Rounding Stage Sizes Without Breaking Design Constraints

Why Integer Rounding Is a Design Decision

A sample-size calculation may return stage sizes such as 20.6 participants in an experimental arm and 41.2 in a shared control arm. These values can be useful for locating a promising design, but participants cannot be enrolled fractionally. More importantly, rounding changes the information available at each analysis. That can change power, the probability of stopping early, expected enrollment, and whether the maximum enrollment is respected.

The goal is therefore not simply to round each number to its nearest integer. It is to find an integer allocation that satisfies the design’s operational constraints and then verify its operating characteristics using the rounded allocations. The numerical search for continuous stage sizes supplies a starting point; the final design must be evaluated as the design that would actually be run.

Definition: Let \(n_{a,k}\) be the integer number assigned to arm \(a\) at stage \(k\), where \(a=0\) denotes the control arm and \(a=1,\ldots,m\) denotes the experimental arms. The rounded design is feasible if all \(n_{a,k}\) are integers and the allocation, power, enrollment, and other prespecified design constraints hold. Write \(\mathcal F\) for the set of feasible integer allocation vectors.

Constraints can include minimum stage sizes, allocation ratios, a maximum total enrollment, and a minimum target power. Some are direct arithmetic conditions. Others, especially power and expected enrollment under early stopping, must be recalculated from the complete MAMS design: its stage sizes, decision rules, outcome model, and multiplicity procedure. A continuous solution does not guarantee that any nearby integer vector satisfies every condition.

What Must Be Rechecked

Before rounding, state the assumptions used to evaluate the design. For example, suppose the planned outcomes follow the specified continuous or binary outcome model, the assumed treatment effects and nuisance parameters are fixed for this calculation, recruitment follows the stated arm allocation, and the interim and final decision rules are fixed. The power calculation must use those same assumptions after rounding. If the outcome model, decision boundaries, or randomization scheme changes, the resulting power is not a check of the original design.

For each candidate integer allocation, check at least the following:

Allocation: all stage sizes are positive integers where required, and arm ratios meet the protocol’s definition (exactly or within a prespecified tolerance).
Power: the relevant target, such as power to reject at least one null hypothesis or power for a specified active arm, remains met under the planned stopping and multiplicity rules.
Enrollment: maximum enrollment is calculated from the actual stagewise allocations and stopping rules. The worst-case total is not the same as expected enrollment.
Other design constraints: any minimum recruitment, stage-size, or expected-enrollment requirements are still satisfied.

A convenient way to organize the task is to define the feasible set before choosing an allocation. For example, let \(\tilde n_{a,k}\) be the continuous candidate, \(P_{\text{target}}\) the power target, \(N_{\max}\) the maximum enrollment, and \(P(n)\) the power evaluated under the specified MAMS model and rules. Then a simplified feasible set is:

\mathcal F = \left\{ \mathbf n: n_{a,k}\in\mathbb Z_{>0},\ \text{allocation constraints hold},\ P(\mathbf n)\ge P_{\text{target}},\ N_{\max}(\mathbf n)\le N_{\max} \right\}.

The notation \(N_{\max}(\mathbf n)\) here means the design’s maximum enrollment under allocation \(\mathbf n\), not a power calculation. The constraints should be expanded to match the actual protocol; the abbreviated set above is not a substitute for specifying them.

A Practical Rounding Procedure

Record the continuous candidate and constraints.
Keep every arm and stage allocation, including the shared control. Write down the target power, allocation-ratio rule, and enrollment limit before rounding.

Generate nearby integer candidates.
Consider floor and ceiling values, ratio-compatible increments, and small coordinated changes across stages. Independent nearest-integer rounding is only one candidate, not a guarantee of feasibility.

Filter by arithmetic constraints.
Reject candidates that violate integer, ratio, minimum-size, or maximum-enrollment rules. Calculate total enrollment from the allocations that can actually be recruited.

Recalculate operating characteristics.
For each remaining candidate, evaluate power and any other required characteristics using the fixed model and decision rules. Do not carry forward the continuous design’s power as if it were unchanged.

Select and document a feasible design.
If several candidates meet every constraint, choose among them using a stated criterion, such as closeness to the continuous solution or lower maximum enrollment. If none do, revise the design rather than labeling an infeasible candidate acceptable.

Worked Rounding Examples

Worked Example: Ratio-Preserving Rounding Can Change the Choice

Hypothetical design. Consider two experimental arms and a shared control over two stages. A continuous search gives stage 1 allocations \((\tilde n_{0,1},\tilde n_{1,1},\tilde n_{2,1})=(40.4,20.2,20.2)\) and stage 2 allocations \((30.4,15.2,15.2)\). The intended control-to-each-treatment ratio is 2:1, the target power is 0.90 for the prespecified design criterion, and maximum enrollment is 150. Assume the outcome model, treatment effects, stopping rules, and multiplicity procedure are fixed. The hypothetical operating-characteristic calculation for each candidate below uses those same assumptions.

Rounding each value to the nearest integer gives stage 1 \((40,20,20)\) and stage 2 \((30,15,15)\). The ratio is exactly 2:1 at both stages. Maximum enrollment is \(40+20+20+30+15+15=140\). Suppose the recalculated power is 0.887. This candidate meets the enrollment limit but fails the 0.90 power target.

A ratio-preserving alternative is stage 1 \((40,20,20)\) and stage 2 \((34,17,17)\). The stage 2 ratio remains \(34:17=2:1\), and maximum enrollment is \(40+20+20+34+17+17=148\), within the limit. Suppose recalculated power is 0.902. This candidate meets both constraints. The increase in stage 2 allocation is a deliberate adjustment to restore power, not a consequence of ordinary rounding.

These power values are hypothetical outputs from the specified operating-characteristic calculation; they are not obtained by rounding 0.90 or by multiplying the original power by an enrollment ratio. The conclusion is that the second integer design is feasible under the stated constraints, while the nearest-integer candidate is not.

Worked Example: Checking Tolerated Allocation Ratios and Enrollment

Hypothetical design. In a two-stage design with two experimental arms, suppose the continuous stage allocations are \((25.3,16.9,16.9)\) and \((20.2,13.5,13.5)\), ordered as control, arm 1, arm 2. The intended control-to-treatment ratio is 1.5:1, with a permitted deviation of at most 10% at each stage. The target power is 0.80 and maximum enrollment is 110. Assume the planned outcome model and all analysis rules are fixed, and use a full recalculation for the rounded design.

Round to stage 1 \((25,17,17)\) and stage 2 \((20,14,14)\). Stage 1 has control-to-treatment ratio \(25/17\approx1.471\), which is \(100((25/17)-1.5)/1.5\approx-1.96\%\) from target. Stage 2 has ratio \(20/14\approx1.429\), or \(100((20/14)-1.5)/1.5\approx-4.76\%\). Both deviations are within 10%.

The maximum enrollment is \(25+17+17+20+14+14=107\), so it is below 110. Suppose the recalculated target power is 0.806. This integer allocation meets the stated ratio tolerance, enrollment limit, and power target. The check is stage-specific: a ratio that is acceptable after pooling stages would not automatically satisfy a requirement imposed separately at each stage.

Worked Example: Choosing Among Feasible Candidates With a Complete Distance

Hypothetical design. A one-stage allocation search gives continuous sizes \((\tilde n_{0,1},\tilde n_{1,1},\tilde n_{2,1})=(30.4,15.2,15.2)\), with a required exact 2:1 control-to-treatment ratio for each experimental arm. Target power is 0.90 and maximum enrollment is 64. A hypothetical operating-characteristic calculation gives power 0.88 for \((30,15,15)\) and 0.91 for \((32,16,16)\).

The first candidate has total enrollment \(30+15+15=60\), but fails the power target. The second has total \(32+16+16=64\), preserves both ratios, and meets the power target. It is feasible, exactly at the enrollment limit.

To quantify closeness to the continuous candidate, use positive weights \(w_{a,k}\) and include every arm, including control:

D(\mathbf n,\tilde{\mathbf n}) = \sum_k w_{0,k}(n_{0,k}-\tilde n_{0,k})^2 + \sum_{a=1}^{m}\sum_k w_{a,k}(n_{a,k}-\tilde n_{a,k})^2.

With all weights equal to 1, the distance for \((30,15,15)\) is \((30-30.4)^2+(15-15.2)^2+(15-15.2)^2=0.16+0.04+0.04=0.24\). For \((32,16,16)\), it is \((32-30.4)^2+(16-15.2)^2+(16-15.2)^2=2.56+0.64+0.64=3.84\). The closer candidate is not feasible because its power is 0.88. If the second candidate is the only feasible one considered, it is the choice despite its larger distance. In general, minimize this distance over \(\mathbf n\in\mathcal F\), not over all integer vectors.

Why the Distance Objective Needs Every Arm

A distance objective is useful when several feasible integer designs remain, but its meaning depends on what it includes. If \(a\) indexes experimental arms only, then a sum over \(a\) omits the control allocations. Two candidate designs with identical experimental sizes but different control sizes would receive the same distance, even if one control allocation is much closer to the continuous solution. That is not a measure of total allocation closeness.

The complete objective in the example includes \(a=0\) explicitly and gives every component a positive weight. Equal weights treat a one-person discrepancy equally in every arm and stage. Other positive weights can reflect a prespecified priority or scaling, but the weighting rule should be reported because changing it can change which feasible vector is selected. The objective is a selection criterion; it does not establish power or feasibility.

Diagnostics, Alternatives, and Limitations

Several checks help identify a rounding process that is going wrong:

Power is copied from the continuous solution. Recompute it at the integer allocations. Stagewise changes can alter information and the chance of crossing stopping boundaries.
Ratios are checked only in aggregate. If ratios are required at each stage, assess each stage separately. Shared control makes it especially important to use the correct control allocation in every comparison.
Expected enrollment is confused with the cap. Expected enrollment can be below a budget while the maximum possible enrollment exceeds it. Check each quantity against the constraint that applies to it.
The control arm is omitted from the search criterion or total. Include control participants in both the distance calculation and enrollment arithmetic.
No nearby candidate meets all conditions. This is a substantive finding, not a rounding error to hide. Expand the search or revise the continuous target, allocation ratio, boundaries, or enrollment limit, then re-evaluate the resulting design.

Independent rounding is quick and may be adequate when constraints are loose, but it can violate exact ratios or miss the power target. Rounding up every component can be conservative for some information measures, but it may exceed a cap and does not guarantee that all MAMS operating characteristics improve in a simple, uniform way. A constrained integer search is more systematic: generate or optimize over integer vectors, enforce the arithmetic constraints, and evaluate operating characteristics for candidate feasible allocations. Its limitation is computational: the search can be large, and its result is only as sound as the model and calculation used to evaluate each candidate.

Rounding also cannot remove uncertainty in assumed effects, outcome variability, recruitment, or adherence. The selected allocation is feasible relative to the assumptions and criteria used; it is not guaranteed to achieve the planned power in every realized trial. The practical importance of the check is that the final protocol is internally consistent: the stated integer recruitment plan, analysis rules, power target, and enrollment limits refer to the same design.

Key takeaway: Treat integer rounding as constrained design selection. Include control allocations, verify ratio and enrollment rules at the level where they apply, and recompute power and other operating characteristics for the rounded MAMS design.

Check Your Understanding

Answer each question using the definitions and checks in this tutorial.

A continuous stage size is 18.6 in an experimental arm and 37.2 in its control. Why does rounding each number independently not, by itself, establish that the final design meets its constraints?
A feasible set requires power at least 0.85 and maximum enrollment at most 100. Candidate A has power 0.84 and maximum enrollment 92; candidate B has power 0.86 and maximum enrollment 101. Which candidate is feasible?
Why should a distance-to-continuous-solution objective include \(n_{0,k}\), even when control is shared across several comparisons?
A two-stage design meets a 2:1 allocation ratio after adding both stages, but stage 1 has ratio 1.7:1. What should be checked before accepting it?

Solutions. (1) Independent rounding can change ratios, power, stagewise information, and enrollment. Each applicable condition must be checked on the integer design, with power recalculated under the fixed model and decision rules. (2) Neither is feasible: A fails power, while B exceeds the enrollment cap. (3) Control participants are part of the allocation vector. Omitting them means designs with different control sizes can have identical reported distance, so the objective does not measure closeness of the complete design. (4) Check whether the ratio constraint applies separately at each stage. If it does, the stage 1 ratio violates the requirement despite the pooled ratio; if the rule is explicitly pooled, assess against that rule and still verify all other constraints.