For generalized linear models, including those for survival analysis, improved asymptotic methods based on saddlepoint and related methods will be developed and evaluated. These improve- ments are a natural extension of the asymptotic methods ordinari- ly used for generalized linear models and should come into routine use to verify or to improve on application of standard methods. Aside from practical import, these developments provide insight into the basic nature of asymptotic methods particularly in regard to the effects of nuisance parameters. Closely related Laplace approximations will also be developed for likelihood calculations in generalized mixed linear models involving additional random effects in linear predictors. Statistical regression models relate a response variable to a set of explanatory variables and factors or attributes. A major development in statistics during the past 15 years has been the expansion of classical regression modeling to a much broader field of applications often involving more complicated dependen- cies on the explanatory variables and factors. The aim of this project is to continue the development of this broader theory for generalized regression modeling, emphasizing two directions: first, improvement on necessary approximations in the analysis calculations and second, allowance for still more general dependencies reflected in the variability of the data.
