Mathematical Sciences: Algebraic Methods in Multivariate Statistical Analysis

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It is proposed to continue the study of the statistical properties of multivariate normal models determined by lattice conditional independence (LCI) assumptions and/or group symmetry assumptions on the covariance matrix, augmented by compatible linear restrictions on the mean. LCI models have been shown to be applicable to the analysis of multivariate normal data sets with nonnested missing data patterns. A new application of LCI models will be emphasized here: their application to the analysis of a collection of nonnested dependent linear regression models, known in econometrics as a seemingly unrelated regression (SUR) model. A SUR model may be thought of as a finite nonnested collection of linear regression subspaces with correlated errors across regressions. For a given SUR model, the least restrictive LCI covariance model compatible with the mean structure can be determined, leading to explicit maximum likelihood estimates for the SUR model. To date, LCI models have been studied only for multivariate normal distributions. Another new aspect of this proposal is the application of LCI models to categorical data in multiway contingency tables. As in the case of normal data, such LCI models should allow explicit maximum likelihood estimates for contingency tables with nonnested missing categories. Many familiar statistical models occurring in classical multivariate analysis (the study of correlated data) can be viewed as special cases of models defined in terms of natural algebraic conditions on the means and/or covariances. This viewpoint will (a) lead to a unified and explicit (non-iterative) analysis of these models, and (b) expand the scope of multivariate analysis by allowing the application of classical methods to many new models, as well as allowing the possibilities of missing data occurring in nonnested patterns and of nonnested regression subspaces.