## Bayesian Isotonic Regression and Trend Analysis

### Brian Neelon^{1} and David B. Dunson^{2}

### ^{1}University of North Carolina, Chapel Hill, and
^{2}National 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.