Peter McCullagh will investigate a number of issues, all bearing directly or indirectly on generalized linear models. The main theme of the proposal is the development of a framework for constructing logically consistent statistical models, whose hallmark is extendability or scope. Every sensible linear model for a two-way layout with 4 treatments and 7 blocks must ordinarily be one element, or vector space, in a sequence of logically related vector spaces, one such subspace for each layout. Logical consistency demands that the restriction of one subspace to a subset of the blocks or treatments must coincide with the subspace associated with this subset. In mathematical terms, not only must the model be invariant under permutation of treatments and permutation of blocks, but the sequence of vector spaces must be closed under restriction to subsets. In algebraic terminology, such a sequence of vector spaces constitutes a representation of the category of injective maps on finite sets. The standard factorial models (hierarchical interaction models) coincide precisely with the regular sub-representations of the product category. The PI has developed this notion of extendability for designs, common in genetic studies of plant and animal breeding, in which two or more factors have the same set of levels. In order to accommodate these new models, it is found necessary to extend the factorial model formulae by the inclusion of several new operators that are relevant only for homologous factors. The aim is to develop a succinct way of specifying category-invariant subspaces, i.e.~models, in a way that is unambiguous and can be understood by the computer.
This proposal aims to study statistical models from the viewpoint of their logical structure, and to develop new models where appropriate. In each new area of application, whether it be genetics, biology or social science, new considerations relevant to the application emerge. For example, in experiments connected with plant breeding, the same set of plants may occur as males contributing pollen and as females contributing ova. Likewise, in citation studies, a given journal may occur as a citing journal or as a cited journal. Homologous factors of this sort rarely arise in agricultural field trials where factorial models were first developed. As a result, standard statistical models are not well suited to such applications. It is the aim of this proposal to study such structures from a logical viewpoint and to develop new statistical models where needed.
