Informative interim adaptations in group-sequential and adaptive designs — early-stopping boundaries and sample-size re-estimation (SSR) — spend part of a study's Fisher information on the interim decision itself: Tarima and Flournoy (2024) decompose the total Fisher information into a design component and decision-conditional estimation components, and use the unspent part to derive a sequential Cramér–Rao-type lower bound on the mean squared error (MSE) of post-adaptation estimators. That machinery quantifies the cost of adaptation but does not produce designs. The open problem is to convert this information accounting into a design principle: formulate an optimality criterion that, subject to testing requirements such as type-I error control and power, balances the information spent by interim decisions against the information remaining for post-test estimation (together with trial efficiency), and derive and justify the optimal decision rules — stopping boundaries and SSR policies — under that criterion. No dual-goal optimality criterion or optimal rule has been established: the most recent successor work (2025) supplies information-based metrics for assessing candidate boundaries and explicitly defers their optimization to future research.

Background

Group-sequential designs (GSDs) and sample-size re-estimation (SSR) are clinical-trial designs in which accumulating data are inspected at interim analyses and used to stop the trial early for efficacy or futility, or to modify the planned sample size. When the interim decision depends on data that carry information about the parameter of interest θ\theta (for example, a treatment effect), the adaptation is called informative: the resulting sample size is random, and the decision itself becomes a component of the sufficient statistic (the statistic that summarizes all information in the data about θ\theta). Estimators that use only the observed responses — including the maximum-likelihood estimator (MLE, the estimator maximizing the likelihood of the observed data) — then ignore part of the statistical evidence. It has been known since Whitehead (1986) that the MLE is biased after a sequential test, and unbiased estimators following a group-sequential test were constructed by Liu and Hall (1999).

Tarima and Flournoy (2024) quantify this phenomenon with Fisher information (FI — the expected curvature of the log-likelihood, the standard measure of how much information data carry about a parameter). Their Theorem 1 decomposes the total FI of the observable data into the design information ID(θ)\mathcal{I}_D(\theta) — the information consumed by the informative interim decision, which is not absorbed by the likelihood function and is therefore lost to likelihood-based inference — plus a weighted sum of FI components conditional on each possible interim decision, which is what remains available for post-adaptation estimation. Their Theorem 2 turns the unspent FI into a Cramér–Rao-type lower bound (CRLB; the classical Cramér–Rao bound states that the variance of any unbiased estimator is at least the reciprocal of the Fisher information) on the MSE of regular post-adaptation estimators; the classical bound and its sequential version due to Wolfowitz (1947) do not apply because the interim adaptation is informative. In one-parameter exponential families (families with densities of the form fθ(x)=h(x)exp⁡{θT(x)−A(θ)}f_\theta(x)=h(x)\exp\{\theta T(x)-A(\theta)\} parametrized by the natural parameter θ\theta), the MLE attains this new bound, so it is sharp. The theory is illustrated for two-stage normal sampling with a possibility of early stopping.

On the analysis side the literature is mature: bias-adjusted point estimators after adaptive designs are catalogued in the methodological review of Robertson et al. (2023); Marschner (2021) developed a general likelihood-based framework for analyzing adaptive experiments; and Flournoy and Tarima (2023) proposed Bayesian posteriors that condition on the informative interim decisions. However, all of this work takes the design — the stopping boundaries and the adaptation rule — as given. Classical optimal-boundary-shape work in group-sequential design optimizes testing objectives only (type-I error, power, expected sample size), not the precision of the post-test estimate. The missing step, stated in the closing sentence of Tarima and Flournoy (2024), is design optimization: using the information decomposition and the sequential MSE lower bound to construct sequential designs whose decision rules jointly serve the trial's testing goal and the precision of the final estimate.

Problem Statement

Using the Fisher-information decomposition and the sequential Cramér–Rao-type mean-squared-error lower bound of Tarima and Flournoy (2024), derive and justify optimal sequential designs with the dual goal of sequential testing and post-test estimation. Concretely: (i) formulate an optimality criterion expressed through the decomposition that, subject to testing requirements such as type-I error rate control and power, trades off the design information ID(θ)\mathcal{I}_D(\theta) spent by interim decisions (equivalently, the unspent decision-conditional estimation information, or the sequential-CRLB lower bound on post-adaptation mean squared error) against trial-efficiency quantities such as expected sample size; and (ii) construct the decision rules that are optimal under this criterion within an explicitly stated class of sequential designs — early-stopping boundaries, sample-size re-estimation policies, or their multi-stage generalizations — together with a proof or rigorous justification of their optimality. Both the criterion and the optimal rules are part of the task, since no dual-goal optimality criterion for informative sequential designs currently exists in the literature.

References

  1. Sergey Tarima and Nancy Flournoy (2024), 'The cost of sequential adaptation and the lower bound for mean squared error', Statistical Papers 65(9):5529–5553, DOI 10.1007/s00362-024-01565-x, https://doi.org/10.1007/s00362-024-01565-x (final preprint arXiv:2209.02436v2, https://arxiv.org/abs/2209.02436)
  2. G. Caruso, W. F. Rosenberger, P. Mozgunov, N. Flournoy (2025), 'An information metric for comparing and assessing informative interim decisions in sequential clinical trials', arXiv:2509.04904v3, https://arxiv.org/abs/2509.04904
  3. David S. Robertson, Babak Choodari-Oskooei, Munya Dimairo, Laura Flight, Philip Pallmann, Thomas Jaki (2023), 'Point estimation for adaptive trial designs I: A methodological review', Statistics in Medicine 42(2):122–145, DOI 10.1002/sim.9605, https://doi.org/10.1002/sim.9605
  4. Ian C. Marschner (2021), 'A General Framework for the Analysis of Adaptive Experiments', Statistical Science 36(3):465–492, DOI 10.1214/20-sts803, https://doi.org/10.1214/20-sts803
  5. A. Liu and W. Hall (1999), 'Unbiased estimation following a group sequential test', Biometrika 86(1):71–78, DOI 10.1093/biomet/86.1.71, https://doi.org/10.1093/biomet/86.1.71
  6. Nancy Flournoy and Sergey Tarima (2023), 'Posterior alternatives with informative early stopping', Statistical Papers 64(4):1329–1341, DOI 10.1007/s00362-023-01429-w, https://doi.org/10.1007/s00362-023-01429-w
  7. J. Whitehead (1986), 'On the bias of maximum likelihood estimation following a sequential test', Biometrika 73(3):573–581, DOI 10.1093/biomet/73.3.573, https://doi.org/10.1093/biomet/73.3.573
  8. J. Wolfowitz (1947), 'The Efficiency of Sequential Estimates and Wald's Equation for Sequential Processes', The Annals of Mathematical Statistics 18(2):215–230, DOI 10.1214/aoms/1177730439, https://doi.org/10.1214/aoms/1177730439