Statistical Theory and Methodology

Stanford University, Stanford CA

Investigators

Abstract

ABSTRACT Prop ID: DMS-050 5673 PI: Efron, Bradley and Diaconis, Persi NSF Program: STATISTICS Institution: Stanford University Title: Statistical Theory and Methodology LARGE-SCALE SIMULTANEOUS TESTING (Bradley Efron) This investigator is studying the analysis of large-scale simultaneous hypothesis testing situations, for example a microarray experiment searching for genes that behave differently in HIV positive or negative subjects. A simple methodology is being developed that requires a minimum of frequentist or Bayesian modelling assumptions, and provides for both the efficient selection of the non-null cases, and the estimation of effect sizes. In classical terminology, both size and power are assessed. This methodology depends on false discovery rate calculations, implemented via empirical Bayes techniques. A typical result might report "there are 200 of the 10,000 genes that can be clearly identified as differentially expressed between the two groups of subjects, but there are also about 800 other non-null genes that this experiment was not powerful enough to detect." Classical 20th Century statistical theory was fashioned to handle small problems, dozens or maybe hundreds of data points, with one or maybe a few unknown parameters. 21st Century scientific technology now provides massive data sets, with millions of individual measurements and thousands of parameters to consider all at once. Microarrays are the prime example, but similar problems arise from a variety of devices: proteomic chips, time of flight spectroscopy, flow cytometry, and fMRI scanners. The goal of this research is an efficient, computationally efficient methodology for analyzing massive simultaneous testing problems, without the need for extensive modelling assumptions. MONTE CARLO METHODS IN PROBABILITY AND STATISTICS (Persi Diaconis) The main focus of this investigation is on rates of convergence of Monte Carlo and Markov chain algorithms for statistical computation. One aspect is phase transitions ( cut-off phenomena), extending Diaconis' recent solution of the Peres conjecture (joint with Saloff-Coste). The work includes creating new algorithms using computational tools such as Grobner bases and combinatorial characterizations such as Tuttes f-factors. This also contributes to Bayesian methodology studying prior distributions for Markov chains, non-parametrics and computational tools. A final focus is the development of group theoretic character theories in non-standard situations such as unipotent groups and Hecke algebras. Probability models underlie many areas of modern scientific study, but they raise puzzling and important questions concerning the connection of the model with real world phenomena. Diaconis studies foundational topics such as `what does it mean to say coin flips are random'? This recently led to the discovery (joint with Susan Holmes and Richard Montgomery) that in fact, natural human coin flips show a small but significant bias (about 51% come up the same as the previous flip). Careful looks at the assumptions, justification and validity of `randomness' apply to widely-used simulation methods (how long should an algorithm be run until its job is done?). Weather forcasting and air pollution are just two of the areas that require the dependability of such simulations.