Generalized Linear Models

广义线性模型

• 批准号: 0071726
• 负责人: Peter McCullagh
• 金额: $ 22.5万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071726&HistoricalAwards=false
• 关键词: Generalized; Linear; Models

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.

Peter McCullagh将研究一些问题，所有这些问题都直接或间接地与广义线性模型有关。 该提案的主题是建立一个框架，用于构建逻辑上一致的统计模型，其特点是可扩展性或范围。 对于具有4个处理和7个块的双向布局，每个合理的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。 逻辑一致性要求一个子空间对块或处理的子集的限制必须与与该子集相关联的子空间一致。 在数学术语中，不仅模型必须在处理的置换和块的置换下不变，而且向量空间序列必须在子集的限制下闭合。 在代数术语中，这样的向量空间序列构成了有限集上内射映射范畴的一种表示。 标准的析因模型（层次交互模型）与产品类别的常规子表示完全一致。 PI已经发展了这种设计的可扩展性概念，在植物和动物育种的遗传研究中很常见，其中两个或更多个因素具有相同的水平。 为了适应这些新的模型，它被发现有必要扩展factorial模型formulaebby包括几个新的运营商是相关的同源因子。 目的是发展一种简洁的方法来指定范畴不变子空间，即~模型，以一种明确的方式，可以被计算机理解。这个建议的目的是从它们的逻辑结构的角度来研究统计模型，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，都出现了与应用有关的新考虑。例如，在与植物育种有关的实验中，同一组植物可能以雄性提供花粉和雌性提供卵子的形式出现。 同样，在引文研究中，一个给定的期刊可能会出现作为一个引用期刊或作为一个被引用期刊。 在农业田间试验中，这种同源因子很少出现，因为因子模型是在田间试验中首次开发的。 因此，标准的统计模型并不适合这种应用。 本提案的目的是从逻辑角度研究这些结构，并在必要时开发新的统计模型。