Discrete Sample Sizes and Rounding

From a Calculated Target to a Feasible Design

A sample-size re-estimation rule may produce a noninteger target, such as 137.4 participants per arm. That value can be meaningful as the solution to a continuous calculation, but a trial cannot enroll 0.4 of a participant. The operational design must use whole numbers and respect its allocation scheme and prespecified limits. Rounding is therefore part of implementing the re-estimation rule, not a cosmetic final step.

This tutorial focuses on converting a calculated target into a feasible final allocation. It assumes that the re-estimation calculation itself has already produced a target under a stated statistical model, and that the design has specified its allocation ratio and sample-size bounds. The key question is: what is the smallest feasible integer design that preserves the intended target, subject to those constraints?

Definition: Let \(n^*\) denote a continuous sample-size target per arm in a two-arm, equally allocated design. A feasible equal allocation uses an integer \(n\) in each arm, for total sample size \(N=2n\). Let \(n_{\min}\) and \(n_{\max}\) be prespecified integer bounds per arm. The re-estimated final allocation must be selected from the feasible set \(n_{\min}\leq n\leq n_{\max}\), with any additional requirement to account for participants already enrolled.

Always label whether a quantity is per arm or total. A target of 140 per arm means 280 participants in total under 1:1 allocation; a target of 140 total does not. Mixing these interpretations can change the design by a factor of two.

Why Rounding Up Is Often the Correct Rule

If \(n^*\) is the smallest continuous sample size meeting a target, and the relevant performance criterion is nondecreasing in \(n\), the smallest integer sample size that does not fall below the target is the ceiling, \(\lceil n^*\rceil\). The ceiling is the least integer greater than or equal to \(n^*\). It does not round every number upward to a different value: if \(n^*\) is already an integer, it stays unchanged.

n_{\mathrm{ceil}}=\lceil n^*\rceil,\qquad N_{\mathrm{ceil}}=2\lceil n^*\rceil

The justification is direct. Every feasible integer \(n\) satisfying \(n\geq n^*\) must be at least \(\lceil n^*\rceil\); and \(\lceil n^*\rceil\) itself is an integer satisfying that inequality. Thus it is the smallest such integer. This argument preserves the calculated target only when the criterion used to obtain \(n^*\) is monotone in sample size. If the calculation has a different objective, nearest-integer rounding may be appropriate, but that choice should follow the objective rather than habit.

Worked Example: Rounding an Equal-Allocation Target

Suppose a hypothetical 1:1 trial’s re-estimation calculation returns \(n^*=137.4\) participants per arm. The design requires between 80 and 180 participants per arm, inclusive, and 112 participants have already been enrolled in each arm. Assume the target is the smallest per-arm size intended to meet a criterion that increases with sample size.

First round up to retain the calculated target:

\lceil 137.4\rceil=138

The lower bound and the number already enrolled do not force a larger value, since \(138\geq80\) and \(138\geq112\). The upper bound is also satisfied, since \(138\leq180\). The final allocation is therefore 138 participants per arm, or

N=2(138)=276\text{ participants in total.}

Rounding to 137 per arm would give 274 participants, but \(137<137.4\); it would not preserve the specified continuous target. This does not by itself tell us the exact power at either integer sample size. It tells us which integer does not undershoot the calculated threshold, under the stated monotonicity condition.

Bounds, Enrollment, and Feasibility

A design’s lower and upper limits are constraints on the final planned allocation, not instructions to alter the underlying calculation silently. For equal allocation, define the lower feasible value as the larger of the prespecified minimum and the number already enrolled per arm, when enrollment is balanced. Then apply the upper limit:

n_{\mathrm{final}} = \min\!\left\{ n_{\max}, \max\!\left(n_{\min},\,n_{\mathrm{current}},\,\lceil n^*\rceil\right) \right\}.

Here \(n_{\mathrm{current}}\) is the number already enrolled in each arm. The formula presumes the current allocation is balanced and the bounds are per arm. If the lower requirement exceeds \(n_{\max}\), the constraints are infeasible: no allocation can both respect the maximum and include those already enrolled. That conflict needs an explicit design decision, not an arithmetic workaround.

If the rounded target exceeds the maximum, setting the final allocation to the maximum respects the cap but may not meet the original target. In particular, it is not valid to state that a conditional-power target or planned power has been reached merely because the maximum sample size was used. The achieved operating characteristics at the capped size must be evaluated under the relevant assumptions.

Worked Example: Applying a Maximum and Reporting Its Consequence

Consider a hypothetical equal-allocation trial with \(n^*=211.6\) participants per arm, \(n_{\min}=100\), \(n_{\max}=200\), and 120 participants already enrolled in each arm. The continuous target is rounded up:

\lceil211.6\rceil=212.

The lower requirement is \(\max(100,120,212)=212\), which exceeds the maximum of 200. Applying the cap gives

n_{\mathrm{final}}=\min(200,212)=200,\qquad N=2(200)=400.

The resulting allocation is 200 participants per arm, 400 in total. It is feasible only if the design’s maximum applies to the final total allocation and the enrollment already accrued is no greater than that cap. Because 200 is below the rounded target of 212, the trial has not retained the calculated target. Its power or conditional-power goal should be described as potentially unmet unless a separate calculation at \(n=200\) shows otherwise.

Preserving a Fixed Allocation Ratio

For a fixed ratio other than 1:1, rounding each arm independently can break the ratio. A useful solution is to define a feasible allocation unit. If the intended treatment-to-control ratio is \(a:b\), with positive integers \(a\) and \(b\), one complete unit contains \(a\) treatment participants and \(b\) control participants. Feasible allocations use an integer number \(m\) of these units:

n_T=am,\qquad n_C=bm,\qquad N=(a+b)m.

If a continuous calculation specifies a required total \(N^*\), the smallest number of complete units meeting that total is \(m=\lceil N^*/(a+b)\rceil\), provided the target is a minimum-total requirement and the performance criterion is monotone in the relevant allocation. Bounds must also be translated into permissible values of \(m\). If the target is specified separately by arm, choose a unit count that satisfies both arm requirements; do not round the arms separately and then assume the ratio remains exact.

Worked Example: Rounding a 2:1 Allocation to Complete Units

Suppose a hypothetical trial uses a treatment-to-control allocation ratio of 2:1, and its continuous calculation gives a total target \(N^*=250\). Randomization is conducted in complete units of three participants: two assigned to treatment and one to control. The design permits totals from 180 to 300.

The required number of complete units is

m=\left\lceil\frac{250}{2+1}\right\rceil =\lceil83.333\ldots\rceil=84.

Thus the feasible allocation is

n_T=2(84)=168,\qquad n_C=84,\qquad N=168+84=252.

The total of 252 is above the target of 250 and within the specified bounds. The ratio remains exactly 2:1. Rounding 250 to the nearest integer would leave the total at 250, which cannot be allocated in complete three-person units while maintaining this ratio.

A Binary-Outcome Target Before Rounding

For a binary outcome, the earlier material on re-estimation for binary outcomes provides model-based ways to obtain a continuous sample-size target. The rounding step is separate from that calculation. The following illustration shows how a familiar equal-allocation approximation leads to an integer design; it does not replace checking the assumptions of the model or any design-specific analysis.

Worked Example: Rounding a Two-Proportion Approximation

Assume a hypothetical two-arm trial compares independent binary outcomes, with expected control event probability \(p_C=0.30\), treatment event probability \(p_T=0.45\), two-sided significance level \(\alpha=0.05\), and target power \(1-\beta=0.80\). Use the normal-approximation formula for equal allocation, treating the stated probabilities as planning values. It assumes independent participants, the specified event probabilities, and an approximation based on the standard normal quantiles.

With \(\bar p=(p_C+p_T)/2=0.375\), \(\delta=p_T-p_C=0.15\), \(z_{1-\alpha/2}=1.96\), and \(z_{1-\beta}=0.8416\), the continuous required number per arm is

n^*= \frac{ \left[ z_{1-\alpha/2}\sqrt{2\bar p(1-\bar p)} + z_{1-\beta}\sqrt{p_C(1-p_C)+p_T(1-p_T)} \right]^2 }{\delta^2}.

Substitute the values:

\begin{aligned} n^* &= \frac{ \left[ 1.96\sqrt{2(0.375)(0.625)} + 0.8416\sqrt{(0.30)(0.70)+(0.45)(0.55)} \right]^2 }{(0.15)^2}\\ &= \frac{\left[1.96\sqrt{0.46875}+0.8416\sqrt{0.4575}\right]^2}{0.0225}\\ &\approx \frac{(1.3419+0.5692)^2}{0.0225} \approx 162.3. \end{aligned}

The calculated target is approximately 162.3 participants per arm. With a per-arm minimum of 120 and maximum of 200, rounding up gives \(n=163\) per arm and \(N=326\) total. This arithmetic implements the approximate formula’s target; it does not make the normal approximation exact. In a real design, the analyst should verify that the approximation is adequate for the planned event probabilities and sample size, and use the design’s prespecified method if it differs.

Diagnostics, Alternatives, and Common Mistakes

Before finalizing an integer allocation, check the units, constraints, and rounding objective. A simple audit is to calculate both the per-arm counts and the total, then verify that the stated allocation ratio and bounds hold. For an adaptive design, also check that the proposed final target is not below the number already enrolled when the design prohibits reducing sample size. If allocations are currently unbalanced, a single \(n_{\mathrm{current}}\) per arm is not enough: determine the additional assignments needed from the actual arm counts and the planned randomization scheme.

Choice	When it is appropriate	Main caution
Ceiling	The calculated value is a minimum threshold and the criterion is nondecreasing with sample size.	Can add participants; apply the design’s allocation unit and bounds.
Nearest integer	The goal is to approximate a continuous optimum, rather than guarantee a minimum threshold.	May round below a target defined as the minimum required size.
Floor	Only when a stated objective or constraint calls for the largest integer not exceeding a value.	Usually undershoots a minimum sample-size target.
Complete allocation units	A fixed ratio or blocked allocation requires specific integer increments.	May increase total sample size beyond simple ceiling; check bounds in units.

A frequent mistake is to round \(137.4\) to 137 because it is the nearest integer, then claim that the planned threshold has been met. The correction is to identify what the calculated value means. If it is the minimum size intended to attain a monotone target, use 138, subject to feasibility. If it is an approximate optimizer, nearest-integer rounding might instead be justified, but the resulting design should be evaluated at that integer.

Another mistake is to treat a sample-size cap as evidence that the target has been achieved. A cap is an operational constraint. When it binds, report the capped allocation and assess the resulting performance; do not describe the uncapped target as attained. Likewise, a fractional allocation cannot be made feasible by rounding treatment and control counts independently if doing so violates a required ratio.

Key takeaway: State the target’s units and objective, choose integer allocations consistent with the allocation scheme, then apply the minimum, maximum, and enrollment constraints. If a constraint forces a sample size below the calculated target, report that limitation rather than implying the target was met.

Check Your Understanding

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

A 1:1 design has a minimum-target calculation of 95.2 participants per arm. What is the smallest integer allocation that does not undershoot the target, and what is the total?
A 1:1 design has a rounded target of 154 per arm, a maximum of 150 per arm, and 130 already enrolled in each arm. What allocation results from the cap, and what must not be claimed without further calculation?
A 3:2 allocation requires complete units of five participants, and the target total is 207. How many units are needed, and how many participants are assigned to each arm?
Why can nearest-integer rounding be inappropriate when a calculated sample size is defined as the smallest size meeting a power target?

Solutions. (1) \(\lceil95.2\rceil=96\) per arm, for \(N=2(96)=192\). (2) The capped allocation is 150 per arm, for \(N=300\). Because the cap is below the rounded target of 154 per arm, one must not claim the target power or conditional-power criterion was met without evaluating performance at 150. (3) There are \(\lceil207/5\rceil=42\) complete units, giving 126 participants in the 3-part arm and 84 in the 2-part arm, for a total of 210. (4) Nearest rounding can produce an integer below the minimum threshold; if meeting the criterion requires \(n\geq n^*\), the smallest qualifying integer is \(\lceil n^*\rceil\), not necessarily the nearest integer.