This article proposes a semiparametric Bayesian approach for inference on an unknown isotonic regression function, f(x), characterizing the relationship between a continuous predictor, X, and a response variable, Y, adjusting for covariates, Z. A novel prior formulation is used, which avoids parametric assumptions on f(x), while enforcing the non-decreasing constraint and assigning positive prior probability to the null hypothesis of no association between X and Y conditional on Z. Through the use of carefully tailored hyperprior distributions, we allow for borrowing of information across different regions of X in estimating of f(x) and in assessing hypotheses about local increases in the function. Due to conjugacy properties, posterior computation is straightforward in a variety of settings, including log-linear models for Poisson data and logistic regression for binary outcomes. The methods are illustrated using a series of simulated data examples.

Keywords: Generalized additive model; Log linear; Logistic regression; Mixture prior; Multiple testing; Nonparametric regression; Smoothing; Trend test.

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