Generalized 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。该提案的主题是开发一个框架，用于构建逻辑上一致的统计模型，其标志是可扩展性或范围。具有4个处理和7个块的双向布局的每个合理线性模型通常必须是逻辑相关向量空间序列中的一个元素或向量空间，每个布局都有一个这样的子空间。逻辑一致性要求对块或处理子集的子空间的限制必须与与该子集相关联的子空间一致。在数学上，不仅在处理置换和块置换下模型必须是不变的，而且向量空间的序列在子集的限制下必须是封闭的。在代数术语中，这样一个向量空间序列构成有限集合上的内射映射范畴的表示。标准的阶乘模型（层次交互模型）与产品类别的规则子表示精确地吻合。PI开发了这种设计可扩展性的概念，在植物和动物育种的遗传研究中很常见，其中两个或多个因素具有相同的一组水平。为了适应这些新模型，发现有必要通过包含几个仅与同源因子相关的新算子来扩展阶乘模型公式。目的是开发一种简洁的方式来指定类别不变子空间，即~模型，以一种明确的方式，可以被计算机理解。本提案旨在从逻辑结构的角度研究统计模型，并在适当的地方开发新的模型。在每一个新的应用领域，无论是遗传学、生物学还是社会科学，都会出现与应用相关的新考虑。例如，在与植物育种有关的实验中，同一组植物可能出现雄性提供花粉和雌性提供卵子的情况。同样，在引文研究中，给定的期刊可能是被引用的期刊，也可能是被引用的期刊。这类同源因子很少出现在农业田间试验中，其中析因模型首先被开发出来。因此，标准统计模型不太适合这样的应用程序。本建议的目的是从逻辑的角度研究这种结构，并在需要时发展新的统计模型。

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 constitutesa 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 formulaeby 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.