Mathematical Sciences: General Linear Models

Peter McCullagh将研究一系列直接或间接与广义线性模型相关的问题。除了专门的模型，如在排序数据分析中出现的模型外，最迫切的需要是找到令人满意的方法来处理具有多个变化分量的非线性模型。这种模型适用的实验实例并不难找到。蝾螈数据（McCullagh and Nelder, 1989, p. 440-444）作为涉及纯二进制数据的典型例子被广泛研究。耶茨和费舍尔早期开发了具有多个方差成分的线性模型，但在前计算机时代，只能解决平衡设计问题。对于不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类型的模型。第二个相关目标是开发纵向数据的新方法，特别是在流行病学背景下。线性和非线性模型之间的一个重要区别是特定主题和总体平均参数之间的区别，后者在流行病学背景下更为相关。建议的最后一部分，一种指数族的对应物，是对转换模型的研究，特别强调莫比乌斯转换和柯西模型。到目前为止所得到的结果有助于阐明条件推理和渐近性。该提案旨在发展统计方法，使科学家能够从模型是非线性的数据中得出可靠的结论，并且设计在实验中有多个变化源。这样的实验设计在生物和农业研究中相当普遍，其中线性模型的分析方法已经相当发达。对于非线性模型，似然计算通常是困难且耗时的。我建议发展有用的解析近似，作为数值蒙特卡罗模拟技术的替代方法。

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.