Theory and Applications of Latent Variable and Mixture Models for Repeated Measurements

这项工作的重点是严格一维潜变量模型，它可以被认为是在数据的边缘分布中诱导条件联想的混合模型(Rosenbaum，1984；Holland和Rosenbaum，1986)，或者是Stout(1987,1990)本质上的一维模型的特例。我建议扩展我过去的努力，将这些想法与文献中关于正依赖和负依赖的概念一起使用(例如Joag-Dev，1983；Joag-Dev和Proschan，1982；Newman和Wright，1981；或最近由Block，Sampson和Savits编辑的集合，1990)来刻画严格的一维模型。我最近对这个问题的探索提出了一种相当简单的方法，作为附带好处，它概括了de Finetti对可交换性的表征，而不需要像Diaconis和Freedman(1984)那样指定足够的统计数据。这一建议的第二条工作路线是，沿着Kass，Tierney和Kadane(1990)以及Clarke和Barron(1990)的路线，使用渐近方法探索严格一维模型下关于潜在特征的推断，该模型断言给定潜在特征的条件独立性，而实际上某种温和形式的条件依赖是成立的。此外，由于Ramsay(Ramsay，1991；Ramsay and Winsburg，1991)的非参数回归思想，MLE类估计的渐近标准误差中的偏差也可以计算出来，在某些情况下，还可以用非参数回归思想进行修正。最后，讨论了应用和计算中的一些问题，包括统一和扩展潜在变量数据分析的非参数技术(例如Molenaar，1991；和Grayson，1988)；以及发展参数统计模型和计算方法(例如，对Lindsay，Clogg和Grego，1991所指定的模型的多分类版本的有效估计)，这些都是在认知科学的小规模实验的数据分析中出现的。这项建议涉及重复测量数据的潜在变量模型的统计和概率特征，这是定量心理学家、心理计量学家和认知科学家以及其他社会科学家感兴趣的。潜变量模型的一个典型应用是心理测量，其中潜变量是一个不可观测的变量，它指示一个人的心理特征的水平-例如抑郁、数学能力、工作满意度或工作记忆能力-我们只能通过人对一系列任务、问卷项目等的反应来间接观察。这类数据可以从精神病学评定表、标准化的学业成绩或能力测试(如SAT和GRE)、社会学的标准化问卷或认知心理学实验中对一组任务的编码反应中获得。这项研究的一个主要成果将是在基本统计理论和实际应用水平上对测量问题的潜变量模型有更深的理解。这项研究产生的实用工具包括：确定这类模型与特定情况或数据集匹配程度的改进方法；根据这些模型调整科学推论的规则，以应对正在使用的模型和生成数据的机制之间不可避免的不匹配，无论失配多么小；以及适用于小规模实验数据的计算和模型构建方法，例如认知心理学，在认知心理学中，这些模型在概念上是自然的，但当前的方法往往会崩溃。这里提出的许多工作都是围绕跨学科合作展开的，特别是与量化心理学家和教育测量专家的合作，目的是开发将在应用中使用的统计理论。

The focus of this work is on strictly unidimensional latent variable models, which may be thought of as mixture models which induce conditional association (Rosenbaum, 1984; Holland and Rosenbaum, 1986) in the marginal distribution of the data, or as a special case of Stout's (1987, 1990) essentially unidimensional models. I propose to extend my past efforts to use these ideas together with notions from the literature on positive and negative dependence (e.g. Joag-Dev, 1983; Joag-Dev and Proschan, 1982; Newman and Wright, 1981; or more recently the collection edited by Block, Sampson and Savits, 1990) to characterize strictly unidimensional models. My recent explorations of this problem suggest a reasonably straightforward approach that, as a side benefit, generalizes de Finetti's characterization of exchangeability, without the need to specify sufficient statistics as in, for example, Diaconis and Freedman (1984). A second line of work in this proposal is the exploration, using asymptotic methods along the lines of Kass, Tierney and Kadane (1990), and Clarke and Barron (1990), of inferences about the latent trait under a strictly unidimensional model, which asserts conditional independence given the latent trait, when in fact some mild form of conditional dependence holds. In addition, biases in the asymptotic standard error of an MLE-like estimator can also be calculated and, in some cases, corrected using nonparametric regression ideas due to Ramsay (Ramsay, 1991; Ramsay and Winsburg, 1991). Finally, some problems in applications and computing will be examined, including unifying and extending nonparametric techniques for latent variables data analysis (e.g. Molenaar, 1991; and Grayson, 1988); and developing parametric statistical models and computational methods (e.g. efficient estimation of a polytomous version of the model specified by Lindsay, Clogg and Grego, 1991) that arise in the analysis of data from small scale experiments in cognitive science. This proposal concerns statistical and probabilistic features of latent variable models for repeated measures data, which is of interest to quantitative psychologists, psychometricians, and cognitive scientists, as well as other social scientists. A typical application for latent variable models is psychological measurement, in which the latent variable is an unobservable variable that indicates the level of a psychological feature of a person---such as depression, mathematical aptitude, job satisfaction, or working memory capacity---that we observe only indirectly through the person's responses to a series of tasks, questionnaire items, etc. Data of this type might be obtained from psychiatric rating forms, standardized academic achievement or aptitude tests like the SAT and GRE, standardized questionnaires in sociology, or coded responses to a set of tasks in experiments in cognitive psychology. A primary outcome of this research will be a deeper understanding of latent variable models for measurement problems, at both the level of fundamental statistical theory and the level of practical applications. Practical tools arising from this research would include: enhanced methods for deciding how well or poorly this class of models matches particular situations or data sets; rules for adjusting scientific inferences based on these models for the inevitable mismatch, however small, between the model being used and the mechanism that generated the data; and computational and model-building methods that are adapted to small-scale experimental data, such as might be found in cognitive psychology, where these models are conceptually natural but current methods tend to break down. Much of the work proposed here is built around interdisciplinary collaboration, especially with quantitative psychologists and educational measurement specialists, with the goal of developing statistical theory that will be of use in applications.