Bayesian Inference for Joint Tail Risk in Paired Biomarkers via Archimedean Copulas with Restricted Jeffreys Priors
Agnideep Aich, Md. Monzur Murshed, Sameera Hewage, Ashit Baran Aich
TL;DR
We address joint extremal dependence between two biomarkers by modeling their uniform margins with a one-parameter Archimedean copula and focusing on tail-risk functionals $R_L(\theta)$, $R_U(\theta)$, and $R_C(\theta)$. A restricted Jeffreys prior on the copula parameter $\theta$ and grid-based posterior yield exact posteriors for these functionals, with explicit forms for Clayton and Gumbel copulas. The approach is validated via simulations showing near-nominal coverage and applied to NHANES data (GLU–GHB), revealing substantial extremal co-movement relative to independence and illustrating how Clayton emphasizes lower-tail co-movement while Gumbel emphasizes upper-tail co-movement. Overall, the framework provides a principled, interpretable Bayesian method for reporting joint tail risks in biomedical settings with uncertainty quantification, and it lays groundwork for extensions to more copula families and higher-dimensional or longitudinal analyses.
Abstract
We propose a Bayesian copula-based framework to quantify clinically interpretable joint tail risks from paired continuous biomarkers. After converting each biomarker margin to rank-based pseudo-observations, we model dependence using one-parameter Archimedean copulas and focus on three probability-scale summaries at tail level $α$: the lower-tail joint risk $R_L(θ)=C_θ(α,α)$, the upper-tail joint risk $R_U(θ)=2α-1+C_θ(1-α,1-α)$, and the conditional lower-tail risk $R_C(θ)=R_L(θ)/α$. Uncertainty is quantified via a restricted Jeffreys prior on the copula parameter and grid-based posterior approximation, which induces an exact posterior for each tail-risk functional. In simulations from Clayton and Gumbel copulas across multiple dependence strengths, posterior credible intervals achieve near-nominal coverage for $R_L$, $R_U$, and $R_C$. We then analyze NHANES 2017--2018 fasting glucose (GLU) and HbA1c (GHB) ($n=2887$) at $α=0.05$, obtaining tight posterior credible intervals for both the dependence parameter and induced tail risks. The results reveal markedly elevated extremal co-movement relative to independence; under the Gumbel model, the posterior mean joint upper-tail risk is $R_U(α)=0.0286$, approximately $11.46\times$ the independence benchmark $α^2=0.0025$. Overall, the proposed approach provides a principled, dependence-aware method for reporting joint and conditional extremal-risk summaries with Bayesian uncertainty quantification in biomedical applications.