Sample Size Re-Estimation With Multiple Endpoints

Why Multiple Endpoints Need a Common Enrollment Plan

A sample-size update is often driven by a prespecified primary endpoint, while the same participants also contribute information about secondary endpoints. The primary endpoint may require a certain sample size to achieve planned power; a secondary endpoint may instead have a planned confidence-interval width. These are different design goals, even though both depend on enrollment.

The re-estimation must preserve the primary question and its planned decision rule. Secondary endpoints should be checked against their stated goals at the revised sample size, not assumed to be adequately supported just because the primary calculation is satisfied. Conversely, a precision target for a secondary endpoint does not automatically make that endpoint a confirmatory test or change the trial’s success criterion.

Primary-Endpoint Requirement and Its Assumptions

For a simple illustration, suppose the primary endpoint is continuous, participants are independently randomized in equal numbers to two groups, outcomes are fully observed, and the outcome has the same standard deviation \(\sigma\) in both groups. Let \(\Delta\) be the prespecified treatment difference the design aims to detect. The standard error of the difference in sample means, with \(n\) participants per group, is \(\sigma\sqrt{2/n}\).

For a two-sided test at level \(\alpha\), the usual normal-approximation planning formula for power \(1-\beta\) is:

Here \(z_q\) is the \(q\)-quantile of the standard normal distribution. This is a planning approximation, including when \(\sigma\) is known; it is not an exact two-sided normal-theory power solution. Under the alternative, the standardized mean has noncentrality \(\lambda=\Delta/(\sigma\sqrt{2/n})\). Exact power for the two-sided z-test is \( \Phi(-z_{1-\alpha/2}-\lambda)+1-\Phi(z_{1-\alpha/2}-\lambda)\). The displayed planning formula uses the upper-tail target and does not solve that full equation exactly. If exact normal-theory power is required, solve the full expression for \(n\).

In a re-estimation, an updated nuisance-parameter estimate, such as an interim estimate of \(\sigma\), may be substituted into the prespecified planning rule. The target difference \(\Delta\), significance level, power target, allocation, and update rule should remain prespecified. The resulting continuous value of \(n\) is rounded upward to a whole participant per group. If the protocol sets a minimum or maximum, apply those bounds as written; do not silently replace the primary design target with an observed interim treatment effect.

Translate Secondary Precision Goals Into Enrollment

For a continuous secondary endpoint with standard deviation \(\sigma_S\), equal allocation, and independent participants, the standard error of the difference between group means is \(\sigma_S\sqrt{2/n}\). A two-sided normal-based confidence interval with individual confidence level \(1-\alpha_S\) has half-width \(z_{1-\alpha_S/2}\sigma_S\sqrt{2/n}\). To keep that half-width no larger than a planned value \(h\), solve the inequality for \(n\):

The calculation depends on the endpoint’s scale and variability, the interval’s confidence level, and the desired half-width. It does not depend on the endpoint’s treatment effect. If the endpoint is binary or time-to-event, its standard error and information requirements are different; the continuous-outcome formula must not be applied to those endpoints.

Combine Requirements Without Changing the Primary Question

When all endpoints use the same participants and allocation, calculate the required sample size for each design goal on a consistent basis. If the protocol requires primary power and specified secondary precision, the common per-group target is the largest of those requirements, subject to any prespecified enrollment bounds. If secondary endpoints are descriptive only and no precision goal was committed to, their calculated precision can still be reported, but it need not determine enrollment.

This maximum rule assumes that each requirement is expressed for the same unit of enrollment and that the endpoints are available from the same randomized participants. It does not mean that all requirements are automatically met if endpoints have different analysis populations, different missing-data rates, or different allocation or follow-up needs. Those differences require their own prespecified adjustments.

Justification, Diagnostics, and Failure Modes

The maximum rule follows directly from the meaning of each requirement. If \(n\) is smaller than any required value \(n_j\), then the corresponding design goal is not met by its calculation. If \(n\) is at least every \(n_j\), all the inequalities hold at once. Therefore the smallest common \(n\) satisfying all the requirements is their maximum:

A practical diagnostic is to recalculate each endpoint’s achieved quantity at the proposed enrollment: primary approximate power under its design assumptions, and secondary confidence-interval half-widths under their stated confidence levels. A table can make the bottleneck visible.

The calculations are only as credible as their assumptions. For the simple formulas, check whether the variance estimate is relevant to the endpoint and analysis population, whether the equal-allocation and common-variance assumptions are reasonable, and whether the endpoint is observed for the anticipated participants. If the updated variance is unstable or the endpoint’s missingness differs materially, a nominal increase in randomized enrollment may not deliver the intended precision. The resulting rule should state how rounding, lower and upper limits, and unavailable endpoint data are handled.

The main alternatives reflect different scientific goals. A precision-based plan specifies interval widths; a power-based plan specifies a detectable effect and a testing error rate. Confirmatory testing of several endpoints needs an explicit multiplicity strategy, such as a prespecified allocation of error or a gatekeeping procedure; simply increasing enrollment does not by itself control the familywise type I error rate. Co-primary endpoints are different again: trial success may require success on more than one endpoint, so their joint success rule must be part of the design rather than treated as secondary precision planning.

Endpoint correlation may inform more efficient joint precision methods, but it is not needed for Bonferroni’s simultaneous-coverage guarantee. A limitation of the maximum rule is that it can be conservative when endpoint requirements are not genuinely separate or when a carefully justified joint method is available. Another limitation is that a plug-in variance estimate does not eliminate uncertainty in that estimate. The update should therefore follow its prespecified rule and be interpreted as a design decision under uncertainty, not as proof that final power or precision is guaranteed.

Common Mistake

A frequent error is to say, “The updated sample size meets primary power, so the secondary endpoints are adequately powered and precise.” That conclusion does not follow: power for a secondary test and precision of its interval are distinct properties, and neither is implied merely by the primary calculation. The correction is to state each secondary goal, calculate its requirement or achieved interval width, and apply any multiplicity adjustment needed for simultaneous claims. A second error is to choose the largest observed interim effect among endpoints and use it to justify reducing enrollment. That changes the scientific target based on interim data; retain the prespecified primary endpoint and clinically meaningful target effect in the update.