Mathematical Sciences: General Linear Models

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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.