Bayesian Isotonic Regression and Trend Analysis

Brian Neelon1 and David B. Dunson2

1University of North Carolina, Chapel Hill, and 2National Institute of Environmental Health Sciences

June 2003

In many applications, the mean of a response variable can be assumed to be a non-decreasing function of a continuous predictor, controlling for covariates. In such cases, interest often focuses on estimating the regression function, while also assessing evidence of an association. This article proposes a new framework for Bayesian isotonic regression and order restricted inference. Approximating the regression function with a high dimensional piecewise linear model, the non-decreasing constraint is incorporated through a prior distribution for the slopes consisting of a product mixture of point masses (accounting for flat regions) and truncated normal densities. To borrow information across the intervals and smooth the curve, the prior is formulated as a latent autoregressive normal process. This structure facilitates efficient posterior computation, since the full conditional distributions of the parameters have simple conjugate forms. Point and interval estimates of the regression function and posterior probabilities of an association for different regions of the predictor can be estimated from a single MCMC run. Generalizations to categorical outcomes and multiple predictors are described, and the approach is applied to an epidemiology application.

Keywords: Additive model; Autoregressive prior; Constrained estimation; Monotonicity; Order restricted inference; Smoothing; Threshold model; Trend test.

The manuscript is available in PostScript and PDF formats.