ALGEBRAIC METHODS IN STATISTICS

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This project develops new methods in statistics, both theoretical and applied, using methods of advanced algebra. Results have been obtained in the systematization of the general linear mixed model and in the analysis of data having a structured pattern of correlation. For biomedical data using repeated measurements on the same case, it is often found that one or more data points are missing or were not obtained. Classical methods for analyzing such data require that such cases (e.g., subjects) be completely dropped from the analysis, despite the usually large amount of data that had been obtained on the same case. In order to satisfy the standard mathematical and statistical conditions for the analysis, such deletions often require that half or more of all cases be deleted. This is an inefficient use of biomedical data that is often difficult and costly to obtain, and using just the reduced data that was collected can lead to spurious findings. On the other hand, the Expectation-Maximization algorithm of Dempster, Laird, and Rubin [1977] has been in use for some time as a broadly successful antidote to this problem of missing data. The basic, iterative algorithm is well-known, but is also well-known to have convergence problems that are hard to diagnose and get around. Using an idea first proposed by Rubin and Szatrowski [1982], we give a complete solution to this problem above using methods of advanced algebra (technically: Jordan algebras). And now some other well-known statistical methods are shown to work precisely because of an implicit use of Jordan algebras, and so are special cases of our results. Our algorithm finds estimates for the total variation in an experiment, even when this variation is known to be constrained by any set of linear restrictions. Combined with rigorous, large-sample statistical approximations, the researchers can more systematically probe for effects in measurements taken over time (e.g, true variation vs. noise,) without having to delete cases. Thus, in the context of biomedical data (frequently having many missing data points), the new methods apply to growth curve models, variance components analysis, genetic linkage analysis, time series data, and to longitudinal data that is often acquired in clinical trials, or in epidemiological case-control studies.