Bayesian Covariance Structure Modeling of Multi-Way Nested Data

TL;DR

This paper introduces a Bayesian multivariate model with a structured covariance matrix for multi-way nested data, enabling flexible dependence modeling and efficient inference, validated through simulations and applied to clinical event time data.

Contribution

It proposes a novel Bayesian covariance modeling framework with conjugate priors and efficient Gibbs sampling for nested data, including unbalanced designs, enhancing analysis of complex hierarchical data.

Findings

Validated the Gibbs sampling procedure with simulations

Applied the model to clinical event time data

Detected differential treatment effects in a medical study

Abstract

A Bayesian multivariate model with a structured covariance matrix for multi-way nested data is proposed. This flexible modeling framework allows for positive and for negative associations among clustered observations, and generalizes the well-known dependence structure implied by random effects. A conjugate shifted-inverse gamma prior is proposed for the covariance parameters which ensures that the covariance matrix remains positive definite under posterior analysis. A numerically efficient Gibbs sampling procedure is defined for balanced nested designs, and is validated using two simulation studies. For a top-layer unbalanced nested design, the procedure requires an additional data augmentation step. The proposed data augmentation procedure facilitates sampling latent variables from (truncated) univariate normal distributions, and avoids numerical computation of the inverse of the structured covariance matrix. The Bayesian multivariate (linear transformation) model is applied to two-way nested interval-censored event times to analyze differences in adverse events between three groups of patients, who were randomly allocated to treatment with different stents (BIO-RESORT). The parameters of the structured covariance matrix represent unobserved heterogeneity in treatment effects and are examined to detect differential treatment effects.