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Optimal interim-analysis timing in a group sequential trial depends only on the error rates and stopping rule, not the effect size or endpoint; switching to it cuts expected sample size by up to 5.4% in O'Brien-Fleming designs.

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Optimal interim analysis schedules that minimize expected sample size under the alternative are independent of effect size and endpoint type, and yield modest gains over equal spacing (up to 5.4% for O'Brien-Fleming designs).

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Normal-endpoint optimization is sound and the tool is useful; the binary generalization is asserted, and the abstract oversells the MSS trade-off and the size of the savings. the 3 major comments →

arxiv 2509.05537 v1 pith:IPRXTFVG submitted 2025-09-05 stat.ME stat.AP

Optimal scheduling of interim analyses in group sequential trials

Zhangyi He , Suzie Cro , Laurent Billot This is my paper
classification stat.ME stat.AP MSC 62L0562K0562P10
keywords group sequential designinterim analysis schedulingexpected sample sizeinformation rateO'Brien-Fleming designoptimal designOptimInterimclinical trial design
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Group sequential trials — the most common adaptive design in confirmatory clinical research — are usually planned with interim analyses at equal intervals of accumulated information, a convention this paper argues leaves sample-size savings on the table. Its central claim is that once a stopping rule (Pocock, O'Brien-Fleming, Haybittle-Peto) and the type I and type II error rates are fixed, a single optimal schedule of interim looks minimizes the expected sample size under the alternative hypothesis, and this schedule is the same for every treatment effect size and for continuous or binary endpoints alike. The paper proves the invariance by rewriting the expected sample size as a function of information fractions alone, implements the optimization in the R tool OptimInterim, and tabulates ready-to-use schedules for up to eight interim analyses. Simulations show the largest gains for O'Brien-Fleming designs, where optimal spacing reduces expected sample size under the alternative by up to 5.4% with at most a 1% increase in maximum sample size; Haybittle-Peto and Pocock designs improve only marginally, since equal spacing is already near-optimal for their error spending. Redesigning the HYPRESS and ADRENAL septic-shock trials with optimal schedules substantially raises the probability of early stopping at the first look while keeping maximum sample size essentially unchanged.

Core claim

The central claim is an invariance: for a prespecified stopping rule with fixed type I and type II error rates, the information rates t1:K−1 that minimize the expected sample size under H1 are the same for every standardized effect size and for every asymptotically normal test statistic, binary endpoints included. The proof runs through Eq. (7), where E(N|H1) factors into an effect-size constant times a bracket of information rates, stopping regions and the drift θ fixed by the error rates — so the minimizer never sees the effect size. Observed consequence: optimal schedules hold early looks later; for O'Brien-Fleming designs they cut expected sample size under H1 by up to 5.4% with at most

What carries the argument

The machinery is the canonical joint multivariate normal distribution of the standardized test statistics Z1:K (mean δ√Ik, covariance √(min/max) of information), with trial progress encoded in the information rate tk = Ik/IK = Nk/NK. The Armitage recursion fixes the drift θ for given error allocation, and Eq. (7) collapses E(N|H1) to N0·θ²/(z1−α+z1−β)² times a bracket of continuation-region probabilities — effect size confined to the constant N0, which yields the invariance. OptimInterim minimizes the bracket via Nelder-Mead from many starting points, evaluating probabilities through rpact, and the paper's tables give the resulting optimal information rates.

Load-bearing premise

The load-bearing premise is that the information accumulated at an interim analysis equals the fraction of the total sample size observed — exact for normally distributed endpoints with known variance, but only asserted as an asymptotic approximation for binary endpoints, where it is neither derived nor simulated.

What would settle it

Simulate a two-arm binary-endpoint trial (e.g., 45% vs 30% event rates) under a three-stage O'Brien-Fleming design, one-sided α=0.025, β=0.1, and compute the expected sample size under H1 directly from the binomial likelihood at the paper's optimal schedule (first look at 54.9% information) and at equal spacing. If the binomial-optimal schedule shifts materially from the table, or the relative gain departs from the normal-approximation prediction, the binary-extension claim fails. Second test: for a logistic-regression score statistic, recompute the optimal schedule at two very different effec

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Reference tables give planners a ready-to-use optimal schedule for up to eight interim analyses: for fixed error rates and stopping rule, the same information rates apply whatever the anticipated treatment effect or endpoint.
  • O'Brien-Fleming designs gain the most — up to 5.4% lower expected sample size under H1 for at most ~1% higher maximum sample size — and can match an equally spaced design's efficiency with one fewer interim look.
  • Haybittle-Peto and Pocock designs improve by at most 1.2% under optimal spacing, implying equal spacing is already close to optimal for those error-spending patterns.
  • Adding a futility boundary to an optimized schedule yields much larger savings under no effect (≥25.9% ESS reduction) and under a halved effect (≥8.8%), at the cost of a larger maximum sample size (up to 29.0% for Pocock, 14.6% for O'Brien-Fleming).
  • Optimal schedules shift the first interim look later — in the HYPRESS redesign from 33% to 57% of information — which raises the probability of early efficacy stopping at the first look from 1.9% to 27.6% under H1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the schedule depends only on error rates and the stopping rule, the reference tables plausibly transfer to trials with different design margins (e.g., non-inferiority or larger/smaller target effects) than those simulated — a transfer the paper does not itself claim.
  • The invariance proof assumes the information fraction equals the sample-size fraction; for binary endpoints that identity is only asymptotic, so the natural stress test is exact binomial enumeration at modest sample sizes to see how far the tables drift from the normal-approximation optimum.
  • Reading across the tables, the optimal schedules share a pattern: the first interim look moves later (roughly 55–65% of information for O'Brien-Fleming designs) and later looks are moderately spaced — suggesting a simple planning heuristic: delay the first look to concentrate stopping probability where the test statistic is informative.
  • The same minimization could be run for survival endpoints through the log-rank statistic — noted by the paper as conceptually straightforward — with information accumulated in events rather than patients; whether the effect-size independence survives that change is a concrete open question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a method and an R tool, OptimInterim, for choosing the timing (information rates) of interim analyses in group sequential trials. The objective is to minimize the expected sample size under the alternative hypothesis H1 while preserving overall type I and type II error rates, given a prespecified stopping rule. For normally distributed endpoints with known variance, the paper derives Eq. (9), showing that the optimal information rates are independent of the standardized effect size, and provides reference tables for up to eight interim analyses for Haybittle-Peto, Pocock, and O'Brien-Fleming designs. The operating characteristics are evaluated by simulation for continuous endpoints, and the tool is illustrated by redesigning the HYPRESS and ADRENAL trials, both of which have binary primary endpoints.

Significance. If the claims hold, this is a practically useful contribution: the public R tool is built on the validated rpact package, and the reference tables give immediate design guidance. The derivation for normal endpoints with known variance is sound, and the effect-size independence is a useful simplification. However, the broad generalization to binary endpoints is asserted rather than demonstrated, and the abstract's claim about not compromising the maximum sample size is inconsistent with the paper's own O'Brien-Fleming results. The paper is likely to be a valuable methods/software contribution after these gaps are addressed.

major comments (3)
  1. [Section 2.2, Eq. (9); Section 2.4] The assertion that the results are 'equally applicable to test statistics that are asymptotically normally distributed, e.g., those arising from binary endpoints' is not supported. Equation (9) is derived under the exact normal, known-variance model with I_k proportional to N_k and t_k = N_k/N_K. For binary endpoints, the asymptotic information is proportional to N_k only under fixed response probabilities and a chosen variance estimator, and the covariance structure and cancellation in Eq. (9) hold only asymptotically. The simulations in Section 2.4 use only normally distributed endpoints, while the case studies (Sections 3.2-3.3) apply the normal-derived optimal schedules to binary endpoints. Because the abstract and discussion claim applicability across endpoint types and the reference tables are recommended for binary trials, this gap is load-bearing. The authors should provide a rig
  2. [Abstract; Sections 3.1.3 and 4] The abstract states that optimal scheduling yields savings 'without compromising the maximum sample size.' This is contradicted by the O'Brien-Fleming results: the Discussion reports that the optimal O'Brien-Fleming design is 'accompanied by a modest increase in the MSS, up to 1.0%,' and the HYPRESS redesign (Table 5) has MSS 310.3 with optimal timing vs. 307.6 with the original timing. The abstract should be revised to state that the maximum sample size may increase slightly for some designs, or the claim should be restricted to the designs and comparisons where it holds.
  3. [Section 2.3; Tables 1-3] The numerical optimization is described as using Nelder-Mead with multiple starting values, but the paper does not report convergence diagnostics, the number of starting values, or any verification that the reported schedules are global optima. Since the reference tables are presented as optimal schedules and are the main practical output, the authors should provide evidence that the reported minima are stable, for example by comparing against a fine grid or alternative optimization methods for at least a subset of the scenarios.
minor comments (5)
  1. [Eq. (8) vs. Section 2.1] The symbol N0 is used both for the fixed-design total sample size in Eq. (8) and for the control-arm cumulative sample size N_{0,k} in Section 2.1. Rename the fixed-design quantity, e.g., N_fixed, to avoid confusion.
  2. [Section 2.1] For two-sided alternatives, the continuation and rejection regions are only described verbally. Please state the explicit regions for the two-sided case, since the reference tables are used for both one- and two-sided designs.
  3. [Section 2.3] The phrase 'a restricted global search' is vague. Please clarify what restriction is imposed on the Nelder-Mead search and how the multiple starting values are generated.
  4. [Eq. (1)] The typeset formula for I_k is hard to read; inserting parentheses or using a display fraction would improve clarity.
  5. [References] The reference to Lewis (2023) is incomplete; please provide the full title if available.

Circularity Check

0 steps flagged

No significant circularity: optimal schedules are obtained by minimizing a well-defined ESS objective with standard error-spending constraints; the effect-size/endpoint-type independence follows from the algebraic structure of Eq. (9), not from fitted inputs.

full rationale

The central derivation chain is self-contained. Section 2.2 defines the information rate t_k = I_k/I_K; for the assumed known-variance normal model this equals N_k/N_K. The optimization in Eq. (9) minimizes E(N|H_1) over t_1:K-1 subject to type I/II error constraints. The claimed independence from the standardized effect size is a direct algebraic cancellation: in Eqs. (6)-(8), delta* enters only through the fixed-design sample size N0, while the canonical drift theta is determined solely by t, alpha, and beta via the error equations. None of the schedule outputs are fitted to the data used for evaluation; the simulations in Section 2.4 are independent operating-characteristic checks. The binary-endpoint extension is asserted via asymptotic normality and an information fraction equal to the sample-size fraction (Section 2.2), and the authors acknowledge in Section 2.4 that the performance evaluation focuses on a continuous endpoint. This is a robustness/evidence limitation, not a circular step: the binary claim is not obtained by renaming a known result or by using the binary endpoint to define the optimization. The only self-citation (Li et al., 2023, with co-author Billot) is cited for background on the value of multiple interim analyses and for a futility re-analysis in ADRENAL; it is not load-bearing for the optimal-scheduling derivation. Per hard rule 4, a non-load-bearing self-citation does not raise the circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The method rests on standard group sequential theory and a few simplifying assumptions about outcome availability and information timing. The binary endpoint generalization is an unproven assertion, and the idealized 'recruitment halted, outcomes immediate' setting limits the direct applicability of the reference tables.

axioms (5)
  • standard math The standardized test statistics Z1:K follow the canonical joint multivariate normal distribution with independent increments (Jennison & Turnbull 1999).
    Invoked in Section 2.1 to compute stopping probabilities and calibrate boundaries.
  • domain assumption The outcome is normally distributed with known common variance sigma^2, and recruitment halts at each interim analysis with all outcomes immediately available.
    Stateed at the start of Section 2.1; the paper acknowledges in Section 4 that this is not common in clinical practice.
  • domain assumption The information rate t_k equals the cumulative sample size fraction N_k/N_K.
    Used in Section 2.2 to simplify the optimization; depends on equal allocation and the variance structure.
  • ad hoc to paper For binary endpoints, the test statistics are asymptotically normally distributed and the same independence and scheduling results hold.
    Asserted in Section 2.2 without derivation; no binary simulations are presented in Section 2.4.
  • standard math The canonical drift parameter theta is uniquely determined by t1:K, alpha1:K and beta1:K via the recursive formulation of Armitage et al. (1969).
    Used in Section 2.2 to express the maximum sample size and the ESS objective.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Optimal scheduling of interim analyses in group sequential trials." pith.science (2026). https://pith.science/paper/IPRXTFVG

@misc{pith2026250905537,
  author       = {Pith},
  title        = {Pith review of: Optimal scheduling of interim analyses in group sequential trials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPRXTFVG}},
  note         = {Machine review of arXiv:2509.05537}
}
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read the original abstract

Group sequential designs (GSDs) are well established and the most commonly used adaptive design in confirmatory clinical trials with interim analyses. However, they remain underutilised, and their implementation involves unique theoretical and practical decisions that demand careful consideration to optimise efficiency. A common practice is to schedule interim analyses at equal intervals based on calendar time or accumulated data. While straightforward, this approach does not completely exploit the potential sample size savings achievable with GSDs. To address this challenge, we develop OptimInterim, an R-based tool that can determine the optimal scheduling of interim analyses to minimise the expected sample size under the alternative hypothesis while controlling overall type I and type II errors. Our method accommodates trials with continuous or binary endpoints, allows multiple interim analyses and supports a range of stopping boundaries. Through extensive simulations, we demonstrate that optimally spaced interim analyses can yield substantial savings in expected sample size compared to equally spaced interim analyses, without compromising the maximum sample size, across various endpoint types, effect sizes, error rates and stopping rules. We illustrate its practical utility with two landmark trials evaluating steroid use in septic shock. Notably, for given type I and type II error rates, the optimal scheduling is independent of endpoint types and effect sizes, ensuring broad applicability across a wide range of trial contexts. To facilitate implementation, we offer a ready-to-use reference table of optimal schedules for up to eight interim analyses under commonly used error rates and stopping rules. Access OptimInterim at https://github.com/zhangyi-he/GSD_OptimInterim.

Figures

Figures reproduced from arXiv: 2509.05537 by Laurent Billot, Suzie Cro, Zhangyi He.

Figure 1
Figure 1. Figure 1: Operating characteristics for the optimal Haybittle-Peto design incorporating up to eight interim [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Operating characteristics for the optimal Pocock design incorporating up to eight interim analyses for [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Operating characteristics for the optimal O’Brien-Fleming design incorporating up to eight interim [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MSS and ESSs under H0, H0/H1 and H1 for the HYPRESS trial across all possible timing for up to two interim analyses for efficacy. 3.3. The ADRENAL trial We redesign the ADRENAL trial using the Haybittle-Peto approach, scheduling two interim analyses at optimally determined information rates of 44.4% and 70.5%. These timings, obtained by OptimInterim, are chosen to minimise the ESS under the alternative hyp… view at source ↗
Figure 5
Figure 5. Figure 5: MSS and ESSs under H0, H0/H1 and H1 for the ADRENAL trial across all possible timing for up to two interim analyses for efficacy. As shown by Li et al. (2023), adding a futility boundary to the ADRENAL trial could have 21 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Bayesian predictive framework for adaptive interim-analysis timing with robust borrowing in confirmatory trials

    stat.AP 2026-07 conditional novelty 6.0

    B²-FIC calibrates Bayesian phase-II borrowing for type I error, then uses IA1 predictive probability to schedule the earliest admissible IA2, yielding earlier decisions than fixed GSD when evidence is favorable while ...

Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.