Bayesian Inferences on Shape Constrained Hormone Trajectories in the Menstrual Cycle

1 SUMMARY. In many biomedical applications, one can assume that the mean of an outcome variable increases monotonically with increases in a predictor to an unknown peak and decreases thereafter. To account for dependency in outcome measurements, one can apply a hierarchical model with random effects and autocorrelated errors. In the absence of shape constraints, Bayesian computation can proceed via a Gibbs sampling algorithm. Unfortunately , standard approaches for incorporating parameter constraints in Bayesian analyses cannot be used when the constraints are on higher level parameters in the hierarchy. To solve this problem, this article proposes a transformation approach in which samples from the unconstrained posterior density for the higher level parameters are transformed to the restricted space using a minimal distance projection. This approach is shown to suggest limited bias induced by the order constraint as well as a potential improvement in efficiency relative to unconstrained analyses and analyses that place constraints on the population parameters. The methods are illustrated through application to progesterone data from the literature.