A Hierarchical Bayesian Analysis of Circular Data with Autoregressive Errors: Modeling the Mechanical Properties of Cortical Bone

Abstract

We study a hierarchical model for circular data with covariate information. The model for the data given the first stage parameters is based on a finite Fourier series expansion of the conditional mean level for each circular curve. The dependence of the response on the covariates is incorporated through a linear regression of the parameters on the covariates. The within curve dependence is accounted for by an autoregressive error structure. In the second and subsequent stages, a hierarchical Bayes structure is imposed on the parameters of the expansion.

We use the model to analyze a set of observational data consisting of CAT scans of cross-sections of human femurs at the lesser trochanter from a group of patients admitted to the Hospital for Special Surgery in New York City. Two responses are considered: the thickness of the cortical bone from a series of equi-angular rays through the centroid of the section, and the components of the area moment of inertia about the bone’s major principal axis. The latter provide a measure of the bone’s mechanical properties under loading. We relate these curves to each subject’s age, gender, and diagnosis. Marginal posterior distributions of the model parameters and predictive distributions of future observations are estimated using the Gibbs Sampler and displayed graphically. The sensitivity of the conclusions to the modeling assumptions is assessed. The necessity of interplay between the computationally rather simple classical methods and the computationally intensive Bayesian methods is stressed.

We would like to acknowledge the advice and encouragement of Dr. Albert Burstein of the Hospital for Special Surgery during the preparation of this manuscript and his permission to use the data analyzed in this paper. We are also indebted to Dr. Sue Leurgans of the Rush Presbyterian — St. Luke’s Medical Center for her valuable comments and suggestions, and her active contribution to the early stages of the project. We would finally like to thank Mr. Rich Kite and Ms. Marya Arfer for technical help extracting the data.