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Biostatistics Advance Access published online on April 14, 2005
Biostatistics, doi:10.1093/biostatistics/kxi020
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© The Author 2005. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oupjournals.org.
Received February 11, 2004
Revised January 26, 2005
Accepted February 2, 2005

Article

A Transformation Approach for Incorporating Monotone or Unimodal Constraints

Laura H. Gunn 1* and David B. Dunson 2

1 Jiann-Ping Hsu School of Public Health, Georgia Southern University, P.O. Box 8076 Statesboro, GA 30460-8076; 912-486-7422 (phone); 912-486-7907 (fax)
2 Biostatistics Branch, National Institute of Environmental Health Sciences, MD A3-03, P.O. Box 12233, Research Triangle Park, NC 27709

* To whom correspondence should be addressed.
Laura H. Gunn, E-mail: lgunn{at}georgiasouthern.edu


  Abstract

Samples of curves are collected in many applications, including studies of reproductive hormone levels in the menstrual cycle. Many approaches have been proposed for correlated functional data of this type, including smoothing spline methods and other flexible parametric modeling strategies. In many cases, the underlying biological processes involved restrict the curve to follow a particular shape. For example, progesterone levels in healthy women increase during the menstrual cycle to a peak achieved at random location with decreases thereafter. Reproductive epidemiologists are interested in studying the distribution of the peak and the trajectory for women in different groups. Motivated by this application, we propose a simple approach for restricting each woman's mean trajectory to follow an umbrella shape. An unconstrained hierarchical Bayesian model is used to characterize the data, and draws from the posterior distribution obtained using a Gibbs sampler are then mapped to the constrained space. Inferences are based on the resulting quasi-posterior distribution for the peak and individual woman trajectories. The methods are applied to a study comparing progesterone trajectories for conception and non-conception cycles.

Keywords: Changepoint; Gibbs sampler; Hormone measurements; Menstrual cycle; Nested data; Order restricted inference; Progesterone; Shape constraint.
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