Mathematical Sciences: General Linear Models

数学科学：一般线性模型

• 批准号: 9403560
• 负责人: Peter McCullagh
• 金额: $ 19万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403560&HistoricalAwards=false
• 关键词: Mathematical; Sciences; General; Linear; Models

Peter McCullagh will investigate a number of issues, all bearing directly or indirectly on generalized linear models. Apart from specialized models such as those arising in the analysis of ranked data, the most pressing need has been for satisfactory methods for dealing with non-linear models having several components of variation. Examples of experiments where such models are appropriate are not hard to find. The salamander data (McCullagh and Nelder, 1989, p. 440-444) has been widely studied as an archetypal example involving purely binary data. Linear models having several variance components were developed early by Yates and Fisher, but in the pre-computer era only balanced designs could be tackled. For the estimation of variance components in unbalanced designs the residual likelihood, i.e. the likelihood based on the residuals, has come to play a key role in recent work. A major aim of the present proposal is to extend this work to models of the generalized linear type. A second and related aim is to develop new methodologies for longitudinal data, particularly where this occurs in an epidemiological context. An important difference between linear and non-linear models is the distinction between subject-specific and population-averaged parameters, the latter being more relevant in epidemiological contexts. The final part of the proposal, a sort of counterpoint to the exponential-family, is the study of transformation models with particular emphasis on Mobius transformation and Cauchy models. The results obtained thus far are helpful for the light they shed on conditional inference and asymptotics. This proposal aims to develop statistical methods that enable a scientist to draw reliable conclusions from data where the model is non-linear and the design is such that there is more than one source of variation in the experiment. Such experimental designs are rather common in biological and agricultural research, where methods of analysis for linear mo dels are fairly well developed. For non-linear models, likelihood calculations are generally difficult and time-consuming. I propose to develop useful analytical approximations as an alternative to numerical Monte-Carlo simulation techniques.

Peter McCullagh将研究一些问题，所有这些问题都直接或间接地与广义线性模型有关。除了专门的模型，如在排名数据的分析中出现的，最迫切的需要是令人满意的方法来处理非线性模型有几个组成部分的变化。 这种模型适用的实验例子并不难找到。 蝾螈数据（McCullagh和Nelder，1989年，第440-444页）作为涉及纯二进制数据的典型例子被广泛研究。 具有多个方差分量的线性模型早期由Yates和Fisher开发，但在前计算机时代，只能处理平衡设计。 对于非平衡设计中方差分量的估计，残差似然，即基于残差的似然，在最近的工作中起着关键作用。 本建议的一个主要目的是将这项工作扩展到广义线性型模型。第二个相关目标是为纵向数据制定新的方法，特别是在流行病学背景下。 线性模型和非线性模型之间的一个重要区别是受试者特异性参数和人群平均参数之间的区别，后者在流行病学背景下更相关。 该建议的最后一部分，一种对应的指数族，是研究的转换模型，特别强调莫比乌斯变换和柯西模型。 到目前为止，所获得的结果是有帮助的光，他们在条件推理和渐近性。 该提案旨在开发统计方法，使科学家能够从非线性模型的数据中得出可靠的结论，并且设计是这样的，实验中有一个以上的变异来源。 这样的实验设计在生物学和农业研究中相当普遍，其中线性模型的分析方法相当发达。 对于非线性模型，似然计算通常是困难和耗时的。 我建议开发有用的分析近似值作为替代数值蒙特-卡罗模拟技术。