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

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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.