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REVIEW 2 major objections 4 minor 169 references

New Equivalence Tests for Approximate Independence in Contingency Tables

T0 review · 2 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Closed-form distances and a randomized boundary estimator yield practical equivalence tests for approximate independence in contingency tables.

desk verdict Clean, usable equivalence tests for approximate independence that fix the practical bottlenecks of Ostrovski 2018; the only real caveats are a geometric condition on the boundary estimator and anti-conservatism near zero margins. read the letter →

arxiv 2607.11130 v1 pith:SPFBN4JH submitted 2026-07-13 stat.ME

Vladimir Ostrovski This is my paper
classification stat.ME MSC 62F0362G10
keywords testingapproximateindependencecontingencytablesbootstrapequivalenceminimumdistanceboundarypointestimator
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 shows that approximate row-column independence in two-way contingency tables can be tested with equivalence hypotheses that ask whether a probability matrix lies inside a fixed tolerance ball around the independence model. Two distances are defined that measure absolute or relative deviation from the product of the observed margins; both are closed-form, so no numerical optimization is required. Asymptotic critical values follow from the delta method; finite-sample accuracy is restored by a bootstrap whose critical values are evaluated at a randomized linear-combination estimator of the boundary of the null. Simulations indicate that the bootstrap version keeps type-I error near the nominal level across table sizes while the asymptotic version becomes overly conservative, and three classic data sets illustrate that the procedure can decide, with a transparent tolerance, whether gender and treatment outcome, eye and hair colour, or children and income may be regarded as approximately independent.

What carries the argument

The randomized linear-combination boundary estimator: given a finite set of exterior points, form convex combinations with the observed table, solve for the unique weight that places each combination exactly on the tolerance boundary, then select the combination closest to the data; this estimator converges almost surely under a mild geometric condition and supplies the bootstrap critical value.

What would settle it

For a known boundary probability matrix p and a fixed set of exterior points that all violate the ray condition of Proposition 4, check whether the Monte-Carlo bootstrap critical value still yields empirical type-I error near the nominal level as sample size grows; systematic under- or over-coverage would refute the claimed justification of the bootstrap.

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Extended reading notes

Core claim

Equivalence tests for approximate independence can be constructed from the two closed-form pseudo-metrics da(p,M)=l(ha(p),0) and dr(p,M)=l(hr(p),1). Their asymptotic null distributions are Gaussian with explicitly estimable variances, and a computationally feasible randomized estimator of boundary points makes a consistent bootstrap feasible, yielding tests that are asymptotically consistent and locally most powerful.

Load-bearing premise

The boundary estimator is guaranteed to converge only when at least one of the randomly chosen exterior points lies on a ray from the true boundary point that never re-enters the null region.

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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 / 4 minor

Summary. The paper proposes equivalence tests for approximate row-column independence in two-way contingency tables. It replaces the usual Euclidean minimum-distance criterion with two closed-form pseudo-metrics da and dr (absolute and relative deviations from the product of the observed margins). Asymptotic critical values are obtained via the delta method (Proposition 1); finite-sample performance is improved by a parametric bootstrap that uses a randomized linear-combination estimator of boundary points (Proposition 4 and Corollary 5). Monte-Carlo experiments for several table sizes and three real-data examples are reported, and R code is supplied.

Significance. The contribution is useful for applied work: closed-form distances remove the need for numerical minimization that plagued earlier minimum-distance procedures, and the randomized boundary estimator makes bootstrap critical values computationally feasible. The asymptotic theory is standard and correctly applied; the extensive simulation tables and open-source implementation strengthen the practical claim. The geometric condition required by the boundary estimator and the documented anti-conservatism near zero margins are genuine limitations, but they are already flagged by the author and do not invalidate the asymptotic justification.

major comments (2)
  1. Proposition 4 (and the algorithm in Section 3) requires that at least one exterior point q satisfies d*(a p+(1-a)q,M)>ε for every a∈[0,1). The paper does not quantify how often randomly sampled exterior points fail this condition, nor does it supply a diagnostic or fallback when all sampled q violate it. Because the almost-sure consistency of the bootstrap critical value rests on this geometric assumption, a short Monte-Carlo check of the failure rate (or a simple safeguard) is needed before the bootstrap procedure can be recommended without reservation.
  2. Tables 2 and 9 show that both the asymptotic and bootstrap tests based on dr become anti-conservative when some ricj are near zero. The manuscript notes the phenomenon but offers only the informal advice to “treat results with caution.” A concrete recommendation (e.g., automatic shrinkage of ε, a continuity correction, or a warning threshold on the smallest margin) should be added so that practitioners know when the nominal level is no longer reliable.
minor comments (4)
  1. The scaling factors chosen for the Euclidean distance l (1/√(k1k2) for dr and √(k1k2) for da) are introduced without motivation; a one-sentence justification would help readers compare results across table sizes.
  2. Remark 3 claims that the asymptotic test extends immediately to multi-way tables, yet no formula or simulation is given; either supply the multi-way expressions or remove the claim.
  3. In Section 4 the phrase “the type II error rate equals 1 minus test power” is redundant and can be deleted.
  4. Table 1 caption should state that the power is evaluated at the uniform matrix; the same information is currently buried in the text.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: closed-form distances and boundary estimator are derived from first principles; self-citations supply only general asymptotic power theory.

  1. self citation load bearing [Section 2, after Proposition 1; Remark 6]
    "The outlined test is locally asymptotically most powerful, see [Ostrovski(2017)], Proposition 3. … Consequently, the test is also locally asymptotically most powerful see [Ostrovski(2017)], Proposition 3."

    Local asymptotic most-powerfulness is asserted solely by citation to the author’s own 2017 paper rather than re-derived for the new distances da/dr. The citation is not circular in the strong sense (the 2017 result is a general theorem for minimum-distance equivalence tests), yet it is the sole justification offered for the optimality claim of the present tests.

full rationale

The paper defines two new pseudo-metrics da and dr by composing a differentiable distance l with the explicit maps ha(p)=(pij-ricj) and hr(p)=(pij/(ricj)). These immediately yield closed-form expressions da(p,M)=l(ha(p),0) and dr(p,M)=l(hr(p),1) without numerical minimization, removing the continuous-minimizer assumption required in the author’s 2018 work. Proposition 1 obtains the asymptotic null distribution of the resulting test statistics by the ordinary delta method under multinomial asymptotics; the local asymptotic most-powerfulness claim is then imported from Ostrovski (2017, Prop. 3), a general result that does not encode the independence model or the particular distances. The bootstrap critical-value construction rests on a new randomized linear-combination estimator of boundary points whose almost-sure consistency is proved in Proposition 4 and Corollary 5 under an explicit geometric condition that is also enforced by the algorithm. None of these steps reduces by construction to a previously fitted quantity or to an unverified uniqueness theorem; the self-citations are therefore non-load-bearing background. Finite-sample simulations and open R code further corroborate the claims independently of the cited theory. The only material caveat—the geometric condition on exterior points—is already stated by the author and does not create circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The paper rests on classical multinomial asymptotics, the delta method, and standard bootstrap consistency theorems; the only free choices are the user-supplied tolerance ε, the number of exterior points m, and the particular scaled Euclidean metric. No new physical or statistical entities are postulated beyond the two pseudo-metrics and the boundary estimator, both of which are explicitly constructed.

free parameters (3)
  • tolerance ε
    User-chosen equivalence margin that defines the hypotheses; all power and size statements are conditional on a fixed ε.
  • number of exterior points m = (k1+k2)*50
    Set by the empirical rule m=(k1+k2)*50; controls the quality of the randomized boundary estimator and is not derived from first principles.
  • scale factor of Euclidean distance l = 1/sqrt(k1 k2) or sqrt(k1 k2)
    Chosen as 1/sqrt(k1 k2) for dr and sqrt(k1 k2) for da so that results remain comparable across table sizes; an arbitrary but fixed normalization.
assumptions (4)
  • standard math Multinomial central-limit theorem: sqrt(n)(qn-q) o N(0,Σ(q)) with Σ=Dq-qq op
    Invoked in the proof of Proposition 1 (Bishop et al. 1975, Thm 14.3-4).
  • standard math Delta-method continuous-mapping theorem for the derivative of d*
    Used to obtain the asymptotic variance σ*(p) (van der Vaart 1998, Thm 3.1).
  • standard math Bootstrap consistency under the boundary estimator (Lehmann-Romano Thm 15.6.1)
    Cited for asymptotic validity of the bootstrap critical value (Remark 6).
  • domain assumption Existence of at least one exterior point q satisfying the strict inequality of Prop. 4
    Required for almost-sure consistency of the linear-combination estimator; not automatically guaranteed by the random rejection sampler.
invented entities (2)
  • absolute-deviation pseudo-metric da and relative-deviation pseudo-metric dr
    purpose: Replace the general minimum-distance functional by closed-form expressions that avoid numerical optimization.
    Defined via the maps ha(p)=(pij-ri cj) and hr(p)=(pij/(ri cj)); they are pseudo-metrics only, yet sufficient for the equivalence problem.
  • randomized linear-combination boundary-point estimator c(pn,q)
    purpose: Produce a feasible estimator of a point on ∂H0 so that bootstrap critical values can be computed without parameterizing the whole boundary.
    Constructed by solving d*(a pn+(1-a)q,M)=ε for the largest a∈[0,1] and then selecting the closest such combination among m random exterior points.

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Cite this review

Pith. "Pith review of New Equivalence Tests for Approximate Independence in Contingency Tables." pith.science (2026). https://pith.science/paper/SPFBN4JH

@misc{pith2026260711130,
  author       = {Pith},
  title        = {Pith review of: New Equivalence Tests for Approximate Independence in Contingency Tables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPFBN4JH}},
  note         = {Machine review of arXiv:2607.11130}
}
read the original abstract

We introduce new equivalence tests for approximate independence in two-way contingency tables. The critical values are calculated asymptotically. The finite sample performance of the tests is improved by means of the bootstrap. An estimator of boundary points is developed to make the bootstrap based tests statistically efficient and computationally feasible. We compare the performance of the proposed tests for different table sizes by simulation. Then we apply the tests to real data sets. The tests are implemented in R and available online, see [https://github.com/TestingEquivalence/EquivalenceTestIndependenceR].

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