Why Allocation Ratios Change
In a multi-arm platform trial, the number of active treatment comparisons can change over calendar time. An arm may enter while other arms remain open, or a treatment may stop recruiting. If the randomization probabilities stay fixed, the number of control participants accruing alongside each open treatment may no longer match the trial’s intended allocation. The key calculation is to translate each comparison’s desired concurrent control accrual into probabilities for the arms open at that time.
A control participant can contribute to more than one comparison when several treatment arms are open concurrently. The control allocation therefore cannot be calculated separately for each treatment and then added up as though each comparison had its own control group. Instead, one shared control probability must satisfy the targets for all open comparisons at once.
Write \(0\) for the shared control arm and let \(J\) denote the set of treatment arms currently open. The allocation probabilities are chosen so that the control-to-treatment probability ratio equals the target for every open comparison. These probabilities describe the planned randomization in that period, not a guarantee that the realized numbers will be exactly in the target ratios.
Deriving the Probabilities
Let \(p_0\) be the probability of assignment to control and \(p_j\) the probability of assignment to open treatment arm \(j\). For each comparison, the target requires \(p_0/p_j=\rho_j\), or equivalently \(p_j=p_0/\rho_j\). Because all assignments must go to one of the open arms or control, the probabilities must sum to 1.
Substitution gives \(p_0\left(1+\sum_{j\in J}1/\rho_j\right)=1\). Solving for each probability yields the allocation rule:
The rule has a direct interpretation: each open treatment arm receives allocation weight \(1/\rho_j\), and control receives weight 1. The weights are divided by their sum to obtain probabilities. In the special case where all \(k\) open treatments have \(\rho_j=1\), control and every treatment arm have equal probabilities \(1/(k+1)\). The combined treatment-to-control allocation ratio is then \(k:1\), although each individual treatment-to-control comparison has a \(1:1\) target.
For \(N\) eligible randomized participants during a period, the expected count in an arm is \(Np\). In particular, \(E(N_0)/E(N_j)=p_0/p_j=\rho_j\). This is a statement about expected accrual; random variation means realized counts need not have precisely that ratio.
Worked Allocation Calculations
Worked Example: Equal Targets as Arms Enter
Suppose all open treatment comparisons target \(\rho_j=1\). In a first period, only arm A is open and 60 eligible participants are expected to be randomized. In a second period, arms A and B are open with 120 eligible participants. Find the allocation probabilities and expected counts in each period.
First period. The active set is \(J=\{A\}\). The control probability is \(p_0=1/(1+1)=0.5\), and \(p_A=0.5\). Expected counts are \(60(0.5)=30\) for control and \(60(0.5)=30\) for A.
Second period. Now \(J=\{A,B\}\), so \(p_0=1/(1+1+1)=1/3\), and \(p_A=p_B=1/3\). The expected counts are \(120/3=40\) in control, 40 in A, and 40 in B. The shared 40 control participants accrue concurrently with both open treatment arms; they are not two separate groups of 40.
The combined treatment-to-control ratio changes from \(1:1\) to \(2:1\), while each open treatment comparison retains its individual \(1:1\) target. The calculations presume the specified participants are eligible for all listed arms and are randomized during the stated period.
Worked Example: Different Control Targets Across Comparisons
Suppose arms A and B are open. The design targets one concurrent control participant per participant in A, so \(\rho_A=1\), and two concurrent control participants per participant in B, so \(\rho_B=2\). Calculate the allocation probabilities and expected counts for 100 eligible participants.
The unnormalized weights are 1 for control, \(1/\rho_A=1\) for A, and \(1/\rho_B=1/2\) for B. Their sum is \(1+1+1/2=2.5\). Therefore,
For 100 participants, the expected counts are 40 control, 40 in A, and 20 in B. The resulting expected ratios are \(40/40=1\) control per A participant and \(40/20=2\) controls per B participant. The same 40 controls support both open comparisons; the different targets are achieved by assigning fewer participants to B relative to the shared control.
Worked Example: Updating Probabilities When an Arm Closes
Suppose arms A, B, and D are initially open, each with a \(1:1\) control-to-treatment target. In one period, 120 eligible participants are randomized. Arm B then closes to new accrual. In the next period, 100 participants are eligible for the still-open arms A and D; the targets are \(\rho_A=1\) and \(\rho_D=2\). Calculate the expected allocations in both periods.
With three open arms and equal targets, each arm, including control, has probability \(1/4\). For the first period, the expected counts are 30 in control and 30 in each of A, B, and D.
After B closes, the active set for new assignments is \(J=\{A,D\}\). The weights are 1 for control, 1 for A, and \(1/2\) for D, with sum 2.5. Thus \(p_0=0.4\), \(p_A=0.4\), and \(p_D=0.2\). The expected counts among the next 100 participants are 40 control, 40 in A, and 20 in D.
The calculation changes prospectively for assignments made while B is closed. The earlier control accrual occurred while B was open; it does not change the new-period probabilities. Whether control participants from a particular period belong in an analysis of a particular comparison depends on the comparison’s concurrent eligibility and analysis rules.
Assumptions, Interpretation, and Practical Checks
This calculation presumes that the trial has specified which treatment arms are open for randomization in each period, the eligible population to which the probabilities apply, and a desired control-to-treatment accrual target for each open comparison. It also presumes that assignments are randomized according to the stated probabilities, and that the shared control is eligible to serve as a concurrent comparison for each listed treatment. If eligibility differs by arm, probabilities need to be calculated for the set of arms each participant can actually enter, rather than applied indiscriminately to everyone.
For example, if an incoming participant is not eligible for arm D, that participant is not part of the set to which a three-arm probability calculation applies. Applying the same probabilities to a population with materially different arm eligibility can produce accrual that does not meet the intended comparison-specific targets. Likewise, a target ratio does not itself guarantee enough control accrual: the total number of eligible participants and the duration of overlapping recruitment still matter.
A practical check is to calculate the implied ratios from the probabilities before using them: for each open arm \(j\), verify \(p_0/p_j=\rho_j\), and verify that all probabilities are positive and sum to 1. After randomization, compare observed counts with expected counts while recognizing that finite samples fluctuate. A discrepancy in realized counts is not, by itself, evidence that the planned probability calculation was wrong; it may reflect ordinary random variation. The randomization record should be checked if observed allocation is inconsistent with the probabilities actually specified.
A common pitfall is to assign control probability \(1/2\) for every \(1:1\) comparison and then add those control allocations across the open arms. With two open arms, that would imply \(1/2\) for control, \(1/2\) for A, and \(1/2\) for B, which sum to \(1.5\) and cannot be probabilities for one assignment. The correction is to normalize the shared-control weights: with two equal targets, the probabilities are \(1/3\) for each of control, A, and B.
How This Choice Compares With Alternatives
| Approach | What it does | Main consideration |
|---|---|---|
| Recalculate ratios for the current open arms | Uses the target for each active comparison to determine the current allocation probabilities. | Directly reflects changes in the number of open arms and can accommodate different comparison targets. |
| Keep a fixed allocation ratio as arms enter or leave | Retains the same arm probabilities even when the active set changes. | The resulting control-to-treatment ratios may no longer match the stated targets for every open comparison. |
| Use a separate control group for every comparison | Allocates control participants separately to individual treatment comparisons. | Avoids sharing the same control accrual across comparisons, but does not use a single shared control allocation. |
The ratio calculation is an allocation rule, not an analysis adjustment. It does not correct for calendar-time differences, changes in eligibility, or other differences between participants recruited in separate periods. The earlier material on concurrent control sets concerns which controls are concurrent for an open comparison; allocation ratios determine how new eligible participants are assigned while those comparisons are open. Those are related but distinct design questions.
There are also limits to interpreting a target as a promise about final sample sizes. A ratio controls assignment probabilities, not how many participants will enroll, how long an arm remains open, or how many participants meet the relevant eligibility criteria. If an arm has few eligible participants or closes soon after opening, its realized concurrent accrual may be small even when the probabilities were correctly set.
For the period being planned, identify the treatment arms open to the eligible participants and the shared control arm.
Record \(\rho_j\) as control accrual divided by treatment accrual for comparison \(j\), so that a value above 1 means more control than treatment.
Use control weight 1 and treatment weight \(1/\rho_j\); divide every weight by the sum of all weights.
Confirm probabilities sum to 1 and \(p_0/p_j=\rho_j\) for each open comparison. Recalculate when the active set or targets change, and apply each calculation only to the eligible participants it describes.
Check Your Understanding
For each question, calculate or explain the allocation using the definitions in this tutorial. Solutions follow the questions.
- Two open treatments each target \(\rho_j=1\). What are the probabilities for control and each treatment? What is the combined treatment-to-control ratio?
- One treatment targets \(\rho_A=2\), and 90 eligible participants will be randomized. Find the probabilities and expected counts.
- Arms A and B are open with \(\rho_A=1\) and \(\rho_B=2\). What are the allocation probabilities? Verify both target ratios.
- Explain why an observed control-to-treatment count ratio need not equal the specified target exactly.
Solutions. (1) The weights are 1 for control and 1 for each treatment, so each probability is \(1/3\). The combined treatment-to-control ratio is \(2:1\). (2) The weights are 1 for control and \(1/2\) for A, summing to 1.5; therefore \(p_0=2/3\) and \(p_A=1/3\). Expected counts are \(90(2/3)=60\) control and \(90(1/3)=30\) in A, giving \(60/30=2\). (3) The weights are 1, 1, and \(1/2\), summing to 2.5, so \(p_0=0.4\), \(p_A=0.4\), and \(p_B=0.2\). The ratios are \(0.4/0.4=1\) for A and \(0.4/0.2=2\) for B. (4) The target specifies assignment probabilities and hence expected accrual ratios. Random allocation produces variable realized counts, especially with small numbers of participants, so observed counts can differ from their expectations.