Regression analysis under the assumption of monotonicity is a well-studied statistical problem and has been used in a wide range of applications. However, there remains a lack of a broadly applicable methodology that permits information borrowing, for efficiency gains, when jointly estimating multiple monotonic regression functions. We introduce such a methodology by extending the isotonic regression problem presented in the article "The isotonic regression problem and its dual" (Barlow and Brunk, 1972). The presented approach can be applied to both fixed and random designs and any number of explanatory variables (regressors). Our framework penalizes pairwise differences in the values (levels) of the monotonic function estimates, with the weight of penalty being determined based on a statistical test, which results in information being shared across data sets if similarities in the regression functions exist. Function estimates are subsequently derived using an iterative optimization routine that uses existing solution algorithms for the isotonic regression problem. Simulation studies for normally and binomially distributed response data illustrate that function estimates are consistently improved if similarities between functions exist, and are not oversmoothed otherwise. We further apply our methodology to analyse two public health data sets: neonatal mortality data for Porto Alegre, Brazil, and stroke patient data for North West England.

翻译：在单调性假设下的回归分析是一个被充分研究的统计问题，并已广泛应用于诸多领域。然而，在联合估计多个单调回归函数时，仍缺乏一种具有广泛适用性的方法，能够通过信息借用提高估计效率。我们通过拓展Barlow和Brunk（1972）论文《保序回归问题及其对偶》中提出的保序回归问题，引入了这样一种方法论。所提出的方法可同时适用于固定设计和随机设计，以及任意数量的解释变量（回归变量）。我们的框架对单调函数估计值（水平）之间的成对差异进行惩罚，惩罚权重基于统计检验确定，从而在回归函数存在相似性时实现数据集之间的信息共享。函数估计值随后通过迭代优化程序获得，该程序利用现有的保序回归问题求解算法。针对正态分布和二项分布响应数据的模拟研究表明，当函数之间存在相似性时，函数估计值能持续得到改进，否则不会产生过度平滑。我们进一步将本方法应用于分析两个公共卫生数据集：巴西阿雷格里港的新生儿死亡率数据和英格兰西北部的卒中患者数据。