everyone to the arm with the highest score may appear more responsive, but can produce sparse comparisons and unstable allocation from noisy early data. The floors, control fraction, and positive \(\epsilon\) reduce some of those risks; they do not eliminate selection bias, time trends, or small-sample limitations.

The rule also depends on the response model, prior, endpoint ascertainment, and review schedule being suitable. If response probabilities change over calendar time, or if follow-up availability differs by arm, the scores may not provide a fair comparison. Sensitivity analyses and simulation should examine operating behavior, including allocation paths, arm-removal frequency, and the consequences of delayed outcomes. These checks assess the whole design; a favorable posterior score at one review is not a substitute for them.

Common Mistakes

Using a posterior probability as the randomization probability. A score \(q_j=0.70\) does not imply \(\pi_j=0.70\). The score must be transformed by the stated allocation formula and constraints.
Leaving the timing vague. “Update as data accrue” does not identify the data cutoff or how pending endpoints are handled. Specify both.
Dropping an arm because its current allocation is small. A low allocation can result from the weight formula and bounds. Removal should follow the separate, prespecified criterion.
Protecting control only in the initial design. After an experimental arm is removed, recompute the experimental allocation but retain the stated control probability.
Choosing incompatible bounds. Check feasibility for each possible number of active arms before enrollment begins.

Key takeaway: A complete adaptive rule specifies when data are reviewed, how posterior evidence becomes allocation probabilities, how bounds are enforced, and what independent criterion removes an arm. The probabilities must be feasible, reproducible, and explicit about control protection.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

With \(\rho_C=0.30\), \(\epsilon=0.05\), \(\gamma=1\), and two active arms having \(q_1=0.50\) and \(q_2=0.25\), calculate the allocation probabilities before bounds.
An arm has \(q_j=0.06\) at one eligible review and \(q_j=0.12\) at the next. If the dropping threshold is \(\eta=0.10\) and two consecutive values below the threshold are required, is the arm dropped?
With two active arms, \(\rho_C=0.30\), and a lower bound \(L=0.40\) for each experimental arm, are the constraints feasible? Explain.
In one sentence, distinguish the meaning of \(q_j\) from the meaning of \(\pi_j\).

Solutions. (1) The weights are \(0.55\) and \(0.30\), with total \(0.85\). Thus \(\pi_1=0.70(0.55/0.85)=0.4529\), \(\pi_2=0.70(0.30/0.85)=0.2471\), and \(\pi_C=0.30\); the probabilities sum to 1. (2) No. The second value is not below 0.10, so the arm has not met the two-consecutive-review condition. (3) No. The lower bounds require at least \(2(0.40)=0.80\) experimental allocation, but only \(1-\rho_C=0.70\) is available. (4) \(q_j\) is the posterior probability that arm \(j\) exceeds control by more than the specified margin, whereas \(\pi_j\) is the probability that the next eligible participant is randomized to arm \(j\).