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arxiv: 2604.19177 · v1 · submitted 2026-04-21 · 📊 stat.ME

Recognition: unknown

Multiscale Cochran-Mantel-Haenszel Scanning for Conditional Dependency

Gyeonghun Kang , Jialiang Mao , Li Ma
Authors on Pith no claims yet

Pith reviewed 2026-05-10 02:43 UTC · model grok-4.3

classification 📊 stat.ME
keywords conditional independenceCochran-Mantel-Haenszel testmultiscale scanningnonparametric testingcontingency tablesconditional associationstratified analysiscontinuous data
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The pith

A multiscale scanning procedure generalizes the Cochran-Mantel-Haenszel test to continuous data, yielding consistent tests of conditional independence without requiring large stratum sizes.

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

The paper introduces a nonparametric method for testing conditional independence and estimating conditional associations in continuous data. It extends the classical CMH test by decomposing the sample space into multiple scales of 2x2xT contingency tables and conditioning on marginal order statistics. This allows the test to remain consistent even when individual stratum sizes stay small, which is common in practice. The resulting statistics are straightforward to compute and follow a known asymptotic distribution under the null, with runtime scaling linearly with the number of observations. Such a tool matters because many real datasets involve continuous variables where traditional discretization either loses power or violates consistency requirements.

Core claim

By applying a multiscale scanning approach to decompose the continuous sample space into a cascade of 2×2×T tables and then conditioning on the marginal order statistics, which are almost ancillary for conditional dependency, the generalized CMH procedure produces test statistics with a known asymptotic null distribution under the conditional sampling model. This yields consistency for detecting conditional dependencies without requiring stratum sample sizes to grow to infinity and supports nearly linear computational scaling with sample size.

What carries the argument

The multiscale scanning decomposition of continuous observations into a hierarchy of stratified 2x2xT tables that lets the CMH conditioning strategy remain valid.

If this is right

  • Reliable type I error control holds even for small samples and high-dimensional conditioning variables.
  • The method supplies summary statistics that indicate both the strength and the direction of local conditional associations.
  • Computation of the test scales almost linearly with sample size.
  • The approach identifies the nature of inferred associations in applications such as ride-share data analysis.

Where Pith is reading between the lines

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

  • The same multiscale conditioning device might adapt to other stratified nonparametric tests for dependence.
  • Local association summaries could feed into algorithms for building conditional independence graphs.
  • Linear scaling opens the possibility of routine dependency screening on very large continuous datasets.

Load-bearing premise

The marginal order statistics remain almost ancillary to the conditional dependency even after the multiscale partitioning into tables.

What would settle it

A large sample with fixed small stratum sizes in which the test statistic deviates from its claimed asymptotic null distribution or fails to control type I error at the nominal level.

Figures

Figures reproduced from arXiv: 2604.19177 by Gyeonghun Kang, Jialiang Mao, Li Ma.

Figure 1
Figure 1. Figure 1: This schematic visualizes P ∈ P along with the induced P˜ and P˜ 0. P(X, Y, Z) is depicted as a continuum of 2×2 tables for binary X and Y . PX, PY and P(0, 1 | Z = z) are shorthands for PX|Z, PY |Z and PXY |Z(0, 1 | Z = z). A stratification S = {St : t ∈ [T]} discretizes Z into Z˜ ∈ [T] where the event {Z ∈ St} is represented as {Z˜ = 1}. As such, each cell probability of P˜ is P(X, Y | Z ∈ St) = EZtP(X, … view at source ↗
Figure 2
Figure 2. Figure 2: A pictorial illustration of the DAG for a three-way table n(I3, J2, S) in Theorem 5. nS(i, j) is a shorthand for a two-way table n(Ii , Jj , S) at stratum S ∈ S. nS(i − 1, j) constitutes row sums, nS(i, j − 1) column sums of all the 2×2 tables of nS(i, j). For example, given the margin totals nS(3, 0) and nS(0, 2), the skyblue colored cell in nS(1, 1) is sampled conditioned on the cells marked in gray in n… view at source ↗
Figure 3
Figure 3. Figure 3: Results of Simulation 1. Probabilities of rejection are truncated at 0.35, and the nominal level 0.05 is indicated as a gray dotted line. In the ECDF and median log CPU time plots, a 45◦ reference line through the origin is also drawn in gray dotted line. 3.1 Simulation 1: T1E and power analysis For a fair comparison, we adopt a standard data-generating procedure from the literature, referred to as the pos… view at source ↗
Figure 4
Figure 4. Figure 4: Results of Simulation 1. In the ROC plots, a 45◦ reference line through the origin is included as a gray dotted line. function (ECDF) of p-values. The former serves as a summary diagnostic, while the latter provides a visual assessment of each method’s behavior. For a well-calibrated test, the p-values under the null should closely follow, or at least be stochastically larger than, the uniform distribution… view at source ↗
Figure 5
Figure 5. Figure 5: Results from Simulation 2. The gray dotted line is a 45◦ reference. The results of each method are plotted only if they could be executed within the 64 GB RAM budget. 4 Case study: Uber ride-share request data We apply our method to rider session data from the Uber ride-share platform, collected from an anonymous U.S. metropolitan area over multiple days. When a rider opens the Uber app and enters a destin… view at source ↗
Figure 6
Figure 6. Figure 6: For each window (row), the empirical distributions of ˆθ(S)—the stratum-specific sample log odds ratios—are plotted against the stratum-specific sample means of the conditioning variables on x-axes. Factors related to the fare components include time, distance, and base fare, among others. The common log odds ratio ˆθS of the window is indicated by the red line. For confidentiality reasons, x-axes tick mar… view at source ↗
read the original abstract

We propose a nonparametric approach to testing conditional independence and estimating conditional association, generalizing the Cochran-Mantel-Haenszel (CMH) test and odds-ratio estimator to continuous sample spaces. It leverages a multiscale scanning approach to decompose the sample space into a cascade of $2\times 2 \times T$ tables. Following the CMH test, we condition on the marginal order statistics, which are "almost ancillary" regarding conditional dependency. This strategy helps overcome a key challenge faced by other methods that discretize the sample space: we achieve consistency without requiring stratum sample sizes to grow to infinity, a constraint often difficult to satisfy in practice. Our method produces easy-to-compute test statistics with a known asymptotic null distribution under the conditional sampling model, scaling almost linearly with the sample size. Our simulation results demonstrate reliable Type I error control, even with small samples and high-dimensional conditioning, and competitive power compared to state-of-the-art tests. Finally, a case study on Uber ride-share data highlights the method's unique dual capability, inherited from the CMH, to both test and identify the nature of the inferred conditional association. By providing summary statistics that capture the strength and direction of local associations, our method offers practitioners a useful tool for learning conditional dependencies.

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.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes a nonparametric multiscale scanning generalization of the Cochran-Mantel-Haenszel (CMH) test and odds-ratio estimator for conditional independence and association in continuous spaces. The approach decomposes the sample into a cascade of 2×2×T tables, conditions on marginal order statistics claimed to be almost ancillary for the conditional dependency, and asserts consistency without requiring per-stratum sample sizes to diverge to infinity, together with an asymptotic null distribution under the conditional model, near-linear computational scaling, reliable Type I error control in simulations (including small samples and high-dimensional conditioning), competitive power, and interpretable local association summaries illustrated on Uber ride-share data.

Significance. If the consistency result and the claimed asymptotic null distribution hold, the work would offer a practical and interpretable extension of the classical CMH framework to continuous data, sidestepping the stratum-size requirement that limits many discretization-based competitors. The computational efficiency, dual testing/estimation capability, and robustness to small samples/high dimensions are clear strengths, as are the simulation evidence for Type I error control and the case-study demonstration of local association identification.

major comments (3)
  1. [Abstract] Abstract: the central claim that the procedure 'produces easy-to-compute test statistics with a known asymptotic null distribution under the conditional sampling model' is load-bearing for the method's practicality and for the assertion of linear scaling, yet no derivation, limiting distribution, or set of regularity conditions is supplied; this must be provided (with explicit reference to the multiscale decomposition) to substantiate the claim.
  2. [Abstract] Abstract: the consistency result without requiring stratum sample sizes to grow to infinity rests on the assertion that marginal order statistics are 'almost ancillary' regarding conditional dependency and that the multiscale decomposition into 2×2×T tables preserves CMH validity; a formal argument or explicit verification of this ancillary property (including any approximation error) is required, as it is identified as the weakest assumption.
  3. [Abstract] Abstract: the statement that the method 'achieves consistency without requiring stratum sample sizes to grow to infinity' is presented as a key practical advantage, but the manuscript must delineate the precise conditions on the multiscale scanning and the continuous-space conditioning under which this holds, since the ancillary property is only approximate.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'cascade of 2×2×T tables' and the precise definition of the multiscale decomposition would benefit from a short clarifying sentence or forward reference to the relevant section for readers new to scanning procedures.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We agree that the abstract claims require additional substantiation and will revise the manuscript to provide the requested derivations, formal arguments, and precise conditions. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the procedure 'produces easy-to-compute test statistics with a known asymptotic null distribution under the conditional sampling model' is load-bearing for the method's practicality and for the assertion of linear scaling, yet no derivation, limiting distribution, or set of regularity conditions is supplied; this must be provided (with explicit reference to the multiscale decomposition) to substantiate the claim.

    Authors: We agree that the abstract would benefit from explicit substantiation of this claim. In the revised manuscript, we will update the abstract to reference the relevant theorem establishing the asymptotic null distribution under the conditional sampling model and will add an explicit pointer to the derivation in the main text, including the regularity conditions and their connection to the multiscale decomposition into a cascade of 2×2×T tables. revision: yes

  2. Referee: [Abstract] Abstract: the consistency result without requiring stratum sample sizes to grow to infinity rests on the assertion that marginal order statistics are 'almost ancillary' regarding conditional dependency and that the multiscale decomposition into 2×2×T tables preserves CMH validity; a formal argument or explicit verification of this ancillary property (including any approximation error) is required, as it is identified as the weakest assumption.

    Authors: The referee correctly highlights this as the key supporting assumption. We will add a formal argument, either as a new subsection or appendix, that verifies the almost ancillary property of the marginal order statistics for conditional dependency. This will include an explicit treatment of the approximation error and how the multiscale decomposition preserves the validity of the CMH conditioning procedure. revision: yes

  3. Referee: [Abstract] Abstract: the statement that the method 'achieves consistency without requiring stratum sample sizes to grow to infinity' is presented as a key practical advantage, but the manuscript must delineate the precise conditions on the multiscale scanning and the continuous-space conditioning under which this holds, since the ancillary property is only approximate.

    Authors: We will revise the manuscript to delineate the precise conditions. This will include a clear statement of the requirements on the multiscale scanning scales and the assumptions on the continuous distributions of the conditioning variables under which consistency holds, with explicit bounds that account for the approximate nature of the ancillarity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a multiscale scanning procedure that decomposes the sample space into cascades of 2x2xT tables and conditions on marginal order statistics described as almost ancillary. This yields test statistics with an asymptotic null distribution under the conditional model and consistency without requiring per-stratum sample sizes to diverge. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central claims follow from the nonparametric generalization of the CMH conditioning strategy applied to the new decomposition, which remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the almost-ancillary property of marginal order statistics and the validity of the multiscale table decomposition for preserving CMH properties in continuous data.

axioms (1)
  • domain assumption Marginal order statistics are almost ancillary regarding conditional dependency
    Invoked to justify conditioning on them to achieve consistency without large stratum sizes.

pith-pipeline@v0.9.0 · 5525 in / 1201 out tokens · 41723 ms · 2026-05-10T02:43:23.496168+00:00 · methodology

discussion (0)

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Reference graph

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