Randomization After an Arm Is Dropped

Why Randomization Must Be Updated After an Arm Is Dropped

When a platform trial stops enrolling to an experimental arm, future participants must not continue to be randomized to it. The randomization probabilities therefore need a new definition over the arms that remain open. This update is not merely bookkeeping: a rule that reallocates all probability mass proportionally can change the control allocation, potentially beyond a protocol limit. The chosen rule should be explicit, feasible, and applied consistently to every subsequent randomization.

The calculations below concern future assignments after the decision to drop an arm. Participants already randomized remain in the arms to which they were assigned; dropping an arm stops future enrollment to it and does not change past assignments. The update assumes that the control arm is retained. If the control is itself removed, the set of permissible comparisons and the allocation policy require a separate design decision.

Definition: Immediately before a drop decision, let \(J\) be the set of open arms and let \(\pi_j\) be the probability of assigning the next eligible participant to arm \(j\), with \(\pi_j\geq 0\) and \(\sum_{j\in J}\pi_j=1\). Let \(D\subset J\) be the set of arms being dropped, and let \(A=J\setminus D\) be the surviving arms. The dropped probability mass is \(r=\sum_{j\in D}\pi_j\), and the surviving mass is \(s=1-r=\sum_{j\in A}\pi_j\). The post-drop probabilities must be nonnegative, sum to one over \(A\), and assign probability zero to every arm in \(D\).

A basic feasibility condition for proportional renormalization is \(s>0\). In ordinary operation, the retained control has positive allocation probability, which ensures this condition. If every arm with positive probability were dropped, no renormalization could produce a valid allocation over the stated survivors; the trial would need a new allocation design or a pause in randomization.

Conditional Renormalization

The most direct update is to condition the old allocation distribution on assignment to a surviving arm. For every \(j\in A\), divide its old probability by the total surviving mass. The result sums to one and preserves the relative allocation probabilities among all surviving arms.

\pi_j^{+}=\frac{\pi_j}{s}=\frac{\pi_j}{1-r},\qquad j\in A, \qquad \pi_j^{+}=0,\qquad j\in D.

This formula follows because each surviving arm's original mass is retained, while the total mass available for survivors is \(s\). Indeed, \(\sum_{j\in A}\pi_j^{+}=\sum_{j\in A}\pi_j/s=s/s=1\). For any two surviving arms with positive probabilities, \(\pi_j^{+}/\pi_k^{+}=\pi_j/\pi_k\), so their allocation ratio is unchanged.

For the control arm \(C\), conditional renormalization gives \(\pi_C^{+}=\pi_C/s\). Since \(s\leq 1\), the control probability does not decrease. If positive mass is removed, then \(s<1\): the control probability rises strictly when \(\pi_C>0\), but remains zero when \(\pi_C=0\). This boundary case matters when interpreting or implementing the formula.

Worked Example: Renormalizing After One Arm Is Dropped

Suppose a hypothetical platform currently assigns the next participant to control with probability \(0.30\), to treatment \(T_1\) with probability \(0.35\), to \(T_2\) with probability \(0.20\), and to \(T_3\) with probability \(0.15\). The protocol drops \(T_3\) and, for this example, imposes no additional control constraint.

The removed mass is \(r=0.15\), so the surviving mass is \(s=1-0.15=0.85\). Conditional renormalization gives:

\pi_C^{+}=\frac{0.30}{0.85}=0.3529,\qquad \pi_{T_1}^{+}=\frac{0.35}{0.85}=0.4118,\qquad \pi_{T_2}^{+}=\frac{0.20}{0.85}=0.2353.

The rounded probabilities sum to \(0.3529+0.4118+0.2353=1.0000\). The control allocation increases from \(0.30\) to about \(0.3529\) because its original probability was positive and positive probability mass was removed. The ratio between the two surviving treatment allocations is preserved: \(0.4118/0.2353\) is approximately \(0.35/0.20=1.75\).

Preserving a Control-Allocation Constraint

A trial may require the control probability to remain below a maximum, above a minimum, within an interval, or at a specified target. Conditional renormalization preserves ratios among all surviving arms, but it does not generally preserve a control bound. For example, an upper limit can be exceeded because positive mass removed from experimental arms is redistributed to control along with the other survivors.

Suppose the protocol requires \(L\leq \pi_C^{+}\leq U\), where \(L\) and \(U\) are specified limits satisfying \(0\leq L\leq U\leq 1\). One transparent constrained rule is to first calculate the control probability from conditional renormalization, then cap it to the permitted interval. The remaining probability is distributed among surviving noncontrol arms in proportion to their original probabilities.

q_{\mathrm{raw}}=\frac{\pi_C}{s}, \qquad q_C^{+}=\min\!\left(U,\max(L,q_{\mathrm{raw}})\right),

\pi_j^{+}=(1-q_C^{+}) \frac{\pi_j}{\sum_{k\in A,\ k\ne C}\pi_k}, \qquad j\in A,\ j\ne C.

This rule requires positive surviving noncontrol mass, \(\sum_{k\in A,\ k\ne C}\pi_k>0\). The control allocation is within the stated interval by construction. The noncontrol probabilities sum to \(1-q_C^{+}\), since their proportional weights sum to one, so all post-drop probabilities sum to one. This constrained rule preserves the relative probabilities among surviving noncontrol arms, but it may change their ratios to control. If the protocol specifies a different redistribution rule, that rule should be used instead; a constraint alone does not uniquely determine how the noncontrol mass is divided.

Worked Example: Renormalization Would Exceed a Control Cap

Use the probabilities from the previous hypothetical setup: control \(0.30\), \(T_1\) \(0.35\), \(T_2\) \(0.20\), and \(T_3\) \(0.15\). After dropping \(T_3\), suppose the protocol requires the control allocation to be no greater than \(0.35\). Take \(L=0\) and \(U=0.35\).

The raw conditional control allocation is \(0.30/0.85=0.3529\), which exceeds the cap. The constrained rule therefore sets \(q_C^{+}=0.35\). The surviving noncontrol mass before redistribution is \(0.35+0.20=0.55\), so their original within-treatment weights are \(0.35/0.55\) and \(0.20/0.55\). The post-drop probabilities are:

\pi_{T_1}^{+}=0.65\left(\frac{0.35}{0.55}\right)=0.4136,\qquad \pi_{T_2}^{+}=0.65\left(\frac{0.20}{0.55}\right)=0.2364,\qquad \pi_C^{+}=0.3500.

The probabilities sum to \(0.3500+0.4136+0.2364=1.0000\), and control is exactly at, not above, its maximum. The \(T_1:T_2\) ratio remains \(0.35:0.20\). The resulting allocation differs from unconstrained conditional renormalization because preserving the cap takes priority over preserving the control-to-treatment ratios.

Several Arms Dropped Together and Boundary Cases

The same calculations apply when multiple arms are removed at one interim: add their probabilities to obtain \(r\), identify the surviving arms, and update their probabilities according to the chosen rule. Do not renormalize sequentially using an intermediate allocation unless the design explicitly defines sequential updates; direct calculation from the allocation vector in force at the decision makes the rule easier to audit.

Worked Example: Multiple Drops With a Control Cap

Consider hypothetical probabilities \(0.25\) for control and \(0.30\), \(0.25\), and \(0.20\) for treatments \(T_1,T_2,T_3\), respectively. Both \(T_2\) and \(T_3\) are dropped. The control allocation is capped at \(0.40\), and the constrained redistribution rule is used.

The removed mass is \(r=0.25+0.20=0.45\), leaving \(s=0.55\). Unconstrained renormalization would give control probability \(0.25/0.55=0.4545\), which exceeds \(0.40\). Thus set the control probability to \(0.40\). Only \(T_1\) survives among the noncontrol arms, so it receives the remaining \(1-0.40=0.60\). The final vector is control \(0.40\), \(T_1\) \(0.60\), \(T_2\) \(0\), and \(T_3\) \(0\); its entries are nonnegative and sum to one.

A zero control probability illustrates why the condition for a strict increase must be stated carefully. If control initially has probability zero, conditional renormalization leaves it at zero even when positive mass is removed. A protocol requiring a positive lower control bound would then need the constrained rule to assign that minimum, provided enough surviving noncontrol probability can be redistributed and the protocol permits this change.

Implementation, Diagnostics, and Alternatives

Before randomization resumes, verify the probability vector and the decision rule against the protocol. A useful sequence is:

Freeze the decision-time allocation.
Record the probabilities in force at the drop decision and list exactly which arms remain open. The update applies to future assignments, not to participants already randomized.

Compute removed and surviving mass.
Calculate \(r=\sum_{j\in D}\pi_j\) and \(s=1-r\). Confirm that \(s>0\) and that the retained control has the status required by the design.

Apply the prespecified update.
Use conditional renormalization only when it satisfies all constraints. Otherwise apply the protocol's constrained redistribution rule, not an improvised adjustment.

Check the result before release.
Confirm that dropped-arm probabilities are zero, surviving probabilities are nonnegative, their sum is one, and every control-allocation limit is satisfied. Check the implemented values, not only the written calculation.

If allocation probabilities vary by covariate stratum, the update must be evaluated for the relevant stratum-specific probability vector whenever the design specifies stratum-specific randomization. A pooled calculation can conceal a violation within a stratum. Similarly, rounding should not be allowed to make the final probabilities fail to sum to one or cross a bound; retain adequate precision internally and define how any displayed rounding is handled.

The principal alternatives are not interchangeable. Conditional renormalization preserves all surviving arm-to-arm ratios, including control-to-treatment ratios, but may violate a control constraint. Constrained redistribution enforces the specified control limit while preserving the relative weights of surviving noncontrol arms under the rule shown; it changes their ratio to control. A fixed-control-target rule is the special case \(L=U\). Other designs might allocate the freed mass equally among surviving experimental arms or recalculate allocation weights from current information, but those choices have different consequences and must be specified in advance.

Common pitfall: “An arm was removed, so divide every surviving probability by the surviving total; the control allocation must increase.” The formula is valid as conditional renormalization, but the conclusion needs qualification: control does not decrease and rises strictly only when positive mass was removed and its original probability was positive. Even then, the new value may exceed a control cap. Check the actual value and use the prespecified constrained rule if needed.

These rules do not determine whether an arm should be dropped; they update the allocation distribution after that decision has been made. Nor does a valid probability vector by itself guarantee that the full design is appropriate. The decision rule, constraints, and update policy should be documented together so that randomization remains reproducible and the allocation actually used can be reconstructed.

Check Your Understanding

For each question, calculate or identify the post-drop rule, then compare your work with the solutions.

Control has probability \(0.20\), and two surviving treatments have probabilities \(0.50\) and \(0.30\). A fourth arm with probability \(0.10\) is dropped, while the other probabilities are unchanged from a normalized vector. What is wrong with this description, and how would you repair the calculation?
Suppose control has probability zero and a dropped arm has positive probability. Under conditional renormalization, does control allocation rise? State the correct condition for a strict rise.
Before a drop, control has probability \(0.24\), treatments \(T_1,T_2\) have probabilities \(0.36,0.24\), and \(T_3\) has probability \(0.16\). Drop \(T_3\), impose a control cap of \(0.30\), and use constrained redistribution. Calculate all surviving probabilities.
Name two checks that should be completed before the updated probabilities are used for future randomization.

Solutions. (1) The stated probabilities sum to \(0.20+0.50+0.30+0.10=1.10\), so they cannot be a probability vector. Repair the input by checking the actual allocation probabilities in force; do not renormalize an inconsistent list as though it were valid. (2) No. If \(\pi_C=0\), then \(\pi_C^{+}=0/s=0\). With \(s<1\), the control probability rises strictly exactly when \(\pi_C>0\). (3) The surviving mass is \(s=1-0.16=0.84\), and the raw control value is \(0.24/0.84=0.2857\), below the cap, so the cap does not bind. Conditional renormalization gives \(\pi_C^{+}=0.2857\), \(\pi_{T_1}^{+}=0.36/0.84=0.4286\), and \(\pi_{T_2}^{+}=0.24/0.84=0.2857\). These sum to \(1.0000\) to the displayed precision. (4) For example, check that the surviving probabilities are nonnegative and sum to one, dropped arms have probability zero, and the control constraint is met; also verify that the implemented vector matches the documented rule.