A New Algorithm for Sampling Parameters in a Structured Correlation Matrix With Application to Estimating Optimal Combinations of Muscles to Quantify Progression in Duchenne Muscular Dystrophy

The goal of this paper is to estimate an optimal combination of biomarkers for individuals with Duchenne muscular dystrophy (DMD), which provides the most sensitive combinations of biomarkers to assess disease progression (in this case, optimal with respect to standardized response mean (SRM) for 4 muscle biomarkers). The biomarker data is incomplete (missing and irregular) multivariate longitudinal data. We propose a normal model with structured covariance designed for our setting. To sample from the posterior distribution of parameters, we develop a Markov Chain Monte Carlo (MCMC) algorithm to address the positive definiteness constraint on the structured correlation matrix. In particular, we propose a novel approach to compute the support of the parameters in the structured correlation matrix; we modify the approach from [1] on the set of the largest possible submatrices of the correlation matrix, where the correlation parameter is a unique element. For each posterior sample, we compute the optimal weights of our construct. We conduct data analysis and simulation studies to evaluate the algorithm and the frequentist properties of the posteriors of correlations and weights. We found that the lower extremities are the most responsive muscles at the early and late ambulatory disease stages, and the biceps brachii is the most responsive at the nonambulatory disease stage.