This continuing project will develop new methods in statistics, using algebraic techniques, that have applications for biomedical research. Two areas of major interest are the problems of statistical estimation in the presence of missing data, or data having a known patterned covariance matrix. Recently solutions to both these problems for multinormally distributed data, as well as for counted data (multiway contingency tables) have been obtained. The method solves for the maximum likelihood estimate of the parameter vector, thereby generating estimates that are unbiased as well as asymptotically fully efficient. The technique invokes the Dempster-Laird-Rubin EM algorithm, published in 1977, and Jordan algebras to iteratively solve for the parameter vector. Known properties of Jordan algebras are invoked to guarantee convergence, so that each iteration only uses a matrix multiplication by a single fixed matrix. Applications are in the areas of longitudinal data and repeated measures analysis.
