Organizer:

Tony Babinec, tony@spss.com

Speakers

10:15 a.m.
Applications of Multivariate Saddleppoint Approximations in Conditional Inference
John Kolassa, University of Rochester
with Bo Yang, University of Rochester
Keywords: Conditional Inference; Saddlepoint Approximations

This talk will illustrate the use of multivariate saddlepoint approximations in conditional inference. Methods used include modifications of bivariate distribution function approximations. These methods will be applied to discrete generalized linear models, including logistic regression.

10:45 a.m.
Sinc Quadrature Rules for Computing Tail Probabilities, with Applications to Conditional Inference
Robert Strawderman, University of Michigan
Keywords: Analytic function; Characteristic function; Fourier methods; Mapped trapezoidal rule

The calculation of tail probabilities $S(x) = P\{X > x\}$ for a given random variable $X$ can be accomplished in a variety of ways. One particularly appealing class of methods relates to exploiting the connection between $S$ and its corresponding characteristic function via Fourier methods. Many recent studies, particularly those appearing in the statistical literature, have focused upon saddlepoint approximations. Strawderman and Wells (1998 {\em JASA}) and the ensuing discussion provide one perspective on the attractivness of this approach in the exact conditional inference setting. Numerical integration methods have been studied to a much lesser extent, though interest in the associated numerical integration problem has been growing. The basic problem is beset with difficulties because of the potentially highly oscillatory nature of the Fourier sine and cosine transforms that need to be evaluated over the positive real line. Sinc function quadrature rules, which are essentially mapped trapezoidal rules, have been shown to be especially effective in settings involving analytic integrands for which singularities occur at the endpoints of the interval of integration, and often lead to very simple quadrature rules that are exponentially accurate in the number of quadrature points. We show how these methods apply to the problem of numerically evaluating the tail probability inversion integral. We then illustrate these methods with a few examples from the exact conditional inference setting.

11:15 a.m.
Smart Monte Carlo Methods for Conditional Logistic Regression
Cyrus Mehta, Cytel Software Corporation and Harvard School of Public Health
with Nitin Patel, Pralay Senchaudhuri

Exact inference for the logistic regression model is based on generating the permutation distribution of the sufficient statistics for the regression parameters of interest conditional on the sufficient statistics for the remaining (nuisance) parameters. Despite the availability of fast numerical algorithms for the exact computations, there are numerous instances where a data set is too large to be analyzed by the exact methods, yet too sparse or unbalanced for the maximum likelihood approach to be reliable. What is needed is a Monte Carlo alternative to the exact conditional approach which can bridge the gap between the exact and asymptotic methods of inference. The problem is technically hard because conventional Monte Carlo methods lead to massive rejection of samples that do not satisfy the linear integer constraints of the conditional distribution. We propose a network sampling approach to the Monte Carlo problem that eliminates rejection entirely. The method involves storing all the linear integer constraints in memory as a network. If the memory requirements for storing the network are too severe, one can relax some of the linear integer constraints at the expense of introducing rejection into the sampling scheme. The advantages of this approach over alternative saddlepoint and Markov Chain Monte Carlo approaches are also discussed.

11:45 a.m.
George Casella (discussant), Cornell University