SID: A Novel Class of Nonparametric Tests of Independence for Censored Outcomes

Jicai Liu; Jinhong Li; Jinhong You; Riquan Zhang

arxiv: 2412.06311 · v2 · submitted 2024-12-09 · 📊 stat.ME

SID: A Novel Class of Nonparametric Tests of Independence for Censored Outcomes

Jinhong Li , Jicai Liu , Jinhong You , Riquan Zhang This is my paper

Pith reviewed 2026-05-23 07:45 UTC · model grok-4.3

classification 📊 stat.ME

keywords survival independence divergencenonparametric independence testright-censored datacounting processwild bootstrapnonlinear dependenceindependence testing

0 comments

The pith

Survival independence divergence equals zero if and only if a right-censored response is independent of its covariates.

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

The paper introduces survival independence divergence metrics to test dependence between a right-censored outcome and covariates. A counting process strategy converts the censored testing problem into an equivalent problem on complete observations. The resulting metrics equal zero exactly at independence and detect nonlinear dependence. Empirical estimators are constructed with asymptotic theory, and a wild bootstrap supplies critical values whose consistency is shown.

Core claim

The survival independence divergence metrics equal zero precisely when the right-censored response and the covariates are independent. They are obtained by applying a counting process strategy that recasts the original censored-data independence test as a test on complete observations, and they remain sensitive to various nonlinear dependence structures.

What carries the argument

The survival independence divergence (SID), constructed via a counting process strategy that converts the censored independence test into an equivalent complete-observation test.

If this is right

The SID tests detect nonlinear dependence between the censored response and covariates.
Empirical SID estimators possess established asymptotic properties under standard regularity conditions.
Wild bootstrap procedures yield consistent critical values for the resulting tests.
The tests achieve competitive power and size control relative to existing methods in simulation studies.

Where Pith is reading between the lines

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

The counting-process reduction may allow similar independence tests to be developed for other censoring mechanisms by suitable adaptation of the process representation.
Feature screening or variable selection procedures in censored regression could directly employ the SID as a dependence measure.
The equivalence to complete-data testing suggests that many existing nonparametric independence tools become applicable once the counting-process transformation is applied.

Load-bearing premise

The counting process strategy equivalently transforms the intractable independence test due to the presence of censoring into a test problem for complete observations.

What would settle it

A collection of right-censored observations in which the response is independent of the covariates yet the estimated SID is statistically distinguishable from zero, or in which dependence is present yet the estimated SID remains zero.

read the original abstract

We propose a new class of metrics, called the survival independence divergence (SID), to test dependence between a right-censored outcome and covariates. A key technique for deriving the SIDs is to use a counting process strategy, which equivalently transforms the intractable independence test due to the presence of censoring into a test problem for complete observations. The SIDs are equal to zero if and only if the right-censored response and covariates are independent, and they are capable of detecting various types of nonlinear dependence. We propose empirical estimates of the SIDs and establish their asymptotic properties. We further develop a wild bootstrap method to estimate the critical values and show the consistency of the bootstrap tests. The numerical studies demonstrate that our SID-based tests are highly competitive with existing methods in a wide range of settings.

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

The paper defines a new SID class of divergences for independence testing with right-censored outcomes by recasting the problem through counting processes, then supplies estimators, asymptotics, and bootstrap.

read the letter

The core contribution is a counting-process route that converts the censored independence test into an equivalent complete-data problem, yielding metrics that are zero exactly under independence and sensitive to nonlinear dependence. They build empirical estimators, establish asymptotic properties, and add a wild bootstrap whose consistency they prove. The simulations position the tests as competitive with existing approaches across a range of settings. That package is the actual new piece: the transformation plus the supporting theory and resampling method. The counting-process step is the part that needs the closest look, because everything else is standard once the equivalence is granted. If the derivation holds without extra restrictions on the censoring mechanism, the rest is clean. The abstract states the equivalence directly and reports the usual consistency results, so the central claim is not obviously circular or self-referential. Minor practical questions remain about performance under heavy censoring or when the censoring distribution is misspecified, but those are the normal checks rather than load-bearing gaps. The work sits squarely inside survival analysis and supplies a nonparametric tool for a recurring problem there. Readers who already work with censored data and need an independence test that avoids parametric assumptions will find the most use. It has the structure, the claims, and the supporting pieces to go to referees rather than be desk-rejected.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the survival independence divergence (SID), a new class of nonparametric metrics for testing independence between a right-censored response and covariates. A counting-process strategy is used to recast the censored-data problem as an equivalent complete-observation independence test; the resulting SID equals zero if and only if independence holds and can detect nonlinear dependence. Empirical estimators are defined, asymptotic properties and wild-bootstrap consistency are established, and simulation studies compare the tests favorably with existing methods.

Significance. If the counting-process equivalence and the accompanying consistency results hold, the work supplies a practical nonparametric tool for dependence testing in survival analysis, an area where right-censoring often precludes direct application of standard independence measures. The explicit zero-under-independence property together with the bootstrap procedure would be useful strengths for applied researchers.

minor comments (3)

The abstract asserts that the counting-process transformation yields an “equivalent” complete-data problem, but the precise mapping (transformed variables, measure, and divergence) is not displayed; a short displayed equation or diagram in the introduction would improve readability.
Notation for the censoring indicator and the at-risk process should be introduced once and used consistently; the current abstract mixes “right-censored response” and “counting process” language without a single reference definition.
The numerical studies section would benefit from an explicit statement of the censoring rates and sample sizes used in each scenario, together with a table summarizing power at fixed nominal levels.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The provided summary accurately captures the key elements of the SID framework, the counting-process equivalence, asymptotic results, and bootstrap procedure.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the SID metrics via a counting-process transformation that converts the censored independence test into an equivalent complete-data problem. The zero-iff-independence property is presented as a direct consequence of this construction rather than a fitted or self-referential result. Asymptotic consistency and bootstrap validity are established as separate analytic steps. No self-citation load-bearing steps, fitted-input predictions, or ansatz smuggling appear in the derivation chain. The central claim remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. Standard survival-analysis assumptions (e.g., independent censoring) are not enumerated.

pith-pipeline@v0.9.0 · 5670 in / 1017 out tokens · 24583 ms · 2026-05-23T07:45:28.033998+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.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

A key technique for deriving the SIDs is to use a counting process strategy, which equivalently transforms the intractable independence test due to the presence of censoring into a test problem for complete observations. The SIDs are equal to zero if and only if the right-censored response and covariates are independent
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

SID_β(T,X) = lim Δt→0+ ∫∫ ||φX|ΔNt,Yt(u,t) − φX|Yt(u,t)||² a(t) dw1(u) dν(t)

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

39 extracted references · 39 canonical work pages

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[1] [1]

Li , Qi Q

barticle [author] Fan , Yanqin Y. Li , Qi Q. ( 1996 ). Consistent model specification tests: Omitted variables and semiparametric functional forms . Econometrica 64 865--890 . barticle

work page 1996

[2] [2]

, Gretton , Arthur A

barticle [author] Fern \'a ndez , Tamara T. , Gretton , Arthur A. , Rindt , David D. Sejdinovic , Dino D. ( 2023 ). A kernel log-rank test of independence for right-censored data . J. Amer. Statist. Assoc. 118 925--936 . barticle

work page 2023

[3] [3]

( 1984 )

barticle [author] Hall , Peter P. ( 1984 ). Central limit theorem for integrated square error of multivariate nonparametric density estimators . J. Multivariate Anal. 14 1--16 . barticle

work page 1984

[4] [4]

( 1990 )

bbook [author] Lee , Justin J. ( 1990 ). U -statistics: Theory and Practice . Marcel Dekker, Inc. , New York . bbook

work page 1990

[5] [5]

Silverman , BW B

barticle [author] Mack , YP Y. Silverman , BW B. ( 1982 ). Weak and strong uniform consistency of kernel regression estimates . Probab. Theory Relat. Fields 61 405--415 . barticle

work page 1982

[6] [6]

barticle [author] Sherman , Robert P R. P. ( 1994 ). Maximal inequalities for degenerate U -processes with applications to optimization estimators . Ann. Statist. 22 439--459 . barticle

work page 1994

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barticle [author] Sz \'e kely , G \'a bor J G. J. , Rizzo , Maria L M. L. Bakirov , Nail K N. K. ( 2007 ). Measuring and testing dependence by correlation of distances . Ann. Statist. 35 2769--2794 . barticle

work page 2007

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( 2007 )

barticle [author] Xia , Yingcun Y. ( 2007 ). A constructive approach to the estimation of dimension reduction directions . Ann. Statist. 35 2654--2690 . barticle

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Villegas , Marco M

binproceedings [author] Belanche , Llu铆s L. Villegas , Marco M. ( 2013 ). Kernel functions for categorical variables with application to problems in the life sciences . In International Conference of the Catalan Association for Artificial Intelligence 171--180 . binproceedings

work page 2013

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( 2021 )

barticle [author] Chatterjee , Sourav S. ( 2021 ). A new coefficient of correlation . J. Amer. Statist. Assoc. 116 2009--2022 . barticle

work page 2021

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, Li , KerChau K

barticle [author] Chen , ChunHouh C. , Li , KerChau K. Wang , JaneLing J. ( 1999 ). Dimension reduction for censored regression data . Ann. Statist. 27 1--23 . barticle

work page 1999

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, Strathmann , Heiko H

binproceedings [author] Chwialkowski , Kacper K. , Strathmann , Heiko H. Gretton , Arthur A. ( 2016 ). A kernel test of goodness of fit . In International Conference on Machine Learning 2606--2615 . binproceedings

work page 2016

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barticle [author] Cox , David R D. R. ( 1972 ). Regression models and life-tables . J. Roy. Statist. Soc. Ser. B 34 187--202 . barticle

work page 1972

[14] [14]

barticle [author] Dugu \'e , Audrey Emmanuelle A. E. , Pulido , Marina M. , Chabaud , Sylvie S. , Belin , Lisa L. Gal , Jocelyn J. ( 2016 ). How to deal with interval-censored data practically while assessing the progression-free survival: a step-by-step guide using SAS and R software . Clin. Cancer. Res. 22 5629--5635 . barticle

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, Welchowski , Thomas T

barticle [author] Edelmann , Dominic D. , Welchowski , Thomas T. Benner , Axel A. ( 2021 ). A consistent version of distance covariance for right-censored survival data and its application in hypothesis testing . Biometrics 78 867--879 . barticle

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bbook [author] Fleming , Thomas R T. R. Harrington , David P D. P. ( 2011 ). Counting Processes and Survival Analysis . John Wiley & Sons . bbook

work page 2011

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, Fukumizu , Kenji K

barticle [author] Gretton , Arthur A. , Fukumizu , Kenji K. , Teo , Choon C. , Song , Le L. , Sch \"o lkopf , Bernhard B. Smola , Alex A. ( 2008 ). A kernel statistical test of independence . In Advances in Neural Information Processing Systems 20 585--592 . barticle

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, Borgwardt , Karsten M K

barticle [author] Gretton , Arthur A. , Borgwardt , Karsten M K. M. , Rasch , Malte J M. J. , Sch \"o lkopf , Bernhard B. Smola , Alexander A. ( 2012 ). A kernel two-sample test . J. Mach. Learn. Res. 13 723--773 . barticle

work page 2012

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( 1999 )

barticle [author] Huang , Jian J. ( 1999 ). Efficient estimation of the partly linear additive Cox model . Ann. Statist. 27 1536--1563 . barticle

work page 1999

[20] [20]

( 1984 )

barticle [author] Janson , Svante S. ( 1984 ). The asymptotic distributions of incomplete U -statistics . Z. Wahrsch. Verw. Gebiete 66 495--505 . barticle

work page 1984

[21] [21]

Yin , Xiangrong X

barticle [author] Ke , Chenlu C. Yin , Xiangrong X. ( 2020 ). Expected conditional characteristic function-based measures for testing independence . J. Amer. Statist. Assoc. 115 985--996 . barticle

work page 2020

[22] [22]

bbook [author] Klein , John P J. P. Moeschberger , Melvin L M. L. ( 2003 ). Survival Analysis: Techniques for Censored and Truncated Data 1230 . Springer . bbook

work page 2003

[23] [23]

barticle [author] Laurie , John A J. A. , Moertel , Charles G C. G. , Fleming , Thomas R T. R. , Wieand , Harry S H. S. , Leigh , John E J. E. , Rubin , Jebal J. , McCormack , Greg W G. W. , Gerstner , James B J. B. , Krook , James E J. E. Malliard , James J. ( 1989 ). Surgical adjuvant therapy of large-bowel carcinoma: An evaluation of levamisole and the...

work page 1989

[24] [24]

( 1990 )

bbook [author] Lee , Justin J. ( 1990 ). U-statistics: Theory and Practice . Marcel Dekker, Inc. , New York . bbook

work page 1990

[25] [25]

, Liu , Jicai J

barticle [author] Li , Jinhong J. , Liu , Jicai J. , You , Jinhong J. Zhang , Riquan R. ( 2025 ). Supplement to ``SID: a novel class of nonparametric tests of independence for censored outcomes'' . barticle

work page 2025

[26] [26]

( 1966 )

barticle [author] Mantel , N N. ( 1966 ). Evaluation of survival data and two new rank order statistics arising in its consideration . Cancer Chemother. Rep. 50 163--170 . barticle

work page 1966

[27] [27]

barticle [author] Rodrigues , Agatha S A. S. , Calsavara , Vinicius F V. F. , Silva , Felipe IB F. I. , Alves , F \'a bio A F. A. Vivas , Ana PM A. P. ( 2018 ). Use of interval-censored survival data as an alternative to Kaplan-Meier survival curves: studies of oral lesion occurrence in liver transplants and cancer recurrence . J. Cancer Res. 38 1--10 . barticle

work page 2018

[28] [28]

, Kim , Ilmun I

barticle [author] Schrab , Antonin A. , Kim , Ilmun I. , Guedj , Benjamin B. Gretton , Arthur A. ( 2022 ). Efficient aggregated kernel tests using incomplete U -statistics . In Advances in Neural Information Processing Systems 35 18793--18807 . barticle

work page 2022

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, Sriperumbudur , Bharath B

barticle [author] Sejdinovic , Dino D. , Sriperumbudur , Bharath B. , Gretton , Arthur A. Fukumizu , Kenji K. ( 2013 ). Equivalence of distance-based and RKHS -based statistics in hypothesis testing . Ann. Statist. 41 2263--2291 . barticle

work page 2013

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bbook [author] Silverman , Bernard W B. W. ( 1986 ). Density Estimation for Statistics and Data Analysis . CRC Press . bbook

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( 1993 )

barticle [author] Stute , Winfried W. ( 1993 ). Consistent estimation under random censorship when covariables are present . J. Multivariate Anal. 45 89--103 . barticle

work page 1993

[32] [32]

White , Halbert H

barticle [author] Su , Liangjun L. White , Halbert H. ( 2007 ). A consistent characteristic function-based test for conditional independence . J. Econometrics 141 807--834 . barticle

work page 2007

[33] [33]

( 1949 )

barticle [author] Wald , Abraham A. ( 1949 ). Note on the consistency of the maximum likelihood estimate . Ann. Math. Stat. 20 595--601 . barticle

work page 1949

[34] [34]

, He , Shuyuan S

barticle [author] Wang , Yanhua Y. , He , Shuyuan S. , Zhu , Lixing L. Yuen , Kam K. ( 2007 ). Asymptotics for a censored generalized linear model with unknown link function . Probab. Theory Relat. Fields 138 235--267 . barticle

work page 2007

[35] [35]

, Pan , Wenliang W

barticle [author] Wang , Xueqin X. , Pan , Wenliang W. , Hu , Wenhao W. , Tian , Yuan Y. Zhang , Heping H. ( 2015 ). Conditional distance correlation . J. Amer. Statist. Assoc. 110 1726--1734 . barticle

work page 2015

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( 1992 )

barticle [author] Wei , LeeJen L. ( 1992 ). The accelerated failure time model: A useful alternative to the Cox regression model in survival analysis . Stat. Med. 11 1871--1879 . barticle

work page 1992

[37] [37]

, Drton , Mathias M

barticle [author] Weihs , Luca L. , Drton , Mathias M. Meinshausen , Nicolai N. ( 2018 ). Symmetric rank covariances: A generalized framework for nonparametric measures of dependence . Biometrika 105 547--562 . barticle

work page 2018

[38] [38]

, Yin , Xiangrong X

barticle [author] Yang , Baoying B. , Yin , Xiangrong X. Zhang , Nan N. ( 2019 ). Sufficient variable selection using independence measures for continuous response . J. Multivariate Anal. 173 480--493 . barticle

work page 2019

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