Sample-Size Re-Estimation With a Fixed Target

Why Enrollment Can Change Without Changing the Target

A clinical trial may need more participants than planned because the outcome is more variable than expected, or fewer because the planned precision is already achievable. Revising enrollment can change the amount of information collected without changing the treatment effect the trial is designed to estimate. That conclusion depends on holding the target population, treatment conditions, outcome, strategy for intercurrent events, and treatment-effect summary fixed.

The key distinction is between the estimand—the precise treatment effect the trial seeks to quantify—and the sample size used to estimate it. For example, adding participants from the same target population under the same treatment conditions can make an estimated mean difference more precise. It does not, by itself, redefine that mean difference. In contrast, changing eligibility criteria or the outcome definition can change which effect is being targeted, even if the final sample size is unchanged.

Definition: A sample-size re-estimation with a fixed target is a prespecified revision to planned enrollment that leaves the estimand unchanged. The population, treatment conditions, outcome variable, intercurrent-event strategy, and summary measure defining the target remain fixed; the number of participants is revised to meet a stated precision or power objective.

To make this concrete, suppose treatment \(a\) and treatment \(b\) are compared in a fixed target population \(P^*\). Let \(Y^a\) and \(Y^b\) denote the outcomes under those treatment conditions, with intercurrent events handled by the trial’s specified strategy. A marginal mean-difference estimand is

\theta = E_{P^*}(Y^a) - E_{P^*}(Y^b).

Changing the planned number of participants does not change this definition if \(P^*\), the meanings of \(a\) and \(b\), the outcome, the event strategy, and the mean-difference summary remain unchanged. It may change the precision of the estimate. Whether the planned estimator, confidence interval, or test remains valid after an adaptive revision is a separate design question; preserving the estimand alone does not guarantee valid inference.

A Sample-Size Model for a Fixed Mean Difference

Consider a two-arm, individually randomized trial with equal allocation. Assume outcomes are independent across participants, follow-up and outcome measurement are the same in both arms, and the outcome in each arm is approximately normally distributed with a common variance \(\sigma^2\). Suppose the target is the marginal mean difference \(\Delta=\mu_a-\mu_b\), where \(\Delta\) is the difference considered clinically important for planning. The variance and \(\Delta\) are planning quantities; they do not redefine the estimand.

For a two-sided test with significance level \(\alpha\) and planned power \(1-\beta\), a normal-approximation formula for the required number per arm is

n_{\text{per arm}} = \frac{2\sigma^2\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{\Delta^2}.

Here \(z_q\) is the \(q\)-quantile of a standard normal distribution. The formula follows from the standard error of the difference between independent arm means, \(\sqrt{2\sigma^2/n}\): the planned effect divided by that standard error must be large enough to cross the two-sided critical value with the desired probability. Because participants are whole numbers, the calculated value is rounded up. Exact or simulation-based calculations may be preferable for small samples, non-normal outcomes, unequal allocation, or more complicated analysis plans.

Worked Example: Revising Enrollment After a Larger Variance Estimate

A hypothetical trial compares two treatments on a continuous symptom score, measured in points. The prespecified target is the mean difference between the treatments in the original eligible population, using the same outcome and intercurrent-event strategy throughout. The planning difference is 5 points, the assumed common standard deviation is 10 points, and the design uses a two-sided \(\alpha=0.05\) with 90% power. How many participants per arm are planned?

Under the stated equal-allocation, common-variance model, \(z_{0.975}=1.960\) and \(z_{0.90}=1.282\). Thus

n_{\text{per arm}} = \frac{2(10^2)(1.960+1.282)^2}{5^2} = \frac{200(3.242)^2}{25} = \frac{200(10.51)}{25} = 84.08.

Rounding up gives 85 participants per arm, or 170 in total. Suppose a prespecified review of interim, blinded information supports revising the planning standard deviation to 12 points, while the target difference and all estimand-defining features stay fixed. Then

n_{\text{per arm}} = \frac{2(12^2)(3.242)^2}{5^2} = \frac{288(10.51)}{25} = 121.1.

The revised target is 122 per arm, or 244 total. The revision reflects greater anticipated variability, so more participants are needed to achieve the same planned power for the same 5-point mean difference. It does not change what population mean difference the trial targets.

Why a Larger Sample Need Not Change the Estimand

The basic justification can be seen from the expectation of the sample mean. Under independent sampling from a fixed population with mean \(\mu\), the mean of \(n\) observations satisfies

E(\bar{x}) = \mu, \qquad \operatorname{Var}(\bar{x}) = \frac{\sigma^2}{n}.

For equal-sized treatment groups sampled under the stated model, the difference in sample means estimates \(\theta=\mu_a-\mu_b\). Its expectation and variance are

E(\bar{x}_a-\bar{x}_b)=\mu_a-\mu_b=\theta, \qquad \operatorname{Var}(\bar{x}_a-\bar{x}_b) = \frac{\sigma^2}{n}+\frac{\sigma^2}{n} = \frac{2\sigma^2}{n}.

These equations show the distinction: changing \(n\) changes the variance of the estimator, not the value of \(\theta\) being estimated. This argument presumes that participants continue to represent the same target population and that treatment conditions, outcomes, and event handling remain consistent. If those conditions fail, a change in enrollment can be accompanied by a change in the effect the study actually estimates.

The same logic applies when the study’s stated summary is not a mean difference: changing sample size alone does not redefine a fixed summary measure. But the calculations must then match the estimator and design for that summary. The mean-difference formula above is not a general formula for risk differences, odds ratios, survival contrasts, or other estimands.

Worked Example: Blinded Variance Re-Estimation

Consider a hypothetical equal-allocation trial with a continuous outcome measured in points. Its prespecified mean-difference target is 4.5 points, with two-sided \(\alpha=0.05\) and 80% power. The original planning standard deviation is 10 points, giving an initial sample-size calculation. After 40 participants per arm have contributed blinded outcome data, an approved review estimates a pooled standard deviation of 11.4 points. The trial has specified a maximum of 110 participants per arm.

First calculate the original plan. The normal quantiles are \(z_{0.975}=1.960\) and \(z_{0.80}=0.842\), so their sum is 2.802:

n_{\text{per arm}} = \frac{2(10^2)(2.802)^2}{4.5^2} = \frac{200(7.851)}{20.25} = 77.54.

Rounding up gives 78 per arm. Replacing only the planning variance estimate gives

n_{\text{per arm}} = \frac{2(11.4^2)(2.802)^2}{4.5^2} = \frac{259.92(7.851)}{20.25} = 100.8.

The revised target is 101 per arm, below the prespecified cap of 110. Since 40 per arm have already been enrolled, the operational plan is to enroll 61 more per arm, for 202 participants total. The planning effect remains 4.5 points; the estimand’s population, treatment conditions, outcome, and event strategy are unchanged.

A blinded pooled variance estimate does not use the observed treatment contrast to choose a new target effect. In some settings, however, the pooled variance can also reflect the treatment difference, so it may be conservative as an estimate of within-arm variability. The re-estimation rule should specify how the estimate is formed and how minimum and maximum sample sizes are handled.

When Enrollment Changes the Target in Practice

A protocol can state a fixed target while recruitment changes in a way that makes the enrolled participants less representative of it. For example, later recruitment might disproportionately include a baseline subgroup with a different treatment effect. A simple unadjusted average treatment contrast across all enrolled participants may then reflect the recruitment mix rather than the original target population. This is not a consequence of the larger sample size itself; it is a consequence of changing who contributes to the estimate.

Worked Example: Recruitment Mix and a Fixed Population Target

Suppose a hypothetical target population consists of equal proportions of patients in two baseline groups. In group 1, the treatment effect is a mean difference of \(-4\) outcome points; in group 2, it is \(-1\) point. The prespecified target is the mean difference standardized to the 50%/50% target-population mix.

The fixed target effect is

\theta = 0.50(-4)+0.50(-1) = -2.50 \text{ points}.

Now suppose an expanded recruitment period yields 20% group 1 and 80% group 2. The unadjusted effect for that recruitment mix is

0.20(-4)+0.80(-1) = -0.80-0.80 = -1.60 \text{ points}.

That \(-1.60\)-point contrast is not the original 50%/50% target. If the trial still intends to estimate the original target, its analysis must account for the fixed target mix using a prespecified approach. Simply calling the additional participants a sample-size increase does not make a changed recruitment mix irrelevant.

Safeguards, Diagnostics, and Alternatives

A fixed-target re-estimation is most defensible when the protocol prespecifies the decision rule before the interim review. The rule should state what information may be examined, how the nuisance quantity is estimated, when the calculation occurs, the permitted sample-size range, and whether enrollment can only increase or can also decrease. An independent or blinded review can limit the risk that decisions are driven by emerging treatment differences, but blinding alone does not establish that a statistical test or interval will retain its intended properties.

The re-estimation must also respect the trial’s original definitions. Check that eligibility criteria and recruitment sources still support the target population; that treatment versions and delivery have not changed; that outcome measurement and follow-up are consistent; and that intercurrent events are handled by the same strategy. A change in any of these can alter the target or the data needed to estimate it. The earlier material on treatment versions and standardizing effects to a fixed population is useful background for these checks.

Inference needs its own assessment. If the new sample size depends on interim outcomes, the final estimator or test may no longer behave as it would under a fixed sample size. Even a variance-based rule can require careful justification for the planned analysis. The trial should use an inference procedure validated for its particular re-estimation rule, rather than assuming that a conventional fixed-sample p-value and confidence interval automatically remain calibrated. This is especially important when the adaptation uses unblinded treatment-effect information.

Useful warning signs include a variance estimate based on a different outcome scale, changing missingness or follow-up patterns, a substantial shift in recruitment composition, and an interim decision rule that was not prespecified. A large discrepancy between the observed estimate and planning assumptions is a reason to investigate, not a reason to change the estimand informally.

Approach	What it changes or uses	Key consideration
Fixed sample size	Enrollment stays at the original plan.	Simpler planning and analysis, but may leave the study under- or over-sized if assumptions are wrong.
Blinded sample-size re-estimation	Updates a planning quantity, commonly variability, without using the interim treatment contrast to reset the target effect.	Prespecify the rule and verify inference for the actual design.
Unblinded effect-based re-estimation	Uses emerging treatment differences as well as, or instead of, nuisance information.	Can create selection and error-control concerns; requires design-specific methods.
Changing the estimand-defining features	Changes such elements as population, outcome, treatment condition, or event strategy.	This is not merely sample-size re-estimation: it can change the clinical question.

Common pitfall: “We added participants, so the estimand must have changed,” or, conversely, “The estimand is unchanged, so the original analysis is automatically valid.” Both conclusions confuse the target with the inferential procedure. Changing \(n\) alone need not change a fixed estimand, but adaptive decisions can affect how reliably the estimator, test, or interval represents that target.

Key takeaway: Sample-size re-estimation can preserve a target when it changes only how many participants are enrolled. Preservation requires keeping the target population, treatment conditions, outcome, intercurrent-event strategy, and summary fixed—and separately checking that the adaptive design supports valid inference.

Check Your Understanding

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

A trial’s mean-difference target, treatment conditions, eligible population, and outcome stay fixed, while enrollment increases because the variance estimate is larger. Has the estimand necessarily changed? What has changed?
With equal allocation, \(\sigma=8\) outcome points, \(\Delta=4\) points, two-sided \(\alpha=0.05\), and 80% power, calculate the approximate number per arm. Use \(z_{0.975}=1.960\) and \(z_{0.80}=0.842\).
Why can a blinded variance-based re-estimation still require a design-specific check of the final p-value or confidence interval?
A target population has equal proportions in two groups, with treatment effects of \(-3\) and \(-1\) points. Recruitment later shifts to 25% in the first group and 75% in the second. Calculate the fixed target effect and the effect for the new recruitment mix. What does the difference show?

Solutions. (1) No. If all estimand-defining features remain fixed, the estimand is unchanged; the planned enrollment and expected precision have changed. (2) \(n=2(8^2)(1.960+0.842)^2/4^2=128(7.851)/16=62.81\), so round up to 63 per arm. (3) The sample-size decision is adaptive, and its dependence on interim information can affect the behavior of the final procedure; blinding does not by itself prove fixed-sample calibration. (4) The fixed target is \(0.50(-3)+0.50(-1)=-2.00\) points. The new recruitment-mix effect is \(0.25(-3)+0.75(-1)=-1.50\) points. The difference shows that a changing participant mix can change the unadjusted contrast, even when subgroup effects are unchanged; enrollment growth alone is not the issue.