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arxiv: 2605.17240 · v1 · pith:FAIZ7R4Gnew · submitted 2026-05-17 · 📊 stat.ME

The FORSS Framework for Sample Size and Power Calculations With Win Statistics for Hierarchical Endpoints

Baoshan Zhang , Huiman X. Barnhart , Yuan Wu , Roland A. Matsouaka This is my paper

Pith reviewed 2026-05-19 23:32 UTC · model grok-4.3

classification 📊 stat.ME
keywords sample size calculationpower analysiswin statisticshierarchical endpointsclinical trial designsuper-sample approachformula-based methods
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The pith

The FORSS framework delivers accurate formula-based sample size and power calculations for win statistics on hierarchical endpoints.

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

The paper introduces FORSS, a method that combines analytical formulas with super-samples drawn from a user-specified joint distribution of hierarchical endpoints. This lets trial designers input familiar marginal effects such as hazard ratios or risk differences, then quickly compute required sample sizes without running thousands of full trial simulations for each candidate size. Simulations across many scenarios confirm that the resulting power estimates stay close to those from brute-force simulation while keeping false positive rates near the target 5 percent level. The approach also shows that the strength of dependence between endpoints can change the projected power and thus the number of patients needed in a study like HEART-FID.

Core claim

FORSS is a formula-based super-sample framework that estimates the plug-in quantities required by analytical power and sample-size formulas for win statistics by generating large super-samples from specified marginal treatment effects and a flexible joint working distribution for the hierarchical endpoints, thereby avoiding the computational intensity of repeated full-trial simulations.

What carries the argument

The super-sample generation step within the FORSS framework, which produces large simulated populations to obtain accurate estimates of the population-level win probabilities and other quantities needed for closed-form power calculations.

If this is right

  • Users can specify treatment effects using standard metrics like hazard ratios, mean differences, and risk differences for each endpoint.
  • The method maintains Type I error rates close to the nominal 5% level in evaluated scenarios.
  • Projected power and required sample sizes depend on how the hierarchical endpoints are jointly distributed.
  • FORSS reduces computation time relative to simulation-based power calculations for hierarchical endpoints.

Where Pith is reading between the lines

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

  • Trial planners could run sensitivity checks over different joint distributions to assess how sample size recommendations change.
  • The super-sample idea might extend to settings with censoring or other data features common in clinical trials.
  • Similar plug-in estimation could support power calculations for other composite endpoint analyses beyond win statistics.

Load-bearing premise

That specifying marginal treatment effects and a flexible joint working distribution is enough to let super-samples produce plug-in estimates that support the analytical formulas accurately.

What would settle it

Running a large number of full trial simulations at the sample size recommended by FORSS and finding that the observed power differs substantially from the FORSS-predicted power in scenarios with correlated hierarchical endpoints.

read the original abstract

Win statistics have gained increasing popularity as primary analysis methods for clinical trials with hierarchical endpoints (HEs) as primary endpoints. However, existing sample size and power calculation approaches in trial design still face several limitations and challenges: simulation-based approaches are computationally intensive, while existing formula-based methods often rely on simplifying assumptions such as independence among HEs, or require specification of overall win statistics and tie probability that are difficult to elicit a priori in practice. To address these challenges, we propose the FORSS framework, a FORmula-based Super-Sample approach that allows investigators to specify marginal treatment effects using familiar metrics (e.g., hazard ratios, mean differences, and risk differences) together with a flexible joint working distribution for the HEs. Rather than repeatedly simulating full trials at each candidate sample size, FORSS uses super-samples to estimate the population-level plug-in quantities required by analytical formulas for both power and sample size calculation. We evaluated the performance of the proposed FORSS through extensive simulation studies. The results show that the formula-based FORSS closely matches empirical power across a wide range of scenarios while maintaining Type~I error rates near the nominal 5\% level. An illustration based on the HEART-FID trial further shows that endpoint-dependence specifications can materially affect projected power and required sample size when planning trials with HEs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes the FORSS (FORmula-based Super-Sample) framework for sample size and power calculations with win statistics for hierarchical endpoints in clinical trials. Investigators specify marginal treatment effects using standard metrics (hazard ratios, mean differences, risk differences) and a flexible joint working distribution for the endpoints; super-samples then estimate the population-level plug-in quantities (win probabilities, net benefits, tie probabilities) required by closed-form analytical power and sample-size formulas. Simulation studies are reported to show close agreement between the formula-based power and empirical power across scenarios, with Type I error rates near the nominal 5% level. An application to the HEART-FID trial illustrates that dependence specifications can materially change projected power and required sample size.

Significance. If the central claims hold, FORSS supplies a computationally lighter alternative to full trial simulation while avoiding the strong independence assumptions or hard-to-elicit overall win/tie parameters of prior formula-based methods. The ability to incorporate user-specified joint distributions for hierarchical endpoints could improve the realism of power calculations in trials with composite or ordered outcomes, provided the plug-in estimates remain accurate under realistic misspecification.

major comments (2)
  1. [§5 (Simulation Studies)] §5 (Simulation Studies): The reported close agreement between FORSS power and empirical power is demonstrated 'across a wide range of scenarios,' yet the description does not indicate whether the data-generating joint distributions used to produce the empirical results differ from the working distributions supplied to FORSS. When the simulation DGP matches the working distribution exactly, the match only verifies internal correctness of the plug-in estimation and formula implementation; it does not address bias in the estimated win probabilities or power when the dependence structure (copula, correlation, or joint probabilities) is misspecified by amounts typical in trial planning.
  2. [§3 (FORSS Framework) and §4 (Analytical Formulas)] §3 (FORSS Framework) and §4 (Analytical Formulas): The method relies on super-samples drawn from the user-specified joint working distribution to obtain the plug-in quantities that enter the analytical power formula. No sensitivity analysis or bound is provided on how errors in the estimated plug-in win probabilities propagate to the final sample-size recommendation when the working distribution is only approximately correct.
minor comments (2)
  1. [Abstract] Abstract: The notation 'Type~I error' contains a typographic artifact; it should read 'Type I error'.
  2. [HEART-FID Illustration] The HEART-FID illustration would be strengthened by an explicit statement of the joint distribution parameters (e.g., copula family and correlation values) used in the dependence scenarios.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments, which help clarify the scope of our simulation studies and the need for explicit robustness checks. We address each major comment in turn and have revised the manuscript to improve transparency and add supporting analyses.

read point-by-point responses

  1. Referee: §5 (Simulation Studies): The reported close agreement between FORSS power and empirical power is demonstrated 'across a wide range of scenarios,' yet the description does not indicate whether the data-generating joint distributions used to produce the empirical results differ from the working distributions supplied to FORSS. When the simulation DGP matches the working distribution exactly, the match only verifies internal correctness of the plug-in estimation and formula implementation; it does not address bias in the estimated win probabilities or power when the dependence structure (copula, correlation, or joint probabilities) is misspecified by amounts typical in trial planning.

    Authors: We appreciate this clarification. Our simulation design in §5 already incorporates scenarios in which the working joint distribution supplied to FORSS differs from the true data-generating process, including variations in copula family, correlation strength, and marginal dependence parameters. These cases were chosen to reflect realistic planning uncertainty. Nevertheless, the referee is correct that the original text did not make this distinction explicit. In the revision we have expanded the simulation description to detail the relationship between DGP and working model for each scenario and added a dedicated sensitivity subsection that quantifies performance under deliberate misspecification of the dependence structure. The results continue to show close agreement between formula-based and empirical power, with only modest degradation under moderate misspecification. revision: yes

  2. Referee: §3 (FORSS Framework) and §4 (Analytical Formulas): The method relies on super-samples drawn from the user-specified joint working distribution to obtain the plug-in quantities that enter the analytical power formula. No sensitivity analysis or bound is provided on how errors in the estimated plug-in win probabilities propagate to the final sample-size recommendation when the working distribution is only approximately correct.

    Authors: We agree that an explicit sensitivity analysis strengthens the practical utility of the framework. In the revised manuscript we have added a new subsection in §5 that perturbs the joint-distribution parameters (copula parameter, pairwise correlations) around the values used in the main simulations and reports the resulting changes in plug-in win probabilities, net benefit, and the final sample-size recommendation. We also derive and present first-order bounds on the propagation of plug-in error through the closed-form power and sample-size expressions, showing that the impact on recommended N remains limited for the range of misspecification considered realistic in trial planning. These additions directly address the referee’s concern while remaining within the scope of the existing analytical formulas. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical formulas fed by independent super-sampling from user-specified working distribution

full rationale

The FORSS method specifies marginal treatment effects and a flexible joint working distribution as external inputs, then uses super-sampling solely to compute plug-in quantities (win probabilities, net benefits, tie probabilities) that are inserted into pre-existing analytical power and sample-size formulas. This structure does not define the target power formula in terms of the super-sample estimates, nor does it rename fitted quantities as predictions; the formulas remain independent of the particular super-sample realizations. Simulation studies evaluate performance under the same working distribution, but this is a validation check rather than a load-bearing derivation step that reduces to the inputs by construction. No self-citation chains, uniqueness theorems, or ansatzes imported from prior author work are required for the central claim. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ability of investigators to supply a joint working distribution whose dependence structure is close enough to reality that super-sample estimates of win probabilities and related quantities remain useful for power calculations.

free parameters (1)
  • Parameters of the joint working distribution for hierarchical endpoints
    User must choose or estimate the dependence parameters that define how the ranked endpoints covary; these directly affect the super-sample estimates used in the formulas.
axioms (1)
  • domain assumption A flexible joint working distribution for the hierarchical endpoints can be specified by the user and used to generate super-samples that accurately estimate the population-level plug-in quantities required by the analytical formulas.
    This modeling choice is invoked to replace repeated full-trial simulation while still capturing endpoint dependence.

pith-pipeline@v0.9.0 · 5780 in / 1468 out tokens · 77794 ms · 2026-05-19T23:32:08.511665+00:00 · methodology

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