A nonparametric changepoint model for stratifying continuous variables under order restrictions and binary outcome

Modelling using monotonic regression can be a useful alternative to parametric approaches when optimal stratification for continuous predictors is of interest. This method is described here in the context of binary response. Within this framework we aim to address two points. First, we propose a method to enhance the parsimony of the model, by applying a reducing procedure based on a sequence of Fisher exact tests and a bootstrap method to select between full monotonic and reduced model. Secondly, we discuss the case of multiple predictors: an iterative algorithm (an extension of the Pool Adjacent Violators Algorithm) can be applied when more than one predictor variable is taken into account. The resulting model is a monotonic surface and can be applied alternatively to the additive monotonic models as described by Morton-Jones and colleagues when the explanatory variables are assumed to interact. The monotonic-surface model provides also a multivariate extension of the monotonic likelihood ratio test. This test is discussed here and an approach based on permutations to assess the p-value is proposed. Finally, we combine both ideas (reduced monotonic regression and monotonic-surface estimation) to a simple and easy to interpret model, which leads to a combination of the predictors in a few constant risk groups. Despite the fact that the proposed approach becomes somewhat cumbersome due to the lack of asymptotic methods to infer, it is attractive because of its simplicity and stability. An application will outline the benefit of using bivariate step functions in modelling.