Generalized monotonic regression using random change points

We introduce a procedure for generalized monotonic curve fitting that is based on a Bayesian analysis of the isotonic regression model. Conventional isotonic regression fits monotonically increasing step functions to data. In our approach we treat the number and location of the steps as random. For each step level we adopt the conjugate prior to the sampling distribution of the data as if the curve was unconstrained. We then propose to use Markov chain Monte Carlo simulation to draw samples from the unconstrained model space and retain only those samples for which the monotonic constraint holds. The proportion of the samples collected for which the constraint holds can be used to provide a value for the weight of evidence in terms of Bayes factors for monotonicity given the data. Using the samples, probability statements can be made about other quantities of interest such as the number of change points in the data and posterior distributions on the location of the change points can be provided. The method is illustrated throughout by a reanalysis of the leukaemia data studied by Schell and Singh.