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.

Keywords: Changepoint; Gibbs sampling; Isotonic regression; Order constraint; Monotonicity; Multi-level model; Transformation; Umbrella order.

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