Robust confidence intervals for generalized linear models

Andrea Panarotto; Livio Finos; Riccardo De Santis

arxiv: 2605.03060 · v1 · submitted 2026-05-04 · 📊 stat.ME

Robust confidence intervals for generalized linear models

Andrea Panarotto , Riccardo De Santis , Livio Finos This is my paper

Pith reviewed 2026-05-08 17:57 UTC · model grok-4.3

classification 📊 stat.ME

keywords generalized linear modelsconfidence intervalsrobust inferencesign-flippingvariance misspecificationscore testsRNA-seq

0 comments

The pith

Sign-flipping individual score contributions yields asymptotically valid confidence intervals for generalized linear models under variance misspecification.

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

Generalized linear models are widely used for biomedical data such as RNA-seq counts, yet standard confidence intervals for their parameters often undercover when variances deviate from the assumed mean-variance link through overdispersion or heteroskedasticity. The paper develops intervals by inverting tests formed through sign-flipping of each observation's contribution to the score function, with bounds found by bisection. It proves these intervals retain correct asymptotic coverage for any variance structure. Simulations confirm reliable performance where Wald intervals fail, and the method is demonstrated on differential expression analysis of cancer RNA-sequencing data exhibiting pervasive heterogeneous variability.

Core claim

Inverting hypothesis tests obtained by sign-flipping the individual score contributions produces confidence intervals whose asymptotic coverage remains valid under general variance misspecification in generalized linear models; the resulting intervals achieve reliable coverage in simulations and outperform standard Wald-type intervals when the mean-variance relationship is violated.

What carries the argument

Inversion via bisection of sign-flipping tests applied to individual score contributions from the GLM estimating equations.

If this is right

The intervals maintain nominal coverage rates in finite samples when variances are misspecified in ways common to count data.
They deliver shorter intervals or higher power than Wald intervals in the same misspecified settings.
The procedure applies directly to high-dimensional differential expression analyses without requiring parametric variance modeling.
Bisection search reliably locates the interval endpoints once the sign-flip test statistic is computable.

Where Pith is reading between the lines

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

The same sign-flip inversion could be applied to score-based tests in other estimating-equation settings such as generalized estimating equations for clustered data.
In practice the method may reduce the need for separate overdispersion parameters or quasi-likelihood adjustments when the goal is interval estimation rather than point estimation.
If the score contributions can be computed efficiently, the approach scales to the sample sizes typical of modern sequencing experiments without additional tuning parameters.

Load-bearing premise

Inverting sign-flipped versions of the individual score contributions produces tests whose acceptance regions, when collected, yield intervals with the claimed asymptotic coverage even under completely arbitrary heteroskedasticity or overdispersion.

What would settle it

A Monte Carlo experiment in which the empirical coverage of the proposed intervals falls materially below the nominal level across repeated samples drawn from a GLM with severe, observation-specific variance inflation would falsify the asymptotic validity result.

Figures

Figures reproduced from arXiv: 2605.03060 by Andrea Panarotto, Livio Finos, Riccardo De Santis.

**Figure 1.** Figure 1: Values of fp(β0) on an equispaced grid of values between βˆ obs − 1 and βˆ obs. The n observations have been generated from a logistic model. The figure shows in red the points with an associated p-value lower than α/2, and in black the points that should belong to the confidence set. With n = 50 (left) the p-value function is monotonic. With n = 20 (right), the function is non-monotonic and the bisection … view at source ↗

**Figure 2.** Figure 2: Coverage probabilities of the confidence intervals built from in view at source ↗

**Figure 3.** Figure 3: Median of the interval width of the confidence intervals built from view at source ↗

**Figure 4.** Figure 4: Distributions of the amplitudes of the confidence intervals accord view at source ↗

**Figure 5.** Figure 5: Amplitude comparison between flip and sandwich-based intervals view at source ↗

**Figure 6.** Figure 6: Comparison of the overlaps between the flip and sandwich-based view at source ↗

**Figure 7.** Figure 7: Overlap between the confidence intervals built with Poisson and view at source ↗

read the original abstract

Reliable uncertainty quantification is a central challenge in the analysis of modern biomedical data, where complex sources of variability often violate standard modeling assumptions. In generalized linear models (GLMs), confidence intervals for regression parameters provide such information, but they typically rely on correct specification of the mean-variance relationship. However, overdispersion, heteroskedasticity, and unobserved biological variability can lead to substantial undercoverage in practice. We propose a method for constructing confidence intervals that remains valid under variance misspecification. The approach is based on the inversion of hypothesis tests obtained by sign-flipping individual score contributions and uses a bisection algorithm to determine the interval bounds. The resulting intervals inherit robustness properties from the underlying tests, and we establish their asymptotic validity under general variance misspecification. Through simulation studies, we show that the proposed method achieves reliable coverage and outperforms standard Wald-type intervals when model assumptions are violated. We illustrate the approach in a differential expression analysis of RNA-sequencing data from a cancer study, where heterogeneous variability is pervasive and parametric methods can yield inconsistent inference. The proposed framework provides a practical and robust alternative to conventional quasi-likelihood or Wald-based methods for interval estimation in GLMs, particularly suited to high-throughput biomedical applications.

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.

Desk Editor's Note private letter to a colleague

Sign-flipping inversion of score tests gives asymptotically valid robust CIs for GLMs under variance misspecification.

read the letter

The main takeaway is that this paper supplies a concrete way to build confidence intervals for GLM parameters that stay valid when the variance function is misspecified, using inversion of sign-flipped score contributions and bisection to locate the bounds. The asymptotic argument holds because the score is mean zero at the true parameter regardless of variance, and the sign-flipped version shares the same limiting normal distribution under standard conditions, so the inverted tests deliver correct coverage even with overdispersion or heteroskedasticity. That part checks out without circularity or extra assumptions beyond Lindeberg-type regularity. What the work does well is back the claim with simulations that show reliable coverage where standard Wald intervals undercover, plus a direct application to RNA-seq differential expression in cancer data where heterogeneous variability is the norm. The method is presented as a practical alternative to quasi-likelihood or sandwich approaches, and the simulations make the advantage visible in realistic misspecification settings. Soft spots are modest and mostly about presentation. The abstract and stress-test logic are clear, but the full regularity conditions for the inversion step could be spelled out more explicitly to cover edge cases like very small samples or high-dimensional covariates common in biomedical work. The bisection routine is efficient in principle but lacks any timing or scaling discussion for large datasets. No load-bearing gaps appear in the core construction. This paper is aimed at applied statisticians and biostatisticians who need reliable intervals for GLMs in settings where variance assumptions routinely fail. A reader working on robust inference or high-throughput data analysis would get usable value from the simulations and example. It shows clear engagement with the literature on misspecification and deserves a serious referee because the idea is distinct, the theory is grounded, and the evidence is proportionate to the claims.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes constructing robust confidence intervals for regression parameters in generalized linear models by inverting hypothesis tests formed via sign-flipping of individual score contributions, with bounds located by bisection. It claims that the resulting intervals are asymptotically valid under general variance misspecification (heteroskedasticity or overdispersion) while the mean model remains correct, supports this with simulation studies showing reliable coverage that outperforms Wald-type intervals, and illustrates the method on RNA-sequencing differential expression data from a cancer study.

Significance. If the asymptotic validity result holds, the approach supplies a practical, randomization-based alternative to quasi-likelihood or sandwich estimators for interval estimation in GLMs when variance assumptions fail, which is common in high-throughput biomedical data. The sign-flipping construction on scores is a clean way to achieve robustness without estimating extra variance parameters, and the simulation evidence plus real-data example add to its applied value.

major comments (2)

[Theoretical results / asymptotic validity section] The central claim of asymptotic validity under arbitrary variance misspecification rests on the sign-flipping tests having correct asymptotic size and the inversion preserving coverage. The manuscript asserts this result but does not list the precise regularity conditions (e.g., Lindeberg-type conditions on the score contributions or moment bounds ensuring the sign-flipped and original scores share the same limiting normal distribution). Please add an explicit statement of these conditions in the theoretical section and a brief proof sketch showing that the test inversion yields intervals with the claimed coverage.
[Simulation studies] In the simulation studies, the data-generating processes used to evaluate coverage under misspecification should be described in sufficient detail (including the exact forms of heteroskedasticity or overdispersion and the range of sample sizes) so that readers can assess whether the reported superior performance is robust to the simulation design choices rather than specific to the chosen scenarios.

minor comments (2)

[Method description] The bisection algorithm for locating interval bounds is mentioned but its convergence tolerance and implementation details (e.g., handling of discrete score flips) are not specified; a short algorithmic description or pseudocode would improve reproducibility.
[Method description] Notation for the sign-flipped score vector and the resulting test statistic should be introduced once and used consistently; currently the transition from the score contributions to the randomization distribution is somewhat terse.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the method's potential value. Below we address each major comment in turn. We will revise the manuscript accordingly to incorporate the suggested improvements.

read point-by-point responses

Referee: [Theoretical results / asymptotic validity section] The central claim of asymptotic validity under arbitrary variance misspecification rests on the sign-flipping tests having correct asymptotic size and the inversion preserving coverage. The manuscript asserts this result but does not list the precise regularity conditions (e.g., Lindeberg-type conditions on the score contributions or moment bounds ensuring the sign-flipped and original scores share the same limiting normal distribution). Please add an explicit statement of these conditions in the theoretical section and a brief proof sketch showing that the test inversion yields intervals with the claimed coverage.

Authors: We agree that the theoretical section would benefit from a more explicit statement of the regularity conditions and a proof sketch. In the revised manuscript, we will add a dedicated subsection outlining the key assumptions, including Lindeberg-type conditions on the score contributions and moment bounds that ensure the sign-flipped scores converge to the same limiting normal distribution as the original scores. We will also provide a brief proof sketch demonstrating that the inversion of the asymptotically valid tests yields intervals with the desired asymptotic coverage under variance misspecification. This will strengthen the presentation without altering the core results. revision: yes
Referee: [Simulation studies] In the simulation studies, the data-generating processes used to evaluate coverage under misspecification should be described in sufficient detail (including the exact forms of heteroskedasticity or overdispersion and the range of sample sizes) so that readers can assess whether the reported superior performance is robust to the simulation design choices rather than specific to the chosen scenarios.

Authors: We acknowledge that additional details on the simulation design would enhance reproducibility and allow readers to better evaluate the robustness of our findings. In the revised version, we will expand the description of the data-generating processes, specifying the exact forms of heteroskedasticity and overdispersion used (e.g., variance functions and parameters), as well as the full range of sample sizes considered. We will also clarify how these choices relate to the real-data applications. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives robust confidence intervals for GLMs by inverting sign-flipping tests applied to individual score contributions, then establishes asymptotic validity under variance misspecification via standard Lindeberg CLT arguments on the score process and its randomized counterpart. No equation or claim reduces the interval bounds to a fitted parameter by construction, nor does the central result depend on a self-citation chain or imported uniqueness theorem. The construction is self-contained against external benchmarks (randomization tests and sandwich variance ideas) and does not rename known patterns or smuggle ansatzes. This is the normal honest outcome for a paper whose core contribution is a new application of existing randomization techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no free parameters, invented entities, or non-standard axioms are explicitly introduced. The approach relies on standard asymptotic theory for score tests.

axioms (1)

standard math Standard regularity conditions for asymptotic normality of score statistics in GLMs
Invoked to establish asymptotic validity of the inverted tests under variance misspecification.

pith-pipeline@v0.9.0 · 5511 in / 1170 out tokens · 36450 ms · 2026-05-08T17:57:05.793468+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation (washburn_uniqueness_aczel) — RS's cost-uniqueness chain is unrelated to score-test construction washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

S(F_g) = n^{-1/2} X^T W^{1/2} (I−H) F_g V^{-1/2} (Y − μ̂) ... S*(F_g) = S(F_g) / V(S(F_g))^{1/2}
Foundation.BranchSelection — α here is a significance level, not the bilinear-branch parameter; no shared structure branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We fix a confidence level 1−α ... invert the one-sided tests H_0: β = β_0 ... a (1−α)-confidence interval for a parameter β is defined as the set of all parameter values β_0 that would not be rejected by a level-α test

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

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Journal of the American Statistical Association , volume =

De Santis, Riccardo and Goeman, Jelle J and Hemerik, Jesse and Davenport, Samuel and Finos, Livio , title =. Journal of the American Statistical Association , volume =. 2025 , publisher =

work page 2025

[2] [2]

2015 , publisher=

Foundations of Linear and Generalized Linear Models , author=. 2015 , publisher=

work page 2015

[3] [3]

Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

Robust testing in generalized linear models by sign flipping score contributions , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2020 , publisher=

work page 2020

[4] [4]

Test , volume=

Exact testing with random permutations , author=. Test , volume=. 2018 , publisher=

work page 2018

[5] [5]

Salvan, Alessandra and Sartori, Nicola and Pace, Luigi , year =. Modelli

work page

[6] [6]

1997 , publisher=

Principles of statistical inference: from a Neo-Fisherian perspective , author=. 1997 , publisher=

work page 1997

[7] [7]

1984 , number=

Confidence intervals for discrete distributions , author=. 1984 , number=

work page 1984

[8] [8]

Multivariate Permutation Tests: with Applications in Biostatistics , isbn =

Pesarin, Fortunato , year =. Multivariate Permutation Tests: with Applications in Biostatistics , isbn =

work page

[9] [9]

and Kirk, Shanah and Lee, Y

Erickson, Bradley J. and Kirk, Shanah and Lee, Y. and Bathe, Oliver and Kearns, Melissa and Gerdes, C. and Rieger-Christ, Kimberly and Lemmerman, John , year =. The

work page

[10] [10]

Współczesna Onkologia , author =

Review. Współczesna Onkologia , author =. 2015 , pages =. doi:10.5114/wo.2014.47136 , urldate =

work page doi:10.5114/wo.2014.47136 2015

[11] [11]

Biometrics , author =

Confidence. Biometrics , author =. 1996 , pages =. doi:10.2307/2532852 , number =

work page doi:10.2307/2532852 1996

[12] [12]

Journal of the American Statistical Association , author =

On Obtaining Permutation Distributions in Polynomial Time , volume =. Journal of the American Statistical Association , author =. 1983 , pages =. doi:10.1080/01621459.1983.10477990 , language =

work page doi:10.1080/01621459.1983.10477990 1983

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Journal of the American Statistical Association , author =

On. Journal of the American Statistical Association , author =. 1984 , pages =. doi:10.1080/01621459.1984.10477085 , language =

work page doi:10.1080/01621459.1984.10477085 1984

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Journal of Computational and Graphical Statistics , author =

Fast Conservative. Journal of Computational and Graphical Statistics , author =. 2025 , pages =. doi:10.1080/10618600.2025.2526416 , language =

work page doi:10.1080/10618600.2025.2526416 2025

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, year =

Van Der Vaart, Aad W. , year =. Asymptotic

work page

[16] [16]

Proceedings of the National Academy of Sciences , volume=

Universal inference , author=. Proceedings of the National Academy of Sciences , volume=. 2020 , publisher=

work page 2020

[17] [17]

Stat , volume=

A note on universal inference , author=. Stat , volume=. 2022 , publisher=

work page 2022

[18] [18]

Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

The HulC: confidence regions from convex hulls , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2024 , publisher=

work page 2024

[19] [19]

Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , pages=

Tests of separate families of hypotheses , author=. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , pages=

work page

[20] [20]

Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

Further results on tests of separate families of hypotheses , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 1962 , publisher=

work page 1962

[21] [21]

, booktitle=

Huber, Peter J. , booktitle=. 1967 , title =

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[22] [22]

Maximum Likelihood Estimation of Misspecified Models , volume =

White, Halbert , journal =. Maximum Likelihood Estimation of Misspecified Models , volume =

work page

[23] [23]

Proceedings of the National Academy of Sciences , volume=

On the role of parameterization in models with a misspecified nuisance component , author=. Proceedings of the National Academy of Sciences , volume=. 2024 , publisher=

work page 2024

[24] [24]

Studies in Nonlinear Dynamics & Econometrics , author =

Likelihood-ratio-based confidence intervals for multiple threshold parameters , volume =. Studies in Nonlinear Dynamics & Econometrics , author =. 2025 , pages =. doi:10.1515/snde-2023-0029 , language =

work page doi:10.1515/snde-2023-0029 2025

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Ecology and Society , author =

Green without envy: how social capital alleviates tensions from a. Ecology and Society , author =. 2018 , pages =. doi:10.5751/ES-10181-230410 , language =

work page doi:10.5751/es-10181-230410 2018

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Communications in Statistics - Theory and Methods , author =

Bootstrapping some. Communications in Statistics - Theory and Methods , author =. 2023 , pages =. doi:10.1080/03610926.2021.1955389 , language =

work page doi:10.1080/03610926.2021.1955389 2023

[27] [27]

Biometrics , author =

Debiased lasso for generalized linear models with a diverging number of covariates , volume =. Biometrics , author =. 2023 , pages =. doi:10.1111/biom.13587 , language =

work page doi:10.1111/biom.13587 2023

[28] [28]

Stats , author =

Improving confidence interval estimation in logistic regression with multicollinear predictors: a comparative study of shrinkage estimators and application to prostate cancer data , volume =. Stats , author =. 2026 , pages =. doi:10.3390/stats9010011 , language =

work page doi:10.3390/stats9010011 2026

[29] [29]

Psychometrika , author =

Accurate confidence and bayesian interval estimation for non-centrality parameters and effect size indices , volume =. Psychometrika , author =. 2023 , pages =. doi:10.1007/s11336-022-09899-x , language =

work page doi:10.1007/s11336-022-09899-x 2023

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BMC Genomics , author =

Technical and biological variance structure in. BMC Genomics , author =. 2012 , pages =. doi:10.1186/1471-2164-13-304 , language =

work page doi:10.1186/1471-2164-13-304 2012

[31] [31]

and Sebastian, Jessalyn and Minin, Volodymyr M

Němcová, Barbora and Goldstein, Isaac H. and Sebastian, Jessalyn and Minin, Volodymyr M. and Bracher, Johannes , copyright =. Unjustified. doi:10.1101/2025.07.31.25332479 , year =

work page doi:10.1101/2025.07.31.25332479 2025

[32] [32]

two-stage summary statistics

Robust inference for generalized linear mixed models: a “two-stage summary statistics” approach based on score sign flipping , author=. Psychometrika , volume=. 2025 , publisher=

work page 2025

[33] [33]

Mathematical proceedings of the cambridge philosophical society , volume=

Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation , author=. Mathematical proceedings of the cambridge philosophical society , volume=. 1948 , organization=

work page 1948

[34] [34]

Communications in Statistics-Theory and Methods , volume=

A comment on locally most powerful tests in the presence of nuisance parameters , author=. Communications in Statistics-Theory and Methods , volume=. 2002 , publisher=

work page 2002

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