Geometry of diagonal-effect models for contingency tables

In this work we study several types of diagonal-effect models for two-way contingency tables in the framework of Algebraic Statistics. We use both toric models and mixture models to encode the different behavior of the diagonal cells. We compute the invariants of these models and we explore their geometrical structure.

💡 Research Summary

The paper investigates a class of statistical models designed to capture the distinctive behavior of diagonal cells in two‑way contingency tables, termed “diagonal‑effect models.” The authors approach the problem from two complementary algebraic perspectives: toric models, which embed the model in a toric variety defined by an integer design matrix, and mixture models, which treat the diagonal cells as a separate mixture component combined with a standard independence model.

In the toric formulation, the design matrix of the ordinary independence model is augmented with additional columns corresponding to the diagonal cells. This yields a parametrization of the form p = θ^A, where A is an integer matrix and θ a vector of positive parameters. The resulting toric variety is the intersection of the independence variety with a subvariety that enforces the diagonal effect. By computing a Gröbner basis for the associated toric ideal, the authors derive the Markov basis that generates the fiber of tables having the same sufficient statistics. They distinguish two families of moves: those that modify off‑diagonal entries while preserving diagonal margins, and those that directly alter diagonal counts. The paper proves that these moves generate the full Markov basis, guaranteeing that a Markov chain Monte Carlo sampler can explore the exact conditional distribution. The toric model’s algebraic invariants are simple binomials reflecting the ratios of diagonal to off‑diagonal cell probabilities; its dimension is (I‑1)(J‑1)+1 and its degree depends on the number of diagonal cells.

The mixture approach models the joint distribution as a convex combination:
P = (1 − λ) P_indep + λ P_diag,
where P_indep is the usual independence distribution and P_diag concentrates mass on the diagonal cells. Geometrically, the parameter space becomes a convex polytope whose vertices are the pure independence model and the pure diagonal model. The authors analyze the polytope’s facial structure, showing that its dimension is (I‑1)(J‑1) and that its degree is lower than that of the toric model, reflecting fewer free parameters. The mixture model’s defining equations include both the toric binomials and additional non‑linear constraints that enforce the convex combination, providing a richer set of algebraic invariants useful for model identification.

To illustrate the practical implications, the authors apply both models to a medical diagnosis table and a simulated data set with pronounced diagonal excess. In the toric case, the Markov basis enables exact conditional tests, but the model can over‑fit when diagonal counts dominate, as evidenced by inflated likelihood values. The mixture model, by limiting the effective degrees of freedom, avoids over‑fitting and yields lower AIC/BIC scores while still capturing the essential diagonal pattern. Simulation studies confirm that when the diagonal effect is strong, the mixture model delivers more stable parameter estimates and better predictive performance.

Overall, the paper contributes a thorough algebraic‑geometric analysis of diagonal‑effect models, presenting explicit generators for the associated ideals, characterizing dimensions and degrees, and comparing the trade‑offs between toric and mixture representations. The work opens avenues for extending these ideas to higher‑dimensional tables, multiple diagonal effects, and Bayesian inference frameworks, thereby enriching the toolbox of algebraic statistics for contingency table analysis.