Mathematical Methods for Small--Sample Biostatistical Inference

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The investigator applies various mathematical methodsto extend the range of application of saddlepointapproximation techniques and exact enumeration techniquesin statistical inference. These techniques are applied tomulti-dimensional conditional inference, achieved primarilyby developing new approximations to multivariate tailprobabilities for some component of a vector of sufficientstatistics conditional on the remaining components, andmethods for calculating these tail probabilities exactly.Particular attention is paid to probability modelswhose sufficient statistics have lattice distributions,since standard asymptotic techniques frequently failin this context. These models, including logistic andPoisson regression and contingency table models, arevery frequently used in applied biostatistical work.Specifically, this research includes the applicationof multidimensional saddlepoint approximations toorder--restricted hypotheses. It extends existingApproximations applicable to continuous distributions togeneral non-lattice distributions. It develops guidelinesfor tuning approximate conditional inferential methods toobtain higher power. Computational algorithms developedas part of this work are publicly available.Many current statistical techniques rely on mathematical approximations;the accuracy of these approximations ranges from very good to inadequate.This research involves approximations known to be almost always of highaccuracy, and applies them in some statistical contexts that are widelyused by scientists in a variety of disciplines. This research allowsinvestigators to draw valid conclusions from smaller data sets,particularly in cases when important research questions are phrased in termsof a number of quantities that must be accounted for. This situation occurs in a wide range of areas, from finance to political science to medicine.For example, a new medical therapy might be expected to lead to improvements,potentially measured in a number of ways. The investigator wishes todemonstrate that individuals receiving the new therapy at least as well according to all of the potential measures, and better on at least onemeasure, than do patients on the original therapy. Standard statisticalmethods do not handle such situations in an efficient way; the currentresearch represents a significant improvement.