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REVIEW 3 major objections 5 minor 58 references

Do more observations bring more information in rare events?

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Rare-event independence tests get their power from the number of cases, not the total sample size.

desk verdict The rare-event independence-test framework is a genuinely useful contribution, but the high-dimensional normality results rest on an unverified condition (2.6) that must be addressed before the paper is fully trustworthy. read the letter →

arxiv 2506.13671 v1 pith:F343NOX4 submitted 2025-06-16 stat.ME

Danyang Huang , Liyuan Wang , Liping Zhu This is my paper · ORCID
classification stat.ME MSC 62G1062G2062H20
keywords independencetestrareeventsimbalanceddatageneralizedU-statisticssubsamplinglocalpowerhigh-dimensionaltestingdistancecorrelation
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

Independence tests are normally expected to improve as more data arrive. This paper shows that when one outcome class is rare, the power of an independence test is governed by the number of rare cases $n_1$, not the total sample size $n$: if $n_1$ is fixed while $n_1/n$ shrinks, correlations shrink to zero and test power plateaus below one. To recover the lost information, the paper rescales common statistics — Pearson correlation, Kendall's tau, distance covariance, and projection covariance — as two-sample U-statistics, and proves that the rescaled statistic converges at a rate set only by $n_1$. It then proposes the Boosted Independence Test (BIT), which keeps every case and only about $s n_1$ subsampled controls; with fixed $s\geq 2$ it has the same convergence rate and nearly the same local power as using all controls, at a fraction of the computation. The theory extends to multi-class rare events.

What carries the argument

The carrying object is the rescaled two-sample U-statistic $T$ of (2.3), with kernel $h$ acting on $m_0$ controls and $m_1$ cases, and its projections $h_{a,b}(X^{(0)}_1,\dots,X^{(0)}_a; X^{(1)}_1,\dots,X^{(1)}_b)$ with variances $\xi_{a,b}=\mathrm{Var}(h_{a,b})$. The proof's leverage is the variance expansion $\mathrm{Var}(T)=m_1^2\xi_{0,1}/n_1 + m_0^2\xi_{1,0}/n_0 + O(n_0^{-1}n_1^{-1})$; under $n_1/n_0\to 0$, the $n_0$ term drops out, so the convergence rate of the whole statistic is $n_1$. The BIT version (2.7) multiplies each control in the kernel by an independent Bernoulli indicator $\delta_i$ with $\mathrm{P}(\delta_i=1)=s n_1/n_0$, so only about $s n_1$ controls are used; this adds the variance term $m_0^2\xi_{1,0}/s$ but leaves the $n_1$-rate intact, reducing computational complexity from $O(p n^2)$ to $O(p s^2 n_1^2)$ for the second-order statistics.

What would settle it

Run the rescaled Kendall tau test with $n_1=50$ fixed and a fixed shift alternative, and compare power at $n_0=500$, $5{,}000$, and $500{,}000$: the paper predicts the power stays essentially constant once $n_1/n_0$ is small. If the power climbs systematically with $n_0$ in that comparison, the rate result would fail; conversely, running the same comparison at $n_1/n_0=0.5$ should show power increasing with $n_0$, confirming that the no-information conclusion is specific to the vanishing-proportion regime.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a rate collapse: in the extreme-imbalance regime $n_1/n_0\to 0$, the asymptotic law of any kernel-based independence statistic of the form (2.3) is fixed by the case count alone. For a first-order kernel, $n_1^{1/2}T \xrightarrow{d} N(0, m_1^2 \xi_{0,1})$; for a second-order (degenerate) kernel with $\xi_{0,1}=0$, $n_1 T$ converges to a weighted chi-square distribution unless a high-dimensional scaling (2.6) turns it normal. The same rate controls the subsampled statistic $T_S$ built from all $n_1$ cases and $s n_1$ controls: $n_1^{1/2}T_S \xrightarrow{d} N(0, m_1^2\xi_{0,1}+m_0^2\xi_{1,0}/s)$. This is why the classical statistics in the toy example plateau: they are effectively using a rate tied to $n_1$, so adding controls alone cannot push their power to one. The paper verifies the phenomenon and the rescaled remedy for Pearson, Kendall's tau, distance covariance, improved projection covariance, and multi-class extensions.

Load-bearing premise

The load-bearing premise is that the case fraction vanishes: $n_1/n_0\to 0$ while $n_1\to\infty$ (plus, for second-order kernels, either $\xi_{1,0}=0$ or $n_1^2/n_0\to 0$); if the class proportion does not vanish, the $m_0^2\xi_{1,0}/n_0$ term survives in the limiting variance and additional controls genuinely add information, so the paper's central answer to its title question no longer holds.

Editorial extensions

If this is right

  • Classical tests such as Pearson correlation, Kendall's tau, distance correlation, and projection correlation lose power in rare-event settings, and the loss cannot be repaired by adding controls.
  • Rescaled statistics detect dependence at local alternatives of size $n_1^{-1/2}$ (first-order) or $n_1^{-1}$ (high-dimensional second-order); the classical $n^{-1/2}$ scale is unattainable.
  • BIT with $s n_1$ controls has the same convergence rate as the full-sample RIT, and as $s\to\infty$ its distribution matches the full statistic.
  • The multi-class extension shows that when several rare classes have comparable sizes, the convergence rate is still set by $n_1$; when one class is rarer than all others, the rate is set by that class alone.
  • In high-dimensional settings, the second-order BIT is asymptotically normal under condition (2.6), giving an explicit null distribution that avoids permutation.

Reading between the lines

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

  • If the conclusion holds, then in study design for rare outcomes the relevant currency is the number of observed cases: a database with few cases and millions of controls is informationally equivalent for independence testing to its control subsample, so retrospective collections should prioritize case accretion.
  • The paper's simulations suggest a practical rule of thumb not stated as a theorem: a sampling ratio around $s=5$ recovers essentially full-sample power for both first- and second-order statistics, so the computational saving can be attained without an explicit optimal-$s$ procedure.
  • The variance formula makes a testable prediction for mildly imbalanced data: at $n_1/n_0$ small but not vanishing, power should improve only by the amount predicted by the $m_0^2\xi_{1,0}/n_0$ term, which is negligible compared with $m_1^2\xi_{0,1}/n_1$; this gives a quantitative boundary for when more controls help.
  • The same projection argument should apply to other kernel-based dependence measures, so the rescaled-statistic construction could be transplanted to, for example, kernel correlation or rank-based measures without new rate calculations.
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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

3 major / 5 minor

Summary. The paper studies independence testing when one binary class is rare, i.e., when n1/n0 goes to 0. It introduces a rescaled independence test (RIT) based on generalized U-statistics and proves that, under this extreme-imbalance regime, the convergence rate of the statistic is determined by the number of rare events n1 rather than the total sample size n. It then proposes a boosted independence test (BIT) that subsamples controls with a user-chosen ratio s and shows that the same convergence rate is preserved. The theoretical results cover first-order and second-order kernels in both fixed and high-dimensional settings, include a local power analysis under mixture alternatives, and extend to multiple rare-event classes. Simulations and two real-data applications illustrate the effectiveness and computational savings of the proposed procedures.

Significance. If the main theorems hold, the paper delivers a practically valuable and counterintuitive message: under extreme class imbalance, adding control observations does not improve the power of independence tests, and a small subsample of controls can nearly match the full-sample performance. The first-order asymptotic proofs are standard and carefully executed, and the subsampling construction uses a user-chosen ratio rather than data-fitted constants, which is a methodological strength. The multi-class extension broadens the framework. However, the high-dimensional normal limits for the distance and projection covariance examples rest on an unverified condition, and the headline claim in the abstract is stated without the extreme-imbalance qualifier that the theorems require. These issues are fixable but currently leave the flagship examples only partially supported.

major comments (3)
  1. [Section 2.3, items 4(2) and 5(2); Theorems 2 and 4; Appendix C.2] Condition (2.6) is not verified for the distance covariance or improved projection covariance kernels. The paper states after Theorem 2 that the condition fails when p is fixed, and Appendix C.2 only uses (2.6) as an assumption in the martingale central limit theorem verification; it does not show that the kernels h_0,2 for T_dcov or T_IPcov satisfy the required moment ratios when p diverges. Since the asymptotic N(0,2) limits in items 4(2) and 5(2) are used in the simulation studies (p = 50) and the real-data analyses (p = 5408), this missing verification directly affects the evidence for the paper's flagship high-dimensional claims. The authors should either provide a proof of (2.6) for these kernels under explicit dimension and moment assumptions, or restrict the statements to kernels for which the condition is verified.
  2. [Appendix C.4, proof of Theorem 4] The proof that Var{n1(TS - V)} tends to 0 uses the claim Var(n1TS) converges to m1^2(m1-1)^2 xi_{0,2}/2 'from formula (C.10)', but formula (C.10) also contains the terms m1^2 m0^2 xi_{1,1}/(s n1^2) and m0^2(m0-1)^2 xi_{2,0}/(2 s^2 n1^2). After multiplying by n1^2, these terms vanish only if xi_{1,1} = o(s) and xi_{2,0} = o(s^2). Since xi_{1,1} and xi_{2,0} may depend on n1 and p, the condition s tending to infinity alone does not ensure these negligibility conditions. The theorem needs an additional explicit condition, or a proof that (2.6) implies these ratios are negligible.
  3. [Abstract and Section 2.2] The central statement that 'the power of the test is determined by the number of rare events rather than the total sample size' is presented without the qualifier n1/n0 -> 0. Theorem 1 is proved only under n1/n -> 0, and for the second-order case with xi_{1,0} not zero it additionally requires n1^2/n0 -> 0. Outside this regime the asymptotic variance of the rescaled statistic includes the control contribution m0^2 xi_{1,0}/n0, so increasing n0 does carry information. The abstract and conclusion should state the extreme-imbalance regime explicitly so that the headline claim is not overgeneralized.
minor comments (5)
  1. [Corollary 3(2)] The first variance term in the statement appears to be a typo: it should be m1^2(m1-1)^2 zeta_{2,1}/2 rather than m_k^2(m1-1)^2 zeta_{2,1}/2.
  2. [Appendix C.1, Step 1] In the count of xi_{0,2} terms in the proof of Theorem 1(2), the combinations involving n1 are written with m0 in the lower entries; they should be (n1-2 choose m1-2) and (n1-m1 choose m1-2).
  3. [Section 3.3] Tables 4 and 5 report p-values as <0.001, but the text does not state how the null distribution was obtained (asymptotic normal approximation via Theorem 2, permutation, or another method). Please specify the procedure.
  4. [Abstract] The phrase 'Key W ords' contains a typo and should be 'Key Words'.
  5. [Figure 1] The text refers to Figure 1, but the figure itself is not shown in the manuscript; please ensure it is included in the published version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rare-event rates follow from the paper's own variance calculations under a stated imbalance regime; the unverified high-dimensional condition (2.6) and the omitted IPcov verification are correctness caveats, not circularity.

full rationale

The paper's derivation chain is self-contained and not circular. Theorem 1 computes Var(T) from the U-statistic covariance identities in Lemma 2 and obtains Var(T) = m0^2 ξ_{1,0}/n0 + m1^2 ξ_{0,1}/n1 + O(1/(n0 n1)); the conclusion that the convergence rate is 'determined solely by n1' is the limit under the explicitly stated n1/n0 → 0 regime, not an identity imposed by definition. The subsampled BIT variance in Theorem 3 follows by counting index overlaps and using the Bernoulli indicators δ_i, with no fitted constant: the sampling ratio s is user-chosen, and the comparison to RIT is a direct variance calculation. The power analysis in Appendix C.6 derives explicit power expressions from these variances; no fitted parameter is renamed as a prediction. The high-dimensional normality results are conditioned on assumption (2.6), which is stated but never verified for the distance-covariance or improved-projection-covariance kernels (Section 2.3 items 4(2) and 5(2), Appendix C.2); Appendix B also omits the IPcov verification with 'its computation is almost identical to that of distance covariance. Therefore, we omit the detailed proof.' These are missing-support correctness concerns, not circularity. The citation of Zhang and Zhu (2024) supplies the projection-covariance kernel definition; it is not used as an external theorem to force the rare-event conclusions. No uniqueness theorem is imported, and no known result is merely renamed.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard U-statistics machinery, the zero-mean kernel assumption, and the extreme-imbalance asymptotics. The only user-chosen tuning parameter is the BIT sampling ratio s; no constants are fitted to data. The high-dimensional theorems additionally require condition (2.6), which is unverified for the showcased kernels.

free parameters (1)
  • s (BIT sampling ratio) = user-selected; 3, 5, 20, 50 in experiments
    BIT retains s n1 controls; the paper suggests s>=5 for near-full-sample power and gives asymptotic guidance on s, but s is not estimated from data and no data-adaptive selection criterion is given.
assumptions (5)
  • standard math Standard U-statistics asymptotic theory (Hoeffding, Gregory, Serfling)
    Used in Lemma 1 and Lemma 3 to establish null distributions of U-statistics and generalized U-statistics.
  • domain assumption Kernel h is symmetric within each class block and has zero mean under the null hypothesis
    Stated in Section 2.2; the variance decomposition (A.1) through (A.5) relies on zero-mean kernels to set all nonshared-index expectations to zero.
  • domain assumption Extreme imbalance n1/n0 -> 0 with n1 -> infinity (and n1^2/n0 -> 0 in some second-order cases)
    Defines the rare-event regime; without it, the controls projection xi_{1,0} contributes to the limiting variance and power depends on n0.
  • domain assumption Condition (2.6) on h_{0,2}: E[G^2] + n1^{-1} E[h^4]/(E[h^2])^2 -> 0 as p diverges
    Required for the martingale central limit theorem in Theorems 2 and 4; not verified for the distance or projection covariance kernels.
  • domain assumption Observations and Bernoulli subsampling indicators are independent, and controls beyond the subsample are exchangeable
    Used in the Theorem 3 variance computation and in the BIT definition (2.7).

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Pith. "Pith review of Do more observations bring more information in rare events?." pith.science (2026). https://pith.science/paper/F343NOX4

@misc{pith2026250613671,
  author       = {Pith},
  title        = {Pith review of: Do more observations bring more information in rare events?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F343NOX4}},
  note         = {Machine review of arXiv:2506.13671}
}
read the original abstract

It is generally believed that more observations provide more information. However, we observe that in the independence test for rare events, the power of the test is, surprisingly, determined by the number of rare events rather than the total sample size. Moreover, the correlations tend to shrink to zero even as the total sample size increases, as long as the proportion of rare events decreases. We demonstrate this phenomenon in both fixed and high-dimensional settings. To address these issues, we first rescale the covariances to account for the presence of rare events. We then propose a boosted procedure that uses only a small subset of non-rare events, yet achieves nearly the same power as using the full set of observations. As a result, computational complexity is significantly reduced. The theoretical properties, including asymptotic distribution and local power analysis, are carefully derived for both the rescaled statistic based on the full sample and the boosted test statistic based on subsampling. Furthermore, we extend the theory to multi-class rare events. Extensive simulations and real-world data analyses confirm the effectiveness and computational efficiency of the proposed approach.

Figures

Figures reproduced from arXiv: 2506.13671 by the authors.

Figure 1
Figure 1. The upper row displays the expected correlation, while the lower row il [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.