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Leave-one-out testing for node-level differences in Gaussian graphical models

Davide Benussi, Ester Alongi, Erika Banzato

TL;DR

This paper addresses two-sample equality testing in Gaussian graphical models by proposing node-level inference through a leave-one-out Bartlett-adjusted global LRT on a fully connected graph. The method forms leave-one-out increments $\Delta_M$ whose null distribution is a chi-square with degrees of freedom $h(l,p)=\frac{l(2p-l+3)}{2}$, enabling calibrated p-values for single nodes and fixed-size subsets; singleton tests yield $\Delta_j \sim \chi^2_{p+1}$ under $H_0$. Multiplicity is managed via Holm adjustments to identify altered nodes, and under alternatives the increments exhibit noncentral chi-square behavior, providing power. Simulations show accurate finite-sample calibration and favorable power properties, while the approach remains computationally light and practically useful, with a proposed two-stage strategy to localize changes within a clique when the true graph is unknown. The work advances localized, valid inference in GGMs beyond decomposable-graph restrictions and suggests future extensions to robustness and unbalanced designs.

Abstract

We study two-sample equality testing in Gaussian graphical models. Classical likelihood ratio tests on decomposable graphs admit clique-wise factorizations, offering limited localization and unstable finite-sample behaviour. We propose node-level inference via a leave-one-out Bartlett-adjusted test on a fully connected graph. The resulting increments have standard chi-square null limits, enabling calibrated significance for single nodes and fixed-size subsets. Simulations confirm validity, and a case study shows practical utility.

Leave-one-out testing for node-level differences in Gaussian graphical models

TL;DR

This paper addresses two-sample equality testing in Gaussian graphical models by proposing node-level inference through a leave-one-out Bartlett-adjusted global LRT on a fully connected graph. The method forms leave-one-out increments whose null distribution is a chi-square with degrees of freedom , enabling calibrated p-values for single nodes and fixed-size subsets; singleton tests yield under . Multiplicity is managed via Holm adjustments to identify altered nodes, and under alternatives the increments exhibit noncentral chi-square behavior, providing power. Simulations show accurate finite-sample calibration and favorable power properties, while the approach remains computationally light and practically useful, with a proposed two-stage strategy to localize changes within a clique when the true graph is unknown. The work advances localized, valid inference in GGMs beyond decomposable-graph restrictions and suggests future extensions to robustness and unbalanced designs.

Abstract

We study two-sample equality testing in Gaussian graphical models. Classical likelihood ratio tests on decomposable graphs admit clique-wise factorizations, offering limited localization and unstable finite-sample behaviour. We propose node-level inference via a leave-one-out Bartlett-adjusted test on a fully connected graph. The resulting increments have standard chi-square null limits, enabling calibrated significance for single nodes and fixed-size subsets. Simulations confirm validity, and a case study shows practical utility.
Paper Structure (4 sections, 2 theorems, 12 equations, 3 figures, 1 table)

This paper contains 4 sections, 2 theorems, 12 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our proposal
  3. Simulation study
  4. Conclusions

Key Result

Theorem 1

Fix $1\le l\le p-1$ and $M\in\mathcal{M}(l)$. Assume the regularity conditions in ERIKA_BANZATO_2023 ensuring chi-square limits for $T_n^{V}$ and $T_n^{V\setminus M}$ hold. Under $H_0$, as $n=n_1+n_2\to\infty$, where

Figures (3)

  • Figure 1: Empirical null distribution of Bartlett-adjusted $\Delta_M$ for $|M|=l\in\{1,2,3\}$. The black line denotes the theoretical null limiting distribution. Rows vary by $l$, columns by sample size $n_1=n_2\in\{10,50,100,250\}$.
  • Figure 2: FWER under $H_0$ for singletons ($l=1$), pairs ($l=2$), and triplets ($l=3$), for $n_1=n_2$. Black and grey lines denote Bartlett and non-Bartlett calibration; point shapes indicate multiplicity adjustment. The horizontal dotted line marks the nominal level $\alpha=0.05$.
  • Figure 3: Empirical non-null distribution of Bartlett-adjusted $\Delta_j$ in scenario (S2) with changed nodes $\mathcal{S}=\{1,2\}$, $\delta_\mu=1.5$, and $\xi=0.5$. The black line denotes the theoretical non-null limiting distribution. Results for $n_1=n_2=100$.

Theorems & Definitions (5)

  • Theorem 1: Null limiting distribution of $\Delta_M$
  • proof
  • Corollary 1: Singleton removal
  • Remark 1
  • Conjecture 1: Behaviour under the alternative in the case of singleton removal