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

这项工作的重点是严格的一维潜变量模型，它可以被认为是混合模型，诱导有条件的关联（罗森鲍姆，1984年;荷兰和罗森鲍姆，1986年）的边际分布的数据，或作为一个特殊的情况下，斯托特（1987年，1990年）基本上是一维模型。我建议扩展我过去的努力，将这些思想与文献中关于正相关和负相关的概念（例如Joag-Dev，1983; Joag-Dev和Proschan，1982;纽曼和赖特，1981;或最近由Block、Sampson和Savits编辑的集合，1990）一起使用，来描述严格的一维模型。我最近对这个问题的探索提出了一个相当直接的方法，作为一个附带的好处，它概括了de Finetti对交换的描述，而不需要像Diaconis和Freedman（1984）那样详细说明足够的统计数据。这个建议的第二条工作路线是沿着卡斯、蒂尔尼和卡丹（Kass，Tierney and Kadane，1990）以及克拉克和巴伦（Clarke and Barron，1990）的路线，使用沿着渐近方法，探索在严格的一维模型下关于潜在特质的推论，该模型断言给定潜在特质的条件独立性，而实际上某种温和形式的条件依赖成立。此外，也可以计算MLE类估计量的渐近标准误的偏差，在某些情况下，可以使用Ramsay的非参数回归思想进行校正（Ramsay，1991; Ramsay and Winsburg，1991）。最后，将研究应用和计算中的一些问题，包括潜变量数据分析的统一和扩展非参数技术（例如Molenaar，1991; Grayson，1988）;开发参数统计模型和计算方法（例如，有效估计由林赛，Clogg和Grego，1991年指定的模型的多分支版本），这在认知科学的小规模实验的数据分析中出现。 该建议涉及重复测量数据的潜变量模型的统计和概率特征，这是定量心理学家，心理测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察的变量，它表明一个人的心理特征水平-如抑郁，数学能力，工作满意度或工作记忆能力-我们只能通过一系列任务，问卷项目，这类数据可以从精神病评定表、标准化的学术成就或能力倾向测试（如SAT和GRE）、社会学中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。这项研究的主要成果将是更深入地了解潜在变量模型的测量问题，在基础统计理论和实际应用的水平。 从这项研究中产生的实用工具将包括：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，以防止所用模型与生成数据的机制之间不可避免的不匹配，无论这种不匹配多么小;以及适用于小规模实验数据的计算和模型构建方法，例如可能在认知心理学中发现的方法，其中这些模型在概念上是自然的，但当前的方法往往会崩溃。这里提出的大部分工作是建立在跨学科合作的基础上，特别是与定量心理学家和教育测量专家的合作，目的是开发在应用中有用的统计理论。

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.