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

回归分析的最终目标是在给定一组解释变量的情况下获得关于反应的条件分布的信息。然而，这一目标很少实现，因为大多数已建立的回归模型只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，儿童平均营养状况的决定因素不能被解释为营养不良的风险因素，因为后者是由营养分布的较低分位数定义的，不能期望仅是均值的移位版本。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内放松这一常见的假设。本文提出的新型半参数回归模型允许变换函数依赖于解释变量。我们将通过对概率预测的评分规则进行正则化优化来研究潜在转换函数的估计，例如连续排序的概率分数。从某种意义上讲，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。所得到的模型在流行病学和生物统计学中很有价值，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、推导预测区间和选择超出平均回归效应的变量方面。这项拨款提案的三个主要目标是评估不同的模型估计算法关于它们的有限样本性能，将基本建模框架扩展到生存分析，从而扩展到可以将集中响应的高阶矩描述为解释变量的模型这些新技术在估计儿童营养和出生体重条件分布函数方面的应用。在这两个应用中，均值回归是一种严重的过于简单化，因为分布特性，例如通过营养状况的较低分位数来评估营养不良或预测出生体重的间隔，是实际感兴趣的目标。

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.