Salanti Georgia, Kurt Ulm
Institute for Medical Statistics and Epidemiology, Klinikum Rechts der Isar, Munich, Germany.
Stat Methods Med Res. 2003 Aug;12(4):351-67. doi: 10.1191/0962280203sm338ra.
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.
当对连续预测变量进行最优分层感兴趣时，使用单调回归进行建模可以作为参数方法的一种有用替代方法。本文在二元响应的背景下描述了这种方法。在此框架内，我们旨在解决两个问题。首先，我们提出一种方法来提高模型的简约性，通过应用基于一系列费舍尔精确检验的约简程序和一种自举方法来在完全单调模型和简约模型之间进行选择。其次，我们讨论多个预测变量的情况：当考虑多个预测变量时，可以应用一种迭代算法（池相邻违反者算法的扩展）。得到的模型是一个单调曲面，当假设解释变量相互作用时，可以替代地应用于莫顿 - 琼斯及其同事所描述的加性单调模型。单调曲面模型还提供了单调似然比检验的多变量扩展。本文讨论了这种检验，并提出了一种基于排列来评估p值的方法。最后，我们将这两个想法（简约单调回归和单调曲面估计）结合到一个简单且易于解释的模型中，这导致在几个恒定风险组中对预测变量进行组合。尽管由于缺乏渐近推断方法，所提出的方法变得有些繁琐，但由于其简单性和稳定性，它具有吸引力。一个应用将概述在建模中使用二元阶梯函数的好处。
