Generalized Linear Models

麦卡拉9705347这项研究将审查一些问题，所有这些问题都直接或间接地与广义线性类型的统计模型有关。这项工作的主要部分是关于多变量模型，要么是图形依赖模型，要么是为流行病学或类似目的而构建的边际模型。我们将考察这些模型在多大程度上能够进行因果解释。除了专门的模型，例如那些在分析排序数据时出现的模型，最迫切的需要是找到令人满意的方法来处理具有几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分量的一种技术，它绕过回归参数。其目的是为广义线性模型开发类似的策略，以便在不影响推断的情况下帮助将注意力集中在参数的子集上。研究的最后一部分涉及模型公式的代数，特别是关于常用代数的局限性，特别是在涉及同源因素的情况下。只有一小部分群不变子空间对应于有趣的统计模型：群不变的一个有前途的替代方案是么半群不变，它与阶乘模型密切对应。其目的是找到一种简明的方法来指定合适的不变子空间，这种方法是明确的，并且可以被统计学家和计算机理解。这项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外，广义线性模型还被广泛应用于社会科学、物理学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的进一步发展将是有益的。其中最重要的是从几个可识别的来源产生随机效应的应用。例子包括纵向研究、动植物育种的遗传模型和农业田间试验。将开发方法来处理由这种随机效应引起的依赖模式。已经取得一些进展的第二个领域是统计模型公式和在代数中被称为么半群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水平集。将发展一种扩展的代数来适应这一现象。

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.