Pith. sign in

REVIEW 2 major objections 7 minor 18 references

Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables

T0 review · 2 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that every strictly positive square probability table's asymmetry decomposes orthogonally into a departure from quasi-symmetry and a geometric marginal heterogeneity term, by viewing the table as a point in a simplex…

desk verdict Sound, modest CoDA contribution: a clean orthogonal split of simplicial skewness into quasi-skewness and geometric marginal heterogeneity, with one fixable proof gap. read the letter →

arxiv 2506.02474 v1 pith:T4SGLTQP submitted 2025-06-03 math.ST stat.TH

Keita Nakamura , Tomoyuki Nakagawa , Kouji Tahata This is my paper
classification math.STstat.TH MSC 62H17
keywords contingencytablesquasi-symmetrygeometricmarginalhomogeneityAitchisongeometrycompositionaldataanalysissimplicialskewnessorthogonaldecomposition
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 classical quasi-symmetry of a square contingency table, defined by symmetric local odds ratios, can be re-expressed in compositional geometry as symmetry of the interaction component of the table, and that this viewpoint yields a new companion concept, geometric marginal homogeneity, which is symmetry of the independence component. The central target is the identity that for any strictly positive square probability table the simplicial skewness measure $E^2(P)$ splits as $Q^2(P) + M^2(P)$, where $Q^2(P)$ measures departure from quasi-symmetry and $M^2(P)$ measures geometric marginal heterogeneity. If the identification is right, analysts get one coherent geometric framework in which a table's asymmetry is apportioned between two interpretable sources, with explicit formulas for the closest quasi-symmetric and closest geometric-marginal-homogeneous tables and cell-level contribution arrays.

What carries the argument

The machinery is the Aitchison geometry on the simplex: tables are combined by perturbation (coordinate-wise products, renormalized), scaled by powering, and measured through the centered log-ratio (clr) transform, which maps the simplex onto a Euclidean hyperplane and induces the Aitchison inner product and norm. Two existing orthogonal decompositions, one into independence and interaction parts and one into symmetric and skew-symmetric parts, are intersected to produce a four-way orthogonal decomposition $P = P_{\mathrm{syind}} \oplus P_{\mathrm{skind}} \oplus P_{\mathrm{syint}} \oplus P_{\mathrm{skint}}$. The identification that carries the argument is that quasi-symmetry equals symmetry of the interaction component, $P_{\mathrm{int}} = T(P_{\mathrm{int}})$, while geometric marginal homogeneity equals symmetry of the independence component, $P_{\mathrm{ind}} = T(P_{\mathrm{ind}})$; the skewness decomposition then follows directly from the orthogonality of the four subspaces.

What would settle it

Take any strictly positive $3 \times 3$ (or larger) table whose local odds ratios satisfy $\theta_{ij} = \theta_{ji}$, compute its interaction component $P_{\mathrm{int}}$ from the cell formula in Section 2.2, and check whether $P_{\mathrm{int}} = T(P_{\mathrm{int}})$; a single such table whose interaction component is not symmetric would refute the converse half of Theorem 2. Independently, the identity $E^2(P) = Q^2(P) + M^2(P)$ is claimed for every strictly positive square table, so a random positive table failing this equality numerically to machine precision would refute Theorem 7.

Watch

Extended reading notes

Core claim

Working inside the Aitchison geometry of the simplex of $I \times I$ probability tables, the paper identifies quasi-symmetry with the condition $P_{\mathrm{int}} = T(P_{\mathrm{int}})$, namely that the interaction component is invariant under transposition, and proves this is equivalent to Caussinus's classical definition via symmetric local odds ratios. It introduces geometric marginal homogeneity as the dual condition $P_{\mathrm{ind}} = T(P_{\mathrm{ind}})$ on the independence component. Both collections form linear subspaces of the simplex, of dimensions $(I-1)(I+4)/2$ and $I(I-1)$ respectively, with explicit orthogonal projection formulas giving the closest quasi-symmetric and closest geometric-marginal-homogeneous table to any given table. The main result is the orthogonal decomposition of simplicial skewness, $$$E^{2}$(P) = $Q^{2}$(P) + $M^{2}$(P),$$ where the two summands are called simplicial quasi-skewness and simplicial geometric marginal heterogeneity.

Load-bearing premise

The converse half of the equivalence between the simplicial and classical definitions of quasi-symmetry relies, without proof or citation, on the classical multiplicative representation $p_{ij} = \alpha_i \beta_j \psi_{ij}$ with symmetric $\psi_{ij}$, and the whole framework assumes all cell probabilities are strictly positive, excluding tables with structural zeros.

Editorial extensions

If this is right

  • Any square table's asymmetry can be apportioned between two interpretable sources, so a practitioner can attribute skewness to association structure versus marginal imbalance.
  • The closest quasi-symmetric table preserves the geometric marginals of the original table, giving a constructive way to impose quasi-symmetry while keeping the marginal structure intact.
  • Both quasi-symmetric and geometric-marginal-homogeneous tables form linear subspaces with explicit dimensions, so projections and distance-based comparisons can be carried out in the same Euclidean geometry.
  • Cell-level quasi-skewness and geometric marginal heterogeneity arrays locate which pairs of cells drive each component of skewness, refining the information in the overall skewness array.

Reading between the lines

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

  • A natural extension the paper leaves implicit is hypothesis testing: the quantities $E^2$, $Q^2$, and $M^2$ are natural test statistics, but the paper does not derive their sampling distributions under multinomial sampling.
  • The strict positivity assumption $p_{ij} > 0$ excludes tables with structural zeros; a zero-handling strategy would be needed before the framework applies to many real sparse tables, and the clr transformation would have to be modified.
  • Because geometric marginal homogeneity is the e-flat counterpart of classical marginal homogeneity, the decomposition likely provides a compositional analogue of the classical result that symmetry holds if and only if both marginal homogeneity and quasi-symmetry hold.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. The manuscript develops Aitchison geometry on the simplex for square probability tables. It defines quasi-symmetry as symmetry of the interaction component (Definition 2), proves its equivalence with the classical local-odds-ratio condition (Theorem 2), introduces geometric marginal homogeneity via symmetry of the independent component (Definition 4), proves that both families form linear subspaces (Theorems 3 and 5), gives explicit orthogonal projections onto them (Theorems 4 and 6), and establishes the orthogonal decomposition E^2(P) = Q^2(P) + M^2(P) of simplicial skewness into quasi-skewness and geometric marginal heterogeneity (Theorem 7). An illustrative application to unaided distance vision data is provided.

Significance. If the results stand, the paper provides a clean compositional-geometric refinement of skewness in square contingency tables: a four-way orthogonal decomposition into symmetric/skew-symmetric and independent/interaction parts, with closed-form projection formulas for quasi-symmetry and geometric marginal homogeneity. The dimension counts and the numerical example (0.080 + 0.139 = 0.219) check out, and the decomposition is exact. The paper builds transparently on published decompositions ([8], [12]) and supplies explicit cell formulas, which are strengths. The main limitation is that no inferential machinery is developed for the proposed measures; the contribution is descriptive and geometric. The central identification of Q^2(P) as departure from quasi-symmetry depends on Theorem 2, whose converse is currently not fully supported in the manuscript.

major comments (2)
  1. [Appendix A.2 (Theorem 2)] The converse direction of Theorem 2 invokes the representation p_ij = α_i β_j ψ_ij with ψ_ij = ψ_ji without proof or citation. This is the classical log-linear characterization of quasi-symmetry, and Theorem 2 is exactly the result that identifies Definition 2 with Caussinus's quasi-symmetry. Because the interpretation of Q^2(P) in Theorem 7 depends on this identification, the step is load-bearing. The manuscript should either cite a source for the representation (e.g., Caussinus 1965 or a standard log-linear model text) or supply a short proof, for instance by showing that θ_ij = θ_ji forces the mixed second differences of log p_ij − log p_ji to vanish, so that log p_ij − log p_ji = c_i − c_j, which yields the representation.
  2. [Appendix A.3 and A.5 (Theorems 3 and 5)] The proofs of Theorems 3 and 5 compute the dimensions of S_QS and S_GMH as dim S_ind + dim S_syint and dim S_syind + dim S_int, respectively, but the dimensions of S_syint and S_syind are never derived in the text. In A.3, the value I(I−1)/2 for dim S_syint is inserted into the final sum; in A.5, dim S_syind is stated as I−1 without justification. These dimensions are correct (they follow from the orthogonality of S_sym/S_skew and S_ind/S_int), but they should be proved or explicitly cited, since Theorems 3 and 5 are load-bearing for the proposed decomposition.
minor comments (7)
  1. [Section 4, paragraph after Table 5] The sentence 'Table 5 shows the skew-symmetric interaction PT orthogonal to Table 6' appears to be a typo; it should refer to Table 6.
  2. [Author byline] The author name 'Nakamgawa' is a typo for 'Nakagawa' as used in reference [12].
  3. [Introduction and Section 3] The claims that quasi-symmetric and geometric-marginal-homogeneous tables are 'e-flat subspaces' are not formally defined or proved in the manuscript; either a definition should be added or the wording should be softened to avoid unsupported terminology.
  4. [Section 2.1] The strict positivity assumption p_ij > 0 is stated but not discussed as a limitation; since structural zeros occur in real square tables, the authors should explicitly acknowledge this restriction and its consequences for the applicability of the method.
  5. [Section 3, Definition 6 and Tables 9-11] The signed percentages in the skewness, quasi-skewness, and geometric marginal heterogeneity arrays sum to zero by skew-symmetry; the text should clarify that the absolute values of the entries sum to 100%, so that readers do not misinterpret the totals.
  6. [References] Reference [7] is incomplete: it lacks the full author list and publication venue, and should be completed before the manuscript is finalized.
  7. [Throughout] There are several LaTeX encoding artifacts in author names and references (e.g., 'Faˇcevicov´a'); these should be corrected in production.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the main orthogonal decomposition is a direct algebraic consequence of definitions and prior published decompositions; only a minor proof gap appears in the converse of Theorem 2, where an uncited classical QS representation is invoked.

  1. other [Appendix A.2, Proof of Theorem 2]
    "We utilize the equivalent representation of the quasi-symmetry model, pij = αiβjψij where αi, βj, ψij > 0 and ψij = ψji for i,j ∈ {1,...,I}."

    Theorem 2 aims to prove that Definition 2 (Pint = T(Pint)) is equivalent to the local-odds-ratio condition (1). In the converse half, the proof does not derive Pint = T(Pint) from (1); instead it replaces (1) by the classical log-linear representation pij = αiβjψij with symmetric ψij, labeled 'the equivalent representation of the quasi-symmetry model.' That representation is itself a standard characterization of QS, so the argument reduces the claimed equivalence to an unproved, uncited external theorem. The missing lemma is true, but as written the identification of Definition 2 with Caussinus' QS is not self-contained.

full rationale

The central decomposition E^2(P)=Q^2(P)+M^2(P) (Theorem 7) follows directly from the four-component orthogonal decomposition (Theorem 1), the definitions of quasi-symmetry and geometric marginal homogeneity as Pint=T(Pint) and Pind=T(Pind), and the Pythagorean relation, since Pskew=Pskind⊕Pskint with orthogonal summands. No parameter is fitted and no prediction is validated with the data used to construct the measures, so no fitted input is renamed as a prediction. The self-citations to Egozcue et al. [8] and Nakamura et al. [12] supply previously published, independent decompositions rather than unverified premises. The only load-bearing weakness is in the converse half of Theorem 2, where the classical multiplicative QS representation is invoked without proof or citation; this is a proof gap and a missing reference, but not a circular fit. The strict positivity assumption pij>0 is a domain restriction that excludes structural zeros but does not invalidate the theorem on its stated domain.

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

No free parameters are fitted anywhere; the only inputs are the prior decompositions [8] and [12] and the classical QS representation. The paper introduces one new concept, geometric marginal homogeneity, but it is a subspace condition (Pind = T(Pind)), not a new entity with hidden degrees of freedom.

assumptions (5)
  • domain assumption All cell probabilities are strictly positive (P in S^(I^2) with p_ij > 0)
    Stated in Section 2.1; the log-ratio machinery and the QS/MH projection formulas are undefined for zero cells, which occur in real square tables.
  • domain assumption Existence and uniqueness of the independence/interaction orthogonal decomposition (Egozcue et al. [8])
    Assumed as prior work; all theorems in Section 3 build on this decomposition.
  • domain assumption Symmetry/skew-symmetry orthogonal decomposition and subspaces S_sym, S_skew (Nakamura et al. [12])
    Assumed as prior work by the same authors; it supplies the projections used in Theorems 1, 4, and 6.
  • standard math Classical log-linear representation of quasi-symmetric tables p_ij = alpha_i beta_j psi_ij with psi_ij = psi_ji
    Invoked without proof or citation in Appendix A.2 to prove the converse direction of Theorem 2; a genuine theorem equating local-odds-ratio symmetry with the symmetric-interaction log-linear form.
  • domain assumption Aitchison geometry on the simplex models e-geodesics and e-flat structure (Erb [14])
    Interpretive claim connecting the algebra to information geometry; not load-bearing for the projection formulas but central to the paper's framing of QS and GMH as e-flat subspaces.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables." pith.science (2026). https://pith.science/paper/T4SGLTQP

@misc{pith2026250602474,
  author       = {Pith},
  title        = {Pith review of: Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4SGLTQP}},
  note         = {Machine review of arXiv:2506.02474}
}
read the original abstract

Square contingency tables are traditionally analyzed with a focus on the symmetric structure of the corresponding probability tables. We view probability tables as elements of a simplex equipped with the Aitchison geometry. This perspective allows us to present a novel approach to analyzing symmetric structure using a compositionally coherent framework. We present a geometric interpretation of quasi-symmetry as an e-flat subspace and introduce a new concept called geometric marginal homogeneity, which is also characterized as an e-flat structure. We prove that both quasi-symmetric tables and geometric marginal homogeneous tables form subspaces in the simplex, and demonstrate that the measure of skew-symmetry in Aitchison geometry can be orthogonally decomposed into measures of departure from quasi-symmetry and geometric marginal homogeneity. We illustrate the application and effectiveness of our proposed methodology using data on unaided distance vision from a sample of women.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

18 extracted references · 14 canonical work pages

  1. [8]

    Communica- tions in Statistics - Theory and Methods 44(18), 3978–3996 (2015) https://doi.org/10.1080/03610926.2013.824980

    Egozcue, J.J., Pawlowsky-Glahn, V., Templ, M., Hron, K.: Indepen- dence in Contingency Tables Using Simplicial Geometry. Communica- tions in Statistics - Theory and Methods 44(18), 3978–3996 (2015) https://doi.org/10.1080/03610926.2013.824980

    arXiv 2015

  2. [12]

    Austrian Journal of Statistics 53(4), 85–98 (2024) https://doi.org/10.17713/ajs.v53i4.1845

    Nakamura, K., Nakagawa, T., Tahata, K.: Symmetry of Square C ontingency Tables Using Simplicial Geometry. Austrian Journal of Statistics 53(4), 85–98 (2024) https://doi.org/10.17713/ajs.v53i4.1845 . Chap. Articles

    work page doi:10.17713/ajs.v53i4.1845 2024

  3. [1]

    Jour- nal of the American Statistical Association 43(244), 572–574 (1948) https://doi.org/10.1080/01621459.1948.10483284

    Bowker, A.H.: A Test for Symmetry in Contingency Tables. Jour- nal of the American Statistical Association 43(244), 572–574 (1948) https://doi.org/10.1080/01621459.1948.10483284

    work page arXiv 1948

  4. [2]

    Biometrika 42(3-4), 412–416 (1955) https://doi.org/10.1093/biomet/42.3-4.412

    Stuart, A.: A Test for Homogeneity of the Marginal Distributions in A Two-Way Classification. Biometrika 42(3-4), 412–416 (1955) https://doi.org/10.1093/biomet/42.3-4.412

    work page doi:10.1093/biomet/42.3-4.412 1955

  5. [3]

    Annales de la Facult´ e des sciences de l’Universit´ e de Toulouse pour le s sciences math´ ematiques et les sciences physiques 29, 77–183 (1965)

    Caussinus, H.: Contribution ` a l’analyse statistique des tableaux d e corr´ elation. Annales de la Facult´ e des sciences de l’Universit´ e de Toulouse pour le s sciences math´ ematiques et les sciences physiques 29, 77–183 (1965)

    work page 1965

  6. [4]

    Jou rnal of the Royal Statistical Society: Series B (Methodological) 44(2), 139–160 (1982) https://doi.org/10.1111/j.2517-6161.1982.tb01195.x

    Aitchison, J.: The Statistical Analysis of Compositional Data. Jou rnal of the Royal Statistical Society: Series B (Methodological) 44(2), 139–160 (1982) https://doi.org/10.1111/j.2517-6161.1982.tb01195.x

    arXiv 1982

  7. [5]

    Mon ographs on Statistics and Applied Probability (Series)

    Aitchison, J.: The Statistical Analysis of Compositional Data. Mon ographs on Statistics and Applied Probability (Series). Chapman and Hall, London (1986)

    work page 1986

  8. [6]

    Stochastic Environmental Research and Risk Asses sment 15(5), 384–398 (2001) https://doi.org/10.1007/s004770100077

    Pawlowsky-Glahn, V., Egozcue, J.J.: Geometric approach to stat istical analysis on the simplex. Stochastic Environmental Research and Risk Asses sment 15(5), 384–398 (2001) https://doi.org/10.1007/s004770100077

    work page doi:10.1007/s004770100077 2001

Show all 18 references
  1. [7]

    Egozcue, J.J., D ´ ıaz-Barrero, Pawlowsky-Glahn, V.: Composition al analysis of bivariate discrete probabilities. (2008)

    2008

  2. [9]

    Journal of Applied Statistics 41(5), 944–958 (2014) https://doi.org/10.1080/02664763.2013.856871 23

    Faˇ cevicov´ a, K., Hron, K., Todorov, V., Guo, D., Templ, M.: Logra tio approach to statistical analysis of 2 × 2 compositional tables. Journal of Applied Statistics 41(5), 944–958 (2014) https://doi.org/10.1080/02664763.2013.856871 23

    2014

  3. [10]

    Scandinavian Journal of Statistics 43(4), 962–977 (2016) https://doi.org/10.1111/sjos.12223

    Faˇ cevicov´ a, K., Hron, K., Todorov, V., Templ, M.: Compositiona l Tables Anal- ysis in Coordinates. Scandinavian Journal of Statistics 43(4), 962–977 (2016) https://doi.org/10.1111/sjos.12223

    2016 doi

  4. [11]

    Scandinavian Journal of Statistics 45(4), 879–899 (2018) https://doi.org/10.1111/sjos.12326

    Faˇ cevicov´ a, K., Hron, K., Todorov, V., Templ, M.: General approach to coordinate representation of compositional tables. Scandinavian Journal of Statistics 45(4), 879–899 (2018) https://doi.org/10.1111/sjos.12326

    2018 doi

  5. [13]

    Journal of the American Statistical Association 96(456), 1205–1214 (2001) https://doi.org/10.1198/016214501753381850

    Billheimer, D., Guttorp, P., Fagan, W.F.: Statistical Interpretat ion of Species Composition. Journal of the American Statistical Association 96(456), 1205–1214 (2001) https://doi.org/10.1198/016214501753381850

    2001 doi

  6. [14]

    Information Geometry 6(1), 327–354 (2023) https://doi.org/10.1007/s41884-023-00104-1

    Erb, I.: Power transformations of relative count data as a shrinkage problem. Information Geometry 6(1), 327–354 (2023) https://doi.org/10.1007/s41884-023-00104-1

    2023 doi

  7. [15]

    Journal of the American Statistical Associat ion 64(328), 1323–1341 (1969) https://doi.org/10.2307/2286071 2286071

    Ireland, C.T., Ku, H.H., Kullback, S.: Symmetry and Marginal Homog eneity of an r × r Contingency Table. Journal of the American Statistical Associat ion 64(328), 1323–1341 (1969) https://doi.org/10.2307/2286071 2286071

    1969 doi

  8. [16]

    In: Filzmoser, P., Hron, K., Mart ´ ın-Fe rn´ andez, J.A., Palarea-Albaladejo, J

    Egozcue, J.J., Maldonado, W.L.: An Interpretable Orthogonal D ecomposition of Positive Square Matrices. In: Filzmoser, P., Hron, K., Mart ´ ın-Fe rn´ andez, J.A., Palarea-Albaladejo, J. (eds.) Advances in Compositional Data A nalysis: Festschrift in Honour of Vera Pawlowsky-Gl...

    2021 doi

  9. [17]

    Mathematical Geology 37(7), 795–828 (2005) https://doi.org/10.1007/s11004-005-7381-9

    Egozcue, J.J., Pawlowsky-Glahn, V.: Groups of Parts and Their B alances in Compositional Data Analysis. Mathematical Geology 37(7), 795–828 (2005) https://doi.org/10.1007/s11004-005-7381-9

    2005 doi

  10. [18]

    Biometrika 40(1/2), 105–110 (1953) https://doi.org/10.2307/2333101 2333101 24

    Stuart, A.: The Estimation and Comparison of Strengths of Ass o- ciation in Contingency Tables. Biometrika 40(1/2), 105–110 (1953) https://doi.org/10.2307/2333101 2333101 24

    1953 doi

Pith tools

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