Mathematical Sciences: Smoothed Nonparametric Hazard Regression

9501893本项目的主要目标是在一个可加风险回归模型(Aalen模型)的框架下，发展具有时间依赖回归系数的非参数风险回归的平滑方法。Aalen模型包含了协变量的价值和贡献都可能随时间变化的可能性。最重要的是，没有假定这种时间依赖的特定参数形状。在本项目中，将使用核平滑来估计回归系数曲线。一个主要目标是开发数据驱动的带宽选择器，并研究其特性。预计由此产生的方法将改进目前使用的经验估计数。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。新的平滑方法将用面向对象的编程语言XLISP-STAT实现；将提供用户友好的界面。将使用动态图形来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，建议为统计学专业的硕士或博士开设一门新的“曲线估计中的平滑方法”讲座。目前提案的研究成果将包括在内。动态图形软件将用于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的当前情况和治疗预测患者的生存。患者的病情是通过“协变量值”来描述的，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，量化血液胆固醇水平升高会增加中风风险的程度。通常情况下，协变量的影响会随着时间的推移而变化。在这种情况下，标准方法往往会失败。拟议的项目基于Aalen模型开发风险估计。这种模式非常灵活。协变量的影响可以随时间改变，而无需预先假定这种时间依赖关系的任何特定形状。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，在统计学中被广泛使用。然而，在生存数据的背景下，使用相应的方法是一个非常新的发展，有许多悬而未决的问题。该项目的一部分是对新开发的估算程序进行用户友好的计算实施。交互式图形将被广泛使用，以支持可视化数据探索。有了提供的软件，Aalen风险回归中的平滑方法将第一次为从业者所用。新的平滑程序将用于分析流行病学数据。在教育方面，建议为统计学专业的硕士或博士开设一门新的“曲线估计中的平滑方法”讲座。目前提案的研究成果将包括在内。将使用动态图形软件在课堂上演示平滑技术，从而使学生能够接触到尖端技术。

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.