Generalized Linear Models

麦卡拉，9705347 本研究将探讨一些问题，所有直接或间接的广义线性类型的统计模型。 工作的一个主要部分是关注多变量模型，无论是图形依赖模型，或边际模型构建流行病学或类似的目的。 在何种程度上，这种模式是能够因果关系的解释将进行审查。 除了专门的模型，如在排名数据的分析中出现的，最迫切的需要是令人满意的方法来处理非线性模型有几个组成部分的变化。 残差似然是线性模型中用于估计方差分量的一种技术，绕过回归参数。 其目的是为广义线性模型开发一种类似的策略，以帮助将注意力集中在参数的子集上，而不影响推断。 研究的最后一个组成部分是有关的代数模型公式，特别是在常用的代数的局限性，特别是在同源因素的参与。 只有少数群不变子空间对应于有趣的统计模型：群不变的一个有希望的替代方案是monoid-invariance，它与阶乘模型密切对应。 目的是找到一种简洁的方式来指定合适的不变子空间的方式是明确的，可以理解的统计学家和计算机一样。 这个练习将涉及代数和计算工作的混合。 除了保险和营销等商业应用之外，广义线性模型还广泛应用于社会、物理和生物科学领域。 尽管取得了这一成功，但仍有一些重要领域需要进一步发展。 其中最重要的是应用程序中的随机效应从几个可识别的来源。 例子包括纵向研究，植物和动物育种的遗传模型，以及农业田间试验。 将制定方法来处理这种随机效应引起的依赖模式。 第二个已经取得了一些进展的领域是统计模型公式和代数中称为幺半群不变子空间之间的联系。 目前使用的统计模型代数无法识别两个因素具有相同的水平集。 一个扩展的代数将被开发，以适应这种现象。

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.