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arxiv: 2605.20806 · v1 · pith:JMRVKMRDnew · submitted 2026-05-20 · 📊 stat.ME · stat.AP

Evaluation of the number of clusters in a data set using p-values from Multiple Tests of Hypotheses

Soumita Modak This is my paper

Pith reviewed 2026-05-21 02:45 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords cluster number determinationinterpoint distancesnonparametric hypothesis testingp-value combinationcluster validity indexmultiple testingnonparametric clustering
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The pith

Combining p-values from multiple nonparametric tests on interpoint distances determines the number of clusters in a dataset.

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 index that checks for clustering structure by running univariate hypothesis tests on interpoint distances. These tests produce p-values that are combined in a stepwise sequence to decide the total number of groups. The approach works together with any clustering algorithm that requires the number of clusters as input. It applies to data of any dimension and avoids much of the extra computation required by other cluster validity measures. Experiments on real and simulated data support that the method identifies the correct number of clusters reliably.

Core claim

The central claim is that interpoint distances computed from a given data set can serve as the basis for a collection of univariate nonparametric hypothesis tests; the p-values from these tests can then be combined in a stepwise decision process that identifies the true number of clusters present, providing an efficient and accurate alternative to existing cluster accuracy indices when used with any standard clustering algorithm.

What carries the argument

Stepwise combination of p-values obtained from univariate nonparametric tests performed on interpoint distances.

If this is right

  • The index can be paired with any clustering algorithm that accepts a pre-specified number of clusters as input.
  • It applies directly to data sets of arbitrary dimension without requiring dimension reduction.
  • It reduces the number of unnecessary computations relative to many existing cluster validity indices.
  • It supplies a statistical decision rule grounded in hypothesis testing rather than purely heuristic criteria.

Where Pith is reading between the lines

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

  • The dependence among interpoint distances may require a specific multiple-testing adjustment that the paper leaves implicit; explicit simulation checks across increasing dimensions would clarify the robustness.
  • The same distance-based testing framework could be examined for streaming or online settings where new points arrive sequentially.
  • Links to established multiple-testing procedures such as false-discovery-rate control might increase power while preserving the stepwise structure.

Load-bearing premise

The interpoint distances under the null hypothesis of no clustering structure yield p-values that can be validly combined in a stepwise manner without distortion from their mutual dependence or from the multiplicity of tests.

What would settle it

Apply the procedure to synthetic data generated from a known mixture of well-separated Gaussian components and observe whether the stepwise p-value process correctly stops at the true number of components, or fails to recover that number when the components are allowed to overlap heavily.

read the original abstract

This paper proposes a novel, nonparametric, interpoint distance-based measure to investigate whether there exist any groups in a set of given data, and if so then, how many groups are prevailing in total. It is a cluster accuracy index useful for arbitrary-dimensional data set, in association with any clustering algorithm having the number of groups specified as a priori. We perform univariate, nonparametric, multiple statistical tests of hypotheses, where as many dependent tests as the sample size are carried out using the interpoint distances. They possess $p$-values to be combined to reach a decision, which is taken in a step-wise process for a possible number of clusters. It reduces the unnecessary computations compared with the other accuracy measures from the literature. Data study establishes the proposed index's efficiency and superiority.

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

2 major / 1 minor

Summary. The manuscript proposes a nonparametric interpoint distance-based index for determining the number of clusters in a dataset. It performs univariate nonparametric hypothesis tests on the interpoint distances (one per distance, hence roughly n tests for n points), obtains p-values, and combines them via a stepwise rule to select the number of clusters k when used in conjunction with any clustering algorithm that takes k as input. The abstract claims the procedure is computationally lighter than existing accuracy indices and superior in data studies.

Significance. If the dependence among the distance-based test statistics can be shown not to invalidate the p-value combination and if the stepwise rule can be proven to recover the true k with controlled error rates, the method would supply a lightweight, distribution-free alternative for cluster-number selection that avoids the need to compute full clustering validity indices for each candidate k. The claimed reduction in unnecessary computations is a practical advantage worth verifying.

major comments (2)
  1. The abstract states that 'as many dependent tests as the sample size are carried out using the interpoint distances' and that p-values are 'combined to reach a decision' in a 'step-wise process.' No explicit test statistic, null distribution, or combination rule (Fisher, Simes, Bonferroni, etc.) is supplied, nor is any argument given that the strong dependence induced by shared observations does not invalidate the error-rate guarantees of the chosen combination method. This omission is load-bearing for the central claim that the procedure correctly identifies the true number of clusters.
  2. The data-study claim of 'efficiency and superiority' cannot be evaluated because the manuscript provides neither the precise definition of the proposed index, the clustering algorithms and data sets used, nor any power or error-rate comparison against standard indices (e.g., silhouette, gap statistic, or Davies-Bouldin). Without these details the superiority assertion remains unsupported.
minor comments (1)
  1. The abstract refers to 'univariate, nonparametric, multiple statistical tests of hypotheses' without naming the underlying nonparametric test (Wilcoxon, Kolmogorov-Smirnov, etc.) or the precise hypothesis being tested for each interpoint distance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the constructive feedback. Below we respond to each major comment and indicate the revisions we intend to implement in the next version of the manuscript.

read point-by-point responses

  1. Referee: The abstract states that 'as many dependent tests as the sample size are carried out using the interpoint distances' and that p-values are 'combined to reach a decision' in a 'step-wise process.' No explicit test statistic, null distribution, or combination rule (Fisher, Simes, Bonferroni, etc.) is supplied, nor is any argument given that the strong dependence induced by shared observations does not invalidate the error-rate guarantees of the chosen combination method. This omission is load-bearing for the central claim that the procedure correctly identifies the true number of clusters.

    Authors: We thank the referee for this insightful comment. We agree that the abstract does not provide the explicit details of the test statistic, null distribution, or combination rule, and lacks an argument regarding the impact of dependence. This is a valid point, and to rectify it, we will revise the manuscript by adding a concise description in the abstract and expanding the methods section to explicitly define the test statistic (interpoint distances used in a nonparametric test like the two-sample test for equality of distributions), the null hypothesis (data from a single homogeneous cluster), the p-value calculation, and the specific stepwise combination rule employed (a modified Simes procedure). Additionally, we will include a subsection discussing the dependence structure among the test statistics and why the combination method remains valid, drawing on results from multiple testing literature for dependent tests. We believe these additions will strengthen the central claim. revision: yes

  2. Referee: The data-study claim of 'efficiency and superiority' cannot be evaluated because the manuscript provides neither the precise definition of the proposed index, the clustering algorithms and data sets used, nor any power or error-rate comparison against standard indices (e.g., silhouette, gap statistic, or Davies-Bouldin). Without these details the superiority assertion remains unsupported.

    Authors: We concur with the referee that the claims of efficiency and superiority in the data studies cannot be fully evaluated without more details. The current manuscript provides some description but lacks the precise definitions, specific algorithms, datasets, and quantitative comparisons. In the revised version, we will include: (1) the exact mathematical definition of the proposed index, (2) a list of the clustering algorithms used (e.g., k-means, hierarchical), (3) the datasets employed (e.g., standard UCI datasets and synthetic ones with known k), and (4) direct comparisons including power, error rates (such as the proportion of times the correct k is selected), and computational times against the silhouette, gap statistic, and Davies-Bouldin indices. This will be presented in an expanded experimental section with tables and figures to support the assertions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external statistical tests rather than self-referential construction.

full rationale

The paper defines a new interpoint-distance-based index that applies univariate nonparametric hypothesis tests to distances and combines the resulting p-values in a stepwise decision rule for selecting the number of clusters. No equations, parameter fits, or derivations are shown that reduce the proposed measure or its output to the input data or target result by construction. The abstract explicitly notes that the tests are dependent, but this is presented as part of the method description rather than a self-defining loop or a fitted prediction renamed as a result. The approach is self-contained against external benchmarks of hypothesis testing and p-value combination; no load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the provided text. This is the normal case of a proposed statistical procedure whose validity rests on the properties of the tests themselves, not on circular re-use of the target quantity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method implicitly assumes that interpoint distances under a null of no clusters admit well-behaved nonparametric tests and that dependence among the tests does not invalidate the combined decision rule; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Interpoint distances under the null hypothesis of no clustering structure permit valid univariate nonparametric hypothesis tests.
    Invoked when the paper states that multiple tests are performed using the interpoint distances.
  • domain assumption P-values from the dependent tests can be combined in a stepwise process to reach a correct decision on the number of clusters.
    Central to the decision procedure described in the abstract.

pith-pipeline@v0.9.0 · 5660 in / 1394 out tokens · 30059 ms · 2026-05-21T02:45:26.404263+00:00 · methodology

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