An evaluation of saddlepoint approximations in the generalized linear model

Abstract

Higher order asymptotic methods based on the saddlepoint approximation to a density provide fast and accurate approximate inference in a variety of situations. The double saddlepoint approximation to the exact conditional density and distribution function in particular is useful in generalized linear model problems which are common in biostatistical applications. These methods give approximations to exact inference that are in general accurate and computationally much less intense than the exact methods. In this work we fully review the theory of the saddlepoint approximation from first principles and then develop and evaluate the approximation in two common discrete problems. Log odds ratio regression in stratified case control studies is an example of a problem where the number of nuisance parameters can increase with sample size, and conditioning on sufficient statistics for the nuisance parameters is essential for asymptotically unbiased inference. Investigation of asymptotic issues and numerical simulation demonstrate that for this model, the saddlepoint approximation accurately approximates exact conditional inference both for point estimation and confidence intervals. For the test for trend in a sequence of binomial random variables, numerical evidence points to the saddlepoint method being an effective approximation to conditional inference in almost all situations. The third part of the work involves applying saddlepoint inference to tests of higher order interaction in logistic regression. We implement the saddlepoint approximation for this problem and through simulation demonstrate that the properties of the saddlepoint approximation are significantly better than those of the commonly used unconditional methods. Finally, we examine and extend the theory for saddlepoint inference in the generalized linear model.