Designing A Sample Size Re-Estimation Rule

Why Specify the Rule Before the Interim

A sample-size re-estimation design uses information collected during a trial to revise the planned number of participants. The purpose may be to improve the chance of answering the study question when a nuisance parameter, such as outcome variability, differs from its planning value. The interim calculation is not an invitation to choose a sample size after seeing what seems favorable. Its inputs, calculation, allowable range, and response to infeasible values should be written down before the interim data are examined.

This tutorial develops a rule for a two-arm comparison of means with equal allocation. It is a planning illustration, not a claim that one formula fits every endpoint or analysis. The re-estimation rule and the final inferential procedure are parts of one design: the latter must account for how the former uses interim information.

Definition: A sample-size re-estimation rule is a prespecified function that maps an interim information set to a final sample-size target. A complete rule identifies the interim look and data used, the nuisance parameter and its estimator, the target effect and error-rate inputs, allowable sample-size limits, the calculation and rounding convention, feasibility conditions, and the action taken when a condition fails.

Model and Planning Calculation

Suppose the primary outcome is continuous, the treatment and control groups are independently randomized, allocation is equal, and the outcome has a common within-group variance $\sigma^2$. Let $\delta_*>0$ be the smallest treatment-minus-control mean difference that the design aims to detect. It is a planning value, not an estimate to be substituted from an interim treatment-effect result. The design specifies a two-sided type I error level $\alpha$ and target power $1-\beta$.

For a normal-approximation comparison with $n$ participants per arm, the standard error of the difference in sample means is approximately $\sigma\sqrt{2/n}$. Under the planning alternative with difference $\delta_*$, the standardized signal is $\delta_*/(\sigma\sqrt{2/n})$. Requiring this signal to exceed the two-sided critical value plus the normal quantile for the target power gives the approximate planning equation below.

\frac{\delta_*}{\sigma\sqrt{2/n}} \approx z_{1-\alpha/2}+z_{1-\beta} \qquad\Longrightarrow\qquad n_{\mathrm{req}} = \frac{2\sigma^2\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{\delta_*^2}.

Here $z_q$ is the $q$-quantile of the standard normal distribution. This is a normal-approximation formula: it treats the variance as known for the calculation and does not account exactly for estimating it in a finite sample. A design using a small sample, a non-normal endpoint, unequal allocation, or a different primary model needs a calculation suited to that setting.

At the interim, replace $\sigma$ with a prespecified estimate $s_{\mathrm{int}}$ of the nuisance standard deviation, obtained from the specified interim information. The estimate could be unstable, biased by the data structure, or poorly matched to the final analysis; its definition and eligibility checks therefore matter. The rule should not use an unplanned interim treatment-effect estimate as though it were a nuisance parameter.

Rule: Let $m$ be the number of participants already observed per arm at the interim, and let $n_{\min}$ and $n_{\max}$ be the permitted lower and upper bounds on the final target per arm. First require $n_{\min}\le n_{\max}$ and $m\le n_{\max}$. If both conditions hold, calculate $$ n_{\mathrm{req,int}} = \frac{2s_{\mathrm{int}}^2\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{\delta_*^2}, \qquad n_{\mathrm{final}} = \min\!\left\{n_{\max},\, \max\!\left[n_{\min},\,m,\,\left\lceil n_{\mathrm{req,int}}\right\rceil\right]\right\}. $$ Here $\lceil x\rceil$ rounds upward to an integer. The rule sets a final target per arm; the additional enrollment is $n_{\mathrm{final}}-m$ participants per arm. It applies only when its feasibility conditions hold.

The maximum and minimum are design constraints, not mathematical guarantees that the target power is achieved. If the unbounded request exceeds $n_{\max}$, the rule returns the cap and the trial accepts that the requested planning target may not be met. If the observed per-arm enrollment already exceeds $n_{\max}$, the expression must not be used: it could return a target smaller than the participants already enrolled.

Worked Calculations

Worked Example: A Feasible Variance-Based Update

Consider a hypothetical equal-allocation trial with $\alpha=0.05$, target power $0.80$, and a planning difference of $\delta_*=5$ outcome units. At the interim there are $m=40$ participants per arm, and the prespecified nuisance estimate is $s_{\mathrm{int}}=12$ outcome units. Let $n_{\min}=60$ and $n_{\max}=130$ per arm. The conditions hold because $60\le130$ and $40\le130$.

For a two-sided test, $z_{1-\alpha/2}=z_{0.975}\approx1.960$, and $z_{1-\beta}=z_{0.80}\approx0.8416$. Substitution gives

n_{\mathrm{req,int}} = \frac{2(12)^2(1.960+0.8416)^2}{5^2} = \frac{288(2.8016)^2}{25} \approx \frac{288(7.849)}{25} \approx90.42.

Rounding upward gives 91 per arm. This lies between the minimum and maximum and exceeds $m=40$, so $n_{\mathrm{final}}=91$. The protocol would therefore add $91-40=51$ participants per arm, for a planned total of 182. This calculation is a plug-in planning target; it does not by itself establish the operating power of the adaptive design.

Worked Example: The Maximum Binds

Suppose another hypothetical trial has $\alpha=0.05$, target power $0.80$, $\delta_*=4$ outcome units, $s_{\mathrm{int}}=15$ outcome units, and $m=50$ participants per arm. Set $n_{\min}=60$ and $n_{\max}=180$. Both feasibility conditions hold.

Using the same quantiles,

n_{\mathrm{req,int}} = \frac{2(15)^2(2.8016)^2}{4^2} = \frac{450(7.849)}{16} \approx220.8.

The upward-rounded request is 221 per arm, above the cap of 180. Thus $n_{\mathrm{final}}=180$, and the rule adds $180-50=130$ participants per arm. The design must report that the cap prevents implementation of the calculated request. It is incorrect to describe the capped design as having the originally requested power unless that claim has been separately justified for the capped adaptive design.

Worked Example: The Minimum Is Not Allowed to Undo Accrual

In a third hypothetical trial, let $\alpha=0.05$, target power $0.80$, $\delta_*=5$ outcome units, and $s_{\mathrm{int}}=6$ outcome units. Suppose $m=75$ participants per arm, $n_{\min}=60$, and $n_{\max}=100$. The feasibility conditions hold: $60\le100$ and $75\le100$.

The interim request is

n_{\mathrm{req,int}} = \frac{2(6)^2(2.8016)^2}{5^2} = \frac{72(7.849)}{25} \approx22.61, \qquad \left\lceil n_{\mathrm{req,int}}\right\rceil=23.

The unbounded requested target is below both the minimum and accrued enrollment. The rule therefore gives $n_{\mathrm{final}}=\min\{100,\max(60,75,23)\}=75$. No further participants are required under this target. The rule does not remove or disregard the 75 already observed per arm.

Feasibility, Justification, and Failure Checks

The key feasibility check follows directly from the meaning of a final target. It cannot be below enrollment already observed, so a target constrained to be no greater than $n_{\max}$ can be feasible only if $m\le n_{\max}$. Similarly, if the specified lower bound exceeds the upper bound, no target can satisfy both. The rule must explicitly reject either case rather than silently return a number.

n_{\min}>n_{\max} \quad\text{or}\quad m>n_{\max} \quad\Longrightarrow\quad \text{the stated bounded rule is infeasible.}

For example, if $m=101$ per arm but $n_{\max}=100$, applying the minimum-and-maximum expression would produce at most 100, which is less than the number already observed. A protocol should specify a contingency before the interim. It might require the sponsor to review and formally amend the cap so that it is at least $m$, if that is allowed, or direct the trial to stop enrollment and proceed under a prespecified analysis plan. The applicable option depends on the protocol and oversight requirements; the rule must not quietly change after seeing the interim result.

A short justification for the size formula also shows its limitations. With known variance, the standard error is $\sigma\sqrt{2/n}$; the normal-approximation test rejects in the direction of a positive alternative when its standardized statistic exceeds $z_{1-\alpha/2}$. At a true difference $\delta_*$, the standardized statistic has approximate mean $\delta_*\sqrt{n/(2\sigma^2)}$. Requiring that mean to be $z_{1-\alpha/2}+z_{1-\beta}$ yields the displayed equation after solving for $n$. Replacing $\sigma$ by $s_{\mathrm{int}}$ makes it a data-dependent plug-in target, not an exact power identity.

Operational diagnostics should be considered alongside the arithmetic. Verify that the interim look occurred at the specified time, the required data are available, the nuisance estimator was computed as written, and its value is in plausible units and range. Check the unrounded request, rounded request, binding limit, resulting target, and additional enrollment separately. Unexpectedly large estimates, missing observations, or a variance estimate near zero can make the calculation unstable; protocol-defined data-quality rules should determine whether the input is usable.

Final Analysis and Alternatives

The final analysis plan should name the estimand, the outcome model, the treatment-effect estimator, its uncertainty measure, and the hypothesis test or interval procedure. Crucially, it must address the fact that the final sample size depends on interim data. A conventional fixed-sample p-value or confidence interval is not automatically valid merely because the sample-size formula is familiar. The procedure should be justified for the complete adaptation rule, through an appropriate theoretical argument or design-specific simulation, and the interim decision and analysis implementation should be documented.

A fixed-design trial is simpler: its sample size does not respond to interim nuisance information, but may be inefficient if the planning variance is substantially wrong. A re-estimation rule can respond to that uncertainty, while introducing operational complexity and a need to validate both the decision rule and final inference. A broader, conservatively chosen fixed sample is another possible response to uncertain variance; it avoids a data-dependent target but may enroll more participants than needed. These are design trade-offs, not interchangeable guarantees.

Do not confuse a rule that updates a nuisance parameter with a rule that changes the target effect after observing interim outcomes. The former still needs prespecification and inferential validation, but it answers a different design question. In all cases, a binding cap, poor nuisance estimate, model mismatch, or failure of feasibility can prevent the intended operating characteristics from being achieved.

Key takeaway: A defensible re-estimation rule states its inputs and limits, checks $n_{\min}\le n_{\max}$ and $m\le n_{\max}$, never sets the target below accrued enrollment, and pairs the adaptation with a final analysis justified for the whole design.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In an equal-allocation trial, what does $n_{\mathrm{final}}$ count, and how many additional participants per arm are needed if $m=35$ and $n_{\mathrm{final}}=82$?
Why does a plug-in calculation using $s_{\mathrm{int}}$ not guarantee the target power?
A protocol sets $n_{\min}=90$, $n_{\max}=120$, and the interim has $m=125$ participants per arm. Can the bounded rule be applied as written? Explain.
If the calculated request is 150 per arm and $n_{\max}=130$, what does the rule return, and what should be said about power?
Why is a fixed-sample analysis not automatically justified after sample size depends on interim information?

Solutions. (1) It is the final target per arm; add $82-35=47$ per arm. (2) The interim estimate is random and may be unstable or mismatched to the assumed model, and the formula is itself an approximation. (3) No. Since $m>n_{\max}$, a target at or below the cap would be smaller than accrued enrollment; follow a prespecified contingency. (4) The target is capped at 130 per arm; the requested target power is not thereby guaranteed. (5) The sample size is data-dependent, so the sampling behavior of the final statistic may differ from its fixed-sample behavior; the final procedure must be justified for the adaptive design.