Generalized Linear Models

这项研究将考察一些问题，所有这些问题都直接或间接地与广义线性型的统计模型有关。这项工作的主要部分是关于多变量模型，或者是图形依赖模型，或者是为流行病学或类似目的构建的边缘模型。这些模型在多大程度上能够进行因果解释将被检验。除了专门的模型，如在排序数据分析中产生的模型外，最迫切的需要是找到令人满意的方法来处理具有多个变化分量的非线性模型。残差似然是线性模型中用于估计方差成分的一种技术，绕过回归参数。目的是为广义线性模型开发类似的策略，以帮助将注意力集中在参数的子集上，而不影响推论。研究的最后一个组成部分与模型公式的代数有关，特别是关于常用代数的局限性，特别是在涉及同源因子的情况下。只有少数群不变量子空间对应于有趣的统计模型：群不变量的一个有希望的替代方案是monoid不变量，它与因子模型密切相关。目的是找到一种简洁的方法，以一种明确的方式指定合适的不变子空间，并且可以被统计学家和计算机理解。这个练习将包括代数和计算作业。除了保险和市场营销等商业应用外，广义线性模型还广泛应用于社会、物理和生物科学的各种应用中。尽管取得了这样的成功，但仍有一些重要领域需要进一步发展。在这些应用中，最重要的是随机效应来自几个可识别的来源。例子包括纵向研究、植物和动物育种的遗传模型以及农业田间实验。将开发方法来处理由这种随机效应引起的依赖模式。第二个已经取得一些进展的领域是统计模型公式与代数中所谓的一元不变子空间之间的联系。目前用于统计模型的代数不能识别两个因素具有相同的一组水平。将发展一种扩展代数来适应这种现象。

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.