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.
The joint behaviour of paired continuous biomarkers is of great interest in medicine, reliability engineering and the social sciences. In a stress-strength model one measures the reliability of a component with strength X subject to a random stress Y through R = Pr(X > Y ); early work by Birnbaum formalised this quantity and connected it with the Mann-Whitney statistic for two independent samples [Birnbaum, 1956]. The literature on estimating R is extensive and includes classical confidence bounds [Nandi and Aich, 1994] as well as Bayesian tests based on a restricted parameter space [Nandi and Aich, 1996]. In biomedical applications one often wishes to assess joint abnormality risk for two biomarkers measured on the same individuals; for example, the probability that both fasting glucose and HbA1c fall simultaneously in their extremal regions is clinically meaningful. A direct interpretation of such extremal co-movement is afforded by tail-risk functionals of the form R T (θ) = Pr θ {(U, V ) ∈ T } for an appropriate tail region T and a dependence parameter θ. When T corresponds to joint lower tails (U ≤ α, V ≤ α), joint upper tails (U ≥ 1 -α, V ≥ 1 -α) or conditional lower tails (U ≤ α given V ≤ α), the resulting probabilities R L , R U and R C provide clinically interpretable summaries that are invariant to marginal transformations. Traditional correlation coefficients do not isolate behaviour in the extremes and therefore may mask clinically relevant tail dependence.
Copula models offer a principled way to separate marginal behaviour from dependence. Sklar’s theorem [Sklar, 1959] states that every multivariate distribution with continuous margins can be expressed as a copula applied to its marginal cumulative distribution functions [Nelsen, 2006]. The second edition of Nelsen’s monograph provides a compendium of parametric copula families and discusses their properties and applications. Tail dependence and measures of association are treated systematically in that text; see also Joe’s monograph for a comprehensive account of multivariate dependence structures [Joe, 1997]. In practice one often restricts attention to one-parameter Archimedean copulas because of their analytical tractability and ability to model either lower or upper tail dependence. Among these, the Clayton family exhibits strong lower-tail association whereas the Gumbel-Hougaard family captures positive upper-tail dependence. Goodness-of-fit procedures and diagnostic tools for copula models are surveyed by Genest et al. [2009], who emphasise that common families such as Clayton, Gumbel, Frank and Farlie-Gumbel-Morgenstern are widely used in actuarial science, survival analysis and finance.
Despite their suitability for modelling extremal co-movement, copulas have been under-utilised in the stress-strength literature. Classical analyses assume either independence of X and Y or impose a specific bivariate normal distribution; for example, Nandi and Aich [1994] derived two-sided confidence bounds for Pr(X > Y ) in bivariate normal samples. Nandi and Aich [1996] proposed a Bayesian hypothesis test for reliability under a positive quadrant restriction, again assuming a particular parametric form. More recent work has introduced specific parametric bivariate distributions via copulas: Kundu and Gupta [2011] constructed an absolutely continuous bivariate generalized exponential distribution using the Clayton copula, while Domma and Giordano [2012] modelled the joint distribution of household income and consumption via a Frank copula to measure financial fragility. Patil et al. [2022] examined the impact of dependence on R = Pr(Y < X) when the margins are exponential and compared estimation methods for several copula families, including Farlie-Gumbel-Morgenstern, Ali-Mikhail-Haq and Gumbel-Hougaard. Outside of reliability, there is growing interest in expanding the class of Archimedean generators and developing robust estimators. Aich et al. [2025a] proposed two new generator functions yielding copulas with flexible dependence properties, and Aich [2026] introduced a neural network estimator that accurately recovers copula parameters when classical likelihood methods become unstable.
In this paper, we develop a Bayesian framework for clinically interpretable tail-risk inference using Archimedean copulas. By transforming each biomarker to the copula scale via rank-based pseudo-observations, our approach isolates the dependence structure from the margins. We derive general generator-based identities for the copula density and its derivatives, which are necessary for likelihood evaluation and posterior computation. A restricted Jeffreys prior is proposed to mitigate impropriety at the boundaries of the parameter space. Posterior summaries for θ are propagated through the tail-risk functionals R L , R U and R C to obtain Bayesian credible intervals that are easy to interpret in clinical terms. We specialise the methodology to the Clayton and Gumbe
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