Mathematical Sciences: Smoothed Nonparametric Hazard Regression

摘要该项目的主要目标是在加性风险回归模型（Aalen模型）的框架下，开发具有时间相关回归系数的非参数风险回归的平滑方法。Aalen模型考虑了协变量的值和贡献随时间变化的可能性。最重要的是，没有假设这种时间依赖性的特定参数形状。在当前的项目中，将使用核平滑来估计回归系数曲线。主要目标是开发数据驱动的带宽选择器，并研究它们的属性。预期所产生的方法将在目前使用的经验估计的基础上得到改进。此外，核方法允许估计回归系数曲线本身，而不是累积系数。这对于图形数据分析尤其重要。新的平滑方法将在面向对象的编程语言XLISP-STAT中实现；将提供用户友好的界面。动态图形将用于支持可视化数据探索。作为该项目的一部分，开发的平滑程序将用于分析流行病学数据。在教育方面，建议为统计专业硕士或博士开设一门新的讲座课程“曲线估计中的平滑方法”。将包括当前提案的研究成果。动态图形软件将在课堂上演示平滑技术，从而使学生能够接触到尖端技术。医学中的一个重要问题是，根据患者目前的状况和治疗方法，预测患者的生存时间。病人的病情由“协变量值”来描述，如血胆固醇、血压、抗体数量等。分析生存数据的一个中心问题是评估协变量的影响；例如，量化血液中胆固醇水平升高会增加中风风险的程度。通常已知协变量的影响会随时间而变化。在这种情况下，标准方法往往会失败。建议的项目基于Aalen模型进行风险评估。这种模式非常灵活。允许协变量的影响随时间变化，而无需事先假定这种时间依赖性的任何特定形状。在这个项目中，将采用现代平滑技术来估计协变量的影响。平滑方法在描述性数据分析中非常有用，在统计学中得到广泛应用。然而，在生存数据的背景下，使用相应的方法是最近的发展，有许多开放的问题。项目的一部分是新开发的估算程序的用户友好计算实现。交互式图形将广泛用于支持可视化数据探索。利用所提供的软件，将首次为从业人员提供Aalen风险回归中的平滑方法。新的平滑程序将用于分析流行病学数据。在教育方面，建议为统计专业硕士或博士开设一门新的讲座课程“曲线估计中的平滑方法”。将包括当前提案的研究成果。动态图形软件将用于演示平滑技术在课堂上，从而使学生获得最前沿的技术。

9501893 Grund Abstract The main objective of the project is to develop smoothing methods for nonparametric hazard regression with time-dependent regression coefficients, in the framework of an additive hazard regression model (Aalen model). The Aalen model incorporates the possibility that both value and contribution of covariates may change over time. Most important, no particular parametric shape of this time-dependence is assumed. In the current project, kernel smoothing will be used to estimate the regression coefficient curves. A major objective is to develop data-driven bandwidth selectors, and to investigate their properties. It is expected that the resulting methods will improve upon currently used empirical estimates. Moreover, kernel methods allow one to estimate the regression coefficient curves themselves, as opposed to cumulative coefficients. This is particularly important for graphical data analysis. The new smoothing methods will be implemented in XLISP-STAT, an object- oriented programming language; a user-friendly interface will be provided. Dynamic graphics will be used to support visual data exploration. As part of the project, the developed smoothing procedures will be used to analyze epidemiological data. On the side of education, the development of a new lecture course on ``Smoothing Methods in Curve Estimation'' for statistics majors at the M.S. or Ph.D. level is proposed. Research results of the current proposal will be included. Dynamic graphics software will be used to demonstrate smoothing techniques in class, thus giving students access to cutting edge technology An important problem in medicine is to predict the survival of patients, given their current condition and treatment. The condition of a patient is described by "covariate values", such as blood cholesterol, blood pressure, number of antibodies, etc. A central problem in analyzing survival data is to assess the influence of covariates; for example, to quantify by how much an elevated level of blood cholesterol increases the risk of stroke. Often the influence of covariates is known to change with time. In this case standard methods tend to fail. The proposed project develops risk estimates based on the Aalen model. This model is extremely flexible. The influence of covariates is allowed to change over time, without assuming any particular shape of this time-dependence beforehand. In this project, modern smoothing techniques will be adapted to estimate the influence of covariates. Smoothing methods are extremely useful for descriptive data analysis, and widely used in statistics. In the context of survival data, however, the use of corresponding methods is a very recent development, with many open problems. Part of the project is the user-friendly computational implementation of the newly developed estimation procedures. Interactive graphics will be used extensively to support visual data exploration. With the provided software, smoothing methods in Aalen hazard regression will be available for practitioners for the first time. The new smoothing procedures will be used to analyze epidemiological data. On the side of education, the development of a new lecture course on ``Smoothing Methods in Curve Estimation'' for statistics majors at the M.S. or Ph.D. level is proposed. Research results of the current proposal will be included. Dynamic graphics software will be used to demonstrate smoothing techniques in class, thus giving students access to cutting edge technology.