More Than Means Can Say: Regressing Entire Distribution Functions in Epidemiology and Biostatistics

回归分析的最终目标是获得给定一组解释变量的响应的一致性分布的信息。然而，这一目标很少实现，因为大多数已建立的回归模型仅估计作为解释变量的函数的条件均值，并假设高阶矩不受回归变量的影响。例如，平均儿童营养状况的决定因素不能被解释为营养不良的风险因素，因为后者的数量是由营养分布的较低分位数定义的，不能预期只是平均值的移动版本。 限制条件均值的根本原因是信号和噪声的可加性假设。 我们计划在转换模型的框架中放宽这一常见假设。 本文提出的一类新的半参数回归模型允许转换函数依赖于解释变量。 我们将研究潜在的 通过概率预测的评分规则的正则化优化，例如连续排名的概率得分，转换函数。 在某种意义上，这些模型可以被看作是“逆分位数回归”，因为我们的目标是估计分布函数而不是分位数函数。由此产生的模型在流行病学和生物统计学中有价值，特别是在描述可能的异方差，比较空间变化的分布，识别极端事件，导出预测区间和选择超出平均回归效应的变量。该资助计划的三个主要目标是评估模型估计的不同算法，有限样本的性能，基本建模框架的扩展到生存分析，从而更高的时刻的censoredresponse模型可以描述为解释变量的函数，最后，这些新技术的应用，估计儿童营养和出生体重的条件分布函数。 在这两种应用中，平均回归是一种严重的过度简化，因为分布特性，如通过营养状况的较低分位数或出生体重的预测区间来评估营养不良，是感兴趣的实际目标。

The ultimate goal of regression analysis is to obtain information about theconditional distribution of a response given a set of explanatory variables.This goal is, however, seldom achieved because most established regressionmodels only estimate the conditional mean as a function of the explanatoryvariables and assume that higher moments are not affected by the regressors.For example, determinants of the mean childhood nutrition status cannot beinterpreted as risk factors for undernutrition since the latter quantity isdefined by lower quantiles of the nutrition distribution that cannot beexpected to be just a shifted version of the mean. The underlying reasonfor the restriction to conditional means is the assumption of additivity ofsignal and noise. We plan to relax this common assumption in the frameworkof transformation models. The novel class of semiparametric regressionmodels proposed herein allows transformation functions to depend onexplanatory variables. We will investigate the estimation of the underlying transformation functions by regularised optimisation of scoring rules forprobabilistic forecasts, e.g. the continuous ranked probability score. In acertain sense, these models can be viewed as ``inverse quantile regression''because we aim at estimating the distribution function instead of thequantile function.The resulting models promise to be valuable in epidemiology andbiostatistics, especially for describing possible heteroscedasticity,comparing spatially varying distributions, identifying extreme events,deriving prediction intervals and selecting variables beyond mean regressioneffects.The three main objectives of this grant proposal are the evaluation ofdifferent algorithms for model estimation with respect to their finitesample performance, the extension of the basic modelling framework tosurvival analysis and thus to models where higher moments of a censoredresponse may be described as functions of explanatory variables and,finally, the application of these novel techniques for the estimation of conditional distribution functions for childhood nutrition and birthweights. In these two applications, mean regression is a severe oversimplification since distributional properties, such as the assessmentof undernutrition by lower quantiles of the nutrition status or predictionintervals for birth weights, are the actual targets of interest.