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arxiv: 2605.20135 · v1 · pith:EM2E4L35new · submitted 2026-05-19 · 📊 stat.ME

Quantile-Based Effectiveness Persistence Function: A Tail-Focused Metric with Theory, Estimation, and Application to Biosimilar Evaluation

Sankaran P.G. , Prasanth V.P. , Midhu N.N This is my paper

Pith reviewed 2026-05-20 03:20 UTC · model grok-4.3

classification 📊 stat.ME
keywords quantile-based persistencetail meanupper tail analysisL-momentbiosimilar evaluationnonparametric estimatorequivalence testclinical adherence
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The pith

A ratio of tail mean to quantile function measures how long high clinical responses persist among top performers.

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

The paper defines a quantile-based effectiveness persistence function as the ratio of the tail mean to the quantile function. This construction parallels expected shortfall and equals the first L-moment of the scaled tail. The authors prove its properties, supply a nonparametric estimator, and build a bootstrap-calibrated two-sample equivalence test. The metric is intended to detect clinically relevant deviations in the upper tail of response distributions that conventional mean or median summaries miss. It is applied to biosimilar evaluation where sustained high performance matters more than average behavior.

Core claim

The quantile-based effectiveness persistence function is defined as the ratio between the tail mean and the quantile function. It is shown to be equivalent to the first L-moment of the scaled tail and therefore supplies robust inference tools for upper-tail analysis. A simple nonparametric estimator is derived and paired with a bootstrap-calibrated two-sample equivalence test that targets clinically meaningful tail differences.

What carries the argument

The quantile-based effectiveness persistence function, defined as the ratio of tail mean to quantile function, quantifies persistence in the upper tail of clinical response distributions.

If this is right

  • The function complements median and mean summaries by focusing on how long high responses persist.
  • It yields a bootstrap-based two-sample test for equivalence in upper-tail behavior.
  • Nonparametric estimation allows application to biosimilar data without parametric tail assumptions.
  • Properties derived from its equivalence to the first L-moment support stable inference in small samples.

Where Pith is reading between the lines

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

  • The same tail-ratio idea could be adapted to other fields that track sustained extreme performance, such as reliability engineering.
  • Regulatory comparisons of biosimilars might incorporate this metric to emphasize long-term top-end outcomes rather than averages.
  • Repeated application across multiple quantiles could reveal whether persistence is uniform or concentrated at particular tail levels.

Load-bearing premise

The upper tail of the response distribution in clinical data captures clinically relevant persistence that this ratio can measure without further distributional assumptions.

What would settle it

In a controlled simulation or real dataset with known upper-tail differences, the new ratio fails to detect the difference while standard mean or median tests succeed, or vice versa.

read the original abstract

In clinical studies, persistence, which measures the duration of time a patient continues to take a prescribed medication without discontinuation, is increasingly recognized as a critical indicator of adherence to medication. Adherence encompasses not only whether a patient takes their medication as prescribed but also the consistency and duration with which they do so. Among the various metrics used to evaluate adherence, persistence stands out as a particularly robust measure because it provides a temporal dimension, reflecting the sustained commitment of patients to their therapeutic regimens. This focus on persistence offers unique insights into adherence-related quality and performance, shedding light on the challenges and opportunities to optimize long-term medication use. The comparison of upper-tail clinical performance, which measures the extent to which very large responses persist among top responders, is often more decisive in therapy evaluation than conventional summaries. In this paper, we introduce the quantile-based effectiveness persistence function defined as the ratio between the tail mean and the quantile function. The notion parallels expected shortfall in risk theory and is tailored to detect clinically meaningful deviations in the upper tail. We establish key properties and show that the function is equivalent to the first L-moment of the scaled tail, yielding robust inference tools. We derive a simple nonparametric estimator of the function and develop a bootstrap-calibrated two-sample (upper-tail) equivalence test. Simulation studies and real-data analysis illustrate that the proposed measures captures clinically relevant tail persistence that complements median and mean-based summaries.

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 / 1 minor

Summary. The paper introduces the quantile-based effectiveness persistence function, defined as the ratio of the tail mean to the quantile function Q(p). It claims this metric parallels expected shortfall, is equivalent to the first L-moment of the scaled tail, and provides a nonparametric estimator along with a bootstrap-calibrated two-sample equivalence test for upper-tail deviations. The approach is motivated by clinical persistence in medication adherence and illustrated via simulations and biosimilar data analysis.

Significance. If the estimator's properties hold in moderate samples, the metric could offer a useful complement to mean- and median-based summaries for detecting clinically relevant tail persistence in clinical trials, particularly for biosimilar evaluations where upper-tail performance matters.

major comments (2)
  1. [Theory section] The equivalence to the first L-moment of the scaled tail is a direct definitional identity once the scaled tail is taken as X/Q(p) conditional on X > Q(p); this should be stated explicitly in the theory section rather than presented as an independent result.
  2. [Estimation and simulation studies] The nonparametric estimator of the tail-mean/quantile ratio inherits the slow convergence of extreme quantiles. In biosimilar trials with n typically 100-300 per arm and p near 1, this risks unstable estimates and degraded bootstrap coverage under heavy tails, which directly affects the reliability of the proposed two-sample equivalence test.
minor comments (1)
  1. Clarify the choice of p in applications and provide guidance on sensitivity to this tuning parameter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the presentation and address practical concerns.

read point-by-point responses

  1. Referee: [Theory section] The equivalence to the first L-moment of the scaled tail is a direct definitional identity once the scaled tail is taken as X/Q(p) conditional on X > Q(p); this should be stated explicitly in the theory section rather than presented as an independent result.

    Authors: We agree that the stated equivalence follows directly once the scaled tail is defined as the conditional random variable X/Q(p) given X > Q(p). In the revised manuscript we will explicitly note this as a definitional identity in the theory section and adjust the surrounding text to avoid any implication that it is an independent result. This change improves clarity without altering the technical content. revision: yes

  2. Referee: [Estimation and simulation studies] The nonparametric estimator of the tail-mean/quantile ratio inherits the slow convergence of extreme quantiles. In biosimilar trials with n typically 100-300 per arm and p near 1, this risks unstable estimates and degraded bootstrap coverage under heavy tails, which directly affects the reliability of the proposed two-sample equivalence test.

    Authors: The referee correctly notes that the estimator inherits the slower convergence rate of extreme-quantile estimators. Our existing simulation design already includes sample sizes in the 100–300 range and several tail behaviors. To respond directly to the concern, we will add a new subsection that reports bootstrap coverage under heavier-tailed distributions (e.g., Student-t with low degrees of freedom) and will include a brief discussion of practical guidance for selecting p when n is moderate. These additions will be presented as supplementary simulation results. revision: partial

Circularity Check

1 steps flagged

Equivalence of persistence function to first L-moment is definitional identity

specific steps
  1. self definitional [Abstract]
    "we introduce the quantile-based effectiveness persistence function defined as the ratio between the tail mean and the quantile function. ... We establish key properties and show that the function is equivalent to the first L-moment of the scaled tail, yielding robust inference tools."

    The function is defined as tail mean / quantile. The scaled tail is X/Q(p) conditional on X > Q(p), so its mean equals the defined ratio. The first L-moment of any distribution is its mean, making the stated equivalence true by construction from the definition plus the known fact that λ1 = mean. This is not an independent result but a direct restatement.

full rationale

The paper defines the quantile-based effectiveness persistence function explicitly as the ratio of tail mean to quantile function. It then presents the equivalence to the first L-moment of the scaled tail as a key property that yields robust inference. Because the first L-moment equals the mean and the ratio is precisely the mean of the scaled tail (X/Q(p) | X > Q(p)), the equivalence holds by algebraic identity from the definition and standard L-moment properties rather than an independent derivation. This makes the central claim of 'yielding robust inference tools' reduce to a renaming of the input definition. The nonparametric estimator and bootstrap test appear to have separate content, so the circularity is partial rather than total. No load-bearing self-citations or other patterns are evident in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim relies on standard properties of quantiles and L-moments from statistical theory, with the new function defined directly.

axioms (1)
  • domain assumption The tail mean and quantile function are well-defined for the positive response distributions in clinical data.
    Required for the ratio to be meaningful.
invented entities (1)
  • quantile-based effectiveness persistence function no independent evidence
    purpose: Measure of upper-tail persistence in effectiveness
    Introduced in this paper as a new metric.

pith-pipeline@v0.9.0 · 5793 in / 1315 out tokens · 58794 ms · 2026-05-20T03:20:54.420743+00:00 · methodology

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