Mathematical Methods for Small--Sample Biostatistical Inference

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

The investigator applies various mathematical methods to extend the range of application of saddlepoint approximation techniques and exact enumeration techniques in statistical inference. These techniques are applied to multi-dimensional conditional inference, achieved primarily by developing new approximations to multivariate tail probabilities for some component of a vector of sufficient statistics conditional on the remaining components, and methods for calculating these tail probabilities exactly. Particular attention is paid to probability models whose sufficient statistics have lattice distributions, since standard asymptotic techniques frequently fail in this context. These models, including logistic and Poisson regression and contingency table models, are very frequently used in applied biostatistical work. Specifically, this research includes the application of multidimensional saddlepoint approximations to order--restricted hypotheses. It extends existing Approximations applicable to continuous distributions to general non-lattice distributions. It develops guidelines for tuning approximate conditional inferential methods to obtain higher power. Computational algorithms developed as 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 high accuracy, and applies them in some statistical contexts that are widely used by scientists in a variety of disciplines. This research allows investigators to draw valid conclusions from smaller data sets, particularly in cases when important research questions are phrased in terms of 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 to demonstrate that individuals receiving the new therapy at least as well according to all of the potential measures, and better on at least one measure, than do patients on the original therapy. Standard statistical methods do not handle such situations in an efficient way; the current research represents a significant improvement.