McCullagh 9705347 This research will examine a number of issues, all bearing directly or indirectly on statistical models of the generalized linear type. A major part of the work is concerned with multivariate models, either graphical dependence models, or marginal models constructed for epidemiological or similar purposes. The extent to which such models are capable of a causal interpretation will be examined. Apart from specialized models such as those arising in the analysis of ranked data, a most pressing need has been for satisfactory methods for dealing with non-linear models having several components of variation. Residual likelihood is one technique used in linear models for the estimation of variance components, by-passing the regression parameters. The intention is to develop a similar strategy for generalized linear models in order to help focus attention on subsets of the parameters without compromising the inferences. The final component of the research is related to the algebra of model formulae, and in particular, on the limitations of the algebra in common use, particularly where homologous factors are involved. Only a minority of group-invariant subspaces correspond to interesting statistical models: A promising alternative to group-invariance is monoid-invariance, which corresponds closely to factorial models. The aim is to find a succinct way of specifying suitable invariant subspaces in a way that is unambiguous and can be understood by statistician and computer alike. This exercise will involve a mixture of algebra and computational work. Generalized linear models have been used in a wide variety of applications in the social, physical and biological sciences, in addition to commercial applications such as insurance and marketing. Despite this success, there are a number of important areas in which further development would be beneficial. Foremost among these are applications in which random effects accrue from several identifiable sources. Examples incl ude longitudinal studies, genetic models for plant and animal breeding, and agricultural field experiments. Methods will be developed to deal with patterns of dependence induced by such random effects. A second area on which some progress has already been made is the connection between statistical model formulas and what are known in algebra as monoid-invariant subspaces. The currently-used algebra for statistical models is incapable of recognizing that two factors have the same set of levels. An extended algebra will be developed to accommodate this phenomenon.
