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REVIEW 2 major objections 5 minor 38 references

Quantile Graph Discovery through QuACC: Quantile Association via Conditional Concordance

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper introduces QuACC, a statistic measuring whether two variables are jointly above (or below) their conditional quantiles, proves asymptotic normality for plug-in estimators, and uses it as a conditional-association test inside a…

desk verdict QuACC is a sensible idea for tail-specific association graphs, but the key variance term in the test statistic is wrong, and the graph semantics are looser than claimed. read the letter →

arxiv 2411.17033 v2 pith:PCP3XGFQ submitted 2024-11-26 stat.ME

Zain Khan , Daniel Malinsky , Martin Picard , Alan A. Cohen , Columbia SOH Group , Ying Wei This is my paper · ORCID
classification stat.ME MSC 62H2062G0562G1062H12
keywords quantileassociationconditionalconcordancegraphicalmodelstaildependenceregressionindependencetestingcross-fittingstructurelearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that tail-specific dependence between two variables, given a covariate set, can be measured by a single concordance probability: the chance that both variables fall on the same side of their conditional quantiles at a chosen level tau. It calls this quantity QuACC, builds plug-in estimators through cross-fitted quantile regression, and claims these estimators are asymptotically normal with a variance that can be estimated, so confidence intervals and hypothesis tests are available. If the claim holds, researchers can run the PC structure-learning algorithm once per quantile level and obtain a graph of which conditional associations live at the tails rather than only at the mean. This would matter for biomedical settings where two biomarkers may look unrelated on average but move together only in extreme states.

What carries the argument

The central object is the QuACC concordance probability, defined as the joint probability that two variables fall on the same side of their conditional quantile functions at level $\tau$. The machinery around it is a cross-fitted plug-in estimator: fit quantile regressions for $Y$ and $X$ on training folds, evaluate indicator concordance on a held-out fold, then pool folds. The proof machinery is a decomposition of estimation error into a testing-set concordance term with variance $V(\tau)$ and a quantile-estimation term with covariance $V_{XY}(\tau)$ weighted by density factors $\kappa_Y,\kappa_X$; Theorem 1 says the sum is asymptotically normal. This normality is what converts QuACC into a $z$-test and lets the PC algorithm use it as a conditional-independence oracle at each quantile level.

What would settle it

Take independent standard normal variables with no conditioning set and $\tau=0.5$: the paper's Lemma 2 gives $V(\tau)=\tau^2(1-\tau)^2=0.0625$, while direct computation gives $\mathrm{Var}(I(Y<0)I(X<0))=(0.25)(0.75)=0.1875$, showing the variance formula in Theorem 1 does not match the variance of the observed concordance statistic.

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Extended reading notes

Core claim

The central claim is that for fixed $\tau$, the QuACC statistic $\rho_\tau(Y,X|Z)=P(Y>Q_Y(\tau|Z),\,X>Q_X(\tau|Z))$ for $\tau\ge 0.5$ (and the analogous lower-tail probability for $\tau<0.5$) is a meaningful, testable quantile-specific conditional association. Under conditions C1-C4, the cross-fitted plug-in estimator $\hat{\rho}^k_\tau$ satisfies $(b_n)^{1/2}(\hat{\rho}^k_\tau-\rho_{\tau 0})\to N(0,\Sigma_\tau)$ with $\Sigma_\tau=\kappa_Y\sigma^2_{Q_Y}\kappa_Y+\kappa_X\sigma^2_{Q_X}\kappa_X+2\kappa_Y\kappa_X V_{XY}(\tau)+V(\tau)$, where $\kappa_Y,\kappa_X$ encode derivatives of the concordance probability with respect to the quantile functions, $V_{XY}(\tau)$ captures the joint estimation error of the two quantile regressions, and $V(\tau)$ is the variance of the concordance indicators at the true quantiles. For linear quantile regression a root-$n$ version is given, yielding a standard-normal test statistic for $H_0:\rho_\tau=(1-\tau)^2$ (upper) or $\tau^2$ (lower). The intended use is not just pairwise testing: feeding this test statistic into the PC algorithm produces a quantile-specific graphical model whose missing edges denote conditional independence at that quantile.

Load-bearing premise

The whole inference rests on the claim that subtracting each variable's exceedance probability before multiplying does not change the variance of the joint-exceedance indicator; the variance formulas in Theorem 1 are valid only if this algebraic identity holds.

Editorial extensions

If this is right

  • If the central claim is right, researchers can test whether two variables move to their extremes together given a covariate set, using a standard normal statistic at any $\tau$ and no assumed copula family.
  • Running the test inside the PC algorithm yields one undirected skeleton per quantile level; edges present at $\tau=0.9$ but absent at $\tau=0.5$ would mark tail-only conditional associations.
  • The same plug-in logic extends the graph framework to nonlinear quantile regressions such as quantile forests, at the cost of a slower $(n/s)^{1/2}$ convergence rate.
  • Applied to biobank biomarker panels, the method can produce quantile-specific graphs that differ at lower and upper tails, which is where dysregulation is often visible.
  • Correct estimation of the conditional quantile functions is a precondition for the null benchmark $(1-\tau)^2$ or $\tau^2$ to be valid, so the test is only as reliable as the fitted quantile models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper tests one pair of quantiles at a time; a natural extension would be a joint test over a grid of $\tau$ values, which would detect dependence appearing at any tail without the multiple-testing burden of separate tests.
  • The concordance definition could be generalized to asymmetric quantile levels $\tau_Y\ne\tau_X$, allowing questions such as whether upper-tail movement in one variable tracks lower-tail movement in another; the variance decomposition would need a new $V(\tau)$ term.
  • If the asymptotic result extends, QuACC could serve as a screening statistic before fitting full conditional copula models, since it targets precisely the corner probabilities that copulas aim to capture.
  • The graph produced by the PC skeleton at one $\tau$ should be read as quantile-specific association structure, not as full causal structure; orienting edges and combining layers across $\tau$ would require additional assumptions the paper does not state.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces QuACC (Quantile Association via Conditional Concordance), a measure of quantile-specific conditional association defined as the probability that two variables jointly exceed (or fall below) their conditional quantile functions at level tau. The authors propose a cross-fitted plug-in estimator, claim asymptotic normality of the estimator (Theorem 1 and Corollary 1.1), and use this result to build Wald tests for the null hypothesis of independence, rho_tau = (1-tau)^2 or tau^2. These tests are then plugged into the PC algorithm to estimate undirected skeletons for what the authors call QuACC graphical models (QGMs). The method is evaluated in simulations and applied to All of Us biobank data for mitochondrial disorders.

Significance. If the asymptotic theory were correct, QuACC would provide a practically useful tool for detecting tail-specific conditional associations and for constructing quantile-level graphical models, complementing existing quantile graphical model approaches. The paper has several strengths: it uses cross-fitting to separate nuisance estimation from inference, it accommodates both linear and random-forest quantile regressions, it includes simulation studies across copula settings, and it demonstrates a real-data application. However, the central variance derivation in Lemma 2 is algebraically incorrect, and the resulting test statistic and graphical outputs are not valid. The paper's main contribution therefore rests on an unsupported claim.

major comments (2)
  1. [Appendix F, Lemma 2; Section 2.2, Theorem 1 and hypothesis test] The proof of Lemma 2 asserts that Var(I(Y<QY)I(X<QX)) = Var((I(Y<QY)-tau)(I(X<QX)-tau)). This identity is false: for independent Bernoulli(tau) indicators A and B, Var(AB) = tau^2(1-tau^2), whereas Var((A-tau)(B-tau)) = tau^2(1-tau)^2. At tau=0.5 the former is 0.1875 and the latter 0.0625, a factor of three. The subsequent algebra also sets (E[W1W2])^2 = tau^2(1-4tau+4tau^2), but E[W1W2] = p(Y<QY, X<QX) - tau^2, which is zero under independence, not tau^2(1-2tau)^2. Because V(tau) enters Theorem 1 and Corollary 1.1, the null variance used in the z-statistic (Section 2.2, Hypothesis test) is too small, leading to inflated Type I error. The simulation rejection rates in Section 3.1 and the QGM graphs in Section 4 therefore do not provide valid evidence of the method's performance.
  2. [Section 2.3, QuACC Graphical Models] The PC algorithm requires a test of conditional independence to decide edge removal. The paper instead tests H0: rho_tau = (1-tau)^2 (or tau^2), which is a single condition on the joint exceedance probability, not a statement of full conditional independence of Y and X given Z. The paper does not prove that accepting this null is equivalent to, or even implies, the conditional independence required for the PC algorithm's edge-removal step. Without such a formal connection, the output of the QGM procedure lacks a well-defined interpretation as a conditional independence graph, and the manuscript's central graphical-model claim is unsupported.
minor comments (5)
  1. [Section 2.1, equation for overall b-rho] The displayed formula for b-rho_tau after defining b-rho^k_tau appears malformed: "bρτ = PK k=1 bρk τqPK k=1 V ar(bρkτ )/nk" is missing an operator and is not a valid expression. Please rewrite it unambiguously.
  2. [Appendix C, last paragraph] The sentence "If they are perfectly discordant, ie. they ever jointly above or below their regression planes, then ρ∗τ is 1" should say "never jointly" and the value should be -1, consistent with the normalization formula.
  3. [Introduction, Section 1] The text states that a negative rho_tau indicates discordance and that the smallest possible rho_tau is 0, but the definition of rho_tau as a probability is always nonnegative. Please clarify that only the normalized version rho*_tau can take negative values, and correct the related discussion.
  4. [Theorem 1 proof, page 21] The variance term for In2 is written as "2κYκX nk VXY(τ)" in the proof but as "2κYκX VXY(τ)" in the theorem statement. The factor 1/nk (if intended) should be specified precisely, since it affects the asymptotic variance expression.
  5. [References and typos] There are several typographical errors: "V owels" in the references should be "Vowels"; "Ceber's optic atrophy" should be "Leber's optic atrophy"; and the notation "≶" in Lemma 2 is unclear without specifying lower versus upper tail cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: QuACC's null values are derived from independence, not fitted; the variance-formula error is algebraic, not circular.

full rationale

The QuACC construction is self-contained: the statistic is defined directly as a joint exceedance concordance probability, and the null value (1-tau)^2 or tau^2 is obtained by multiplying Bernoulli exceedance probabilities under independence, not by fitting any parameter to data. The asymptotic distribution in Theorem 1 is built from a Taylor expansion of the concordance function plus a central limit theorem applied to test-fold indicators, with the quantile-regression nuisance components supplied by standard external results (Koenker and Hallock, Athey et al.). The paper does not invoke a prior uniqueness theorem, does not rely on a self-citation chain, and does not rename a fitted quantity as a prediction; the simulations use known copula parameters to assess power rather than to calibrate the method. The reader-identified flaw in Lemma 2, that Var(I(Y<QY)I(X<QX)) is not equal to Var((I(Y<QY)-tau)(I(X<QX)-tau)), is a genuine algebraic error affecting the Wald denominator and downstream Type I error, but it is a mathematical mistake rather than a circular reduction of the conclusion to the inputs. Similarly, the unproven semantic step from accepting the single-probability null rho_tau = tau^2 to PC edge removal concerns statistical validity, not circularity. No circular step can be exhibited by quoting the paper's equations as equivalent by construction, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method introduces a new statistic but no new unobserved entity. The core measure is a conditional tail probability, a standard copula concept, so no invented entities are introduced.

free parameters (3)
  • number of cross-fitting folds K = 5
    Chosen by hand in Section 2.1 to balance variance and computation; not fitted to data.
  • bandwidth hn for density estimation = Hall-Sheather O(n^{-1/3}) or Bofinger O(n^{-1/5})
    Section 2.2; standard tuning choice for the Hendricks-Koenker sandwich, not data-fitted.
  • quantile level tau = user-specified, e.g., 0.1, 0.5, 0.9
    The analysis is performed at chosen quantiles; this is a question parameter, not a fitted constant.
assumptions (4)
  • domain assumption Conditions C1-C4 (bounded moments, continuous densities, consistent quantile estimators, differentiable joint CDF) hold.
    Invoked in Theorem 1 to obtain asymptotic normality; these are regularity conditions on the data-generating process.
  • domain assumption The conditional quantile models QY(tau|Z) and QX(tau|Z) are correctly specified.
    The paper states 'When quantile models are correctly specified, rho_tau0 is the limit of b-rho_tau'; if misspecified, the target parameter changes.
  • ad hoc to paper In the PC algorithm, failing to reject H0: rho_tau=c is treated as evidence for conditional independence.
    Section 2.3 claims missing edges correspond to conditional independence, but the tested null is only a single tail probability, which is necessary but not sufficient for independence. This unproven step licenses graph estimation.
  • domain assumption Faithfulness and the other standard PC assumptions hold for the quantile-concordance graph.
    The PC algorithm's consistency requires faithfulness; the paper does not verify this for its new graph semantics.

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Cite this review

Pith. "Pith review of Quantile Graph Discovery through QuACC: Quantile Association via Conditional Concordance." pith.science (2026). https://pith.science/paper/PCP3XGFQ

@misc{pith2026241117033,
  author       = {Pith},
  title        = {Pith review of: Quantile Graph Discovery through QuACC: Quantile Association via Conditional Concordance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCP3XGFQ}},
  note         = {Machine review of arXiv:2411.17033}
}
read the original abstract

Graphical structure learning is an effective way to assess and visualize cross-biomarker dependencies in biomedical settings. Standard approaches to estimating graphs rely on conditional independence tests that may not be sensitive to associations that manifest at the tails of joint distributions, i.e., they may miss connections among variables that exhibit associations mainly at lower or upper quantiles. In this work, we propose a novel measure of quantile-specific conditional association called QuACC: Quantile Association via Conditional Concordance. For a pair of variables and a conditioning set, QuACC quantifies agreement between the residuals from two quantile regression models, which may be linear or more complex, e.g., quantile forests. Using this measure as the basis for a test of null (quantile) association, we introduce a new class of quantile-specific graphical models. Through simulation we show our method is powerful for detecting dependencies under dependencies that manifest at the tails of distributions. We apply our method to biobank data from All of Us and identify quantile-specific patterns of conditional association in a multivariate setting.

Figures

Figures reproduced from arXiv: 2411.17033 by the authors.

Figure 1
Figure 1. Example estimates of the marginal QuACC for two standard normal variables related via a Clayton copula [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. We evaluate the empirical rejection rate of the null hypothesis, that ρτ (Y, X|Z1, Z2) = τ 2 if τ ≤ 0.5 or ρτ (Y, X|Z1, Z2) = (1 − τ ) 2 otherwise, conditioned on Z1, Z2. The null hypothesis is tested using the linear QuACC statistic across 500 iterations with sample sizes of n = 200 and n = 400 for a grid of copula parameters from 0 to 10 with a spacing of 0.25. Results are presented in [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 2
Figure 2. Example draws from simulation settings S1, S2, S3 alongside the copula relationship present in each. In S1 and S3 there is strong quantile concordance at each quantile level, although it is heightened at the tails. In S2, the concordance increases with the quantile level. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figures from the paper (11 more)
Figure 3
Figure 3. Figure 3: Rejection rate of null hypothesis versus copula parameter across different simulation settings [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]
Figure 4
Figure 4. Figure 4: Rejection rates across τ levels for simulation settings S1, S2, S3. Empirical ρb ∗ visualized alongside rejection rate showcasing the dependence across τ and how it aligns with rejection rate. For S1 and S3, there is strong concordance at each quantile level and we fin…
Figure 5
Figure 5. Figure 5: True graph underlying the data generating process in simulation studies. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 5
Figure 5. Figure 5: n Precision Recall Structural Hamming Distance 500 0.792 (0.140) 0.593 (0.154) 0.114 (0.043) 1000 0.765 (0.141) 0.67 (0.141) 0.110 (0.044) 5000 0.689 (0.098) 0.797 (0.132) 0.114 (0.041) [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 5
Figure 5. Figure 5: The linear coefficients αi ∼ Uniform(−0.4, 0.4). n Precision Recall Structural Hamming Distance 500 0.705 (0.114) 0.717 (0.146) 0.118 (0.043) 1000 0.684 (0.097) 0.797 (0.137) 0.116 (0.042) 5000 0.599 (0.097) 0.877 (0.105) 0.146 (0.51) [PITH_FULL_IMAGE:figures/full_fig…
Figure 6
Figure 6. Figure 6: The graphs on the left are the pairwise marginal ( [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: The graph on the left is the result of aggregating 15 partial correlation PC graph replicates with majority [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: The graphs on the left are the result of aggregating 15 replicates with majority voting for edges in the All of [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: The graphs on the left are the result of aggregating 15 replicates with majority voting for edges in the All of [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: The graphs on the left are the result of aggregating 15 replicates with majority voting for edges in the All of [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: The graphs on the left are the pairwise maximally conditional ( [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.