The ultimate goal of the proposed research is the application of saddlepoint methodology to biostatistical problems. The main aim of this work is to refine and extend the Gibbs-Skovgaard algorithm of Kolassa and Tanner (1192). This method implements Markov chain Monte Carlo methods in conditional inference by using the Gibbs sampler to construct a Markov chain whose equilibrium distribution is the null conditional distribution of interest. Each step in the Markov chain constructed by the Gibbs sampler is accomplished by cycling through all of the variables whose values are to simulated, and for each of these variables sampling a new value conditional on all other variable held fixed. To date, the methodology has been applied to two-way and three-way contingency tables, and to logistic regression, with applications to cancer research. The double-saddlepoint conditional cumulative distribution, which is often difficult to sample from. This algorithm will be formulated and applied in cases including that of general hierarchical models for contingency tables. Furthermore, a variety of extensions of the double-saddlepoint approximation of Skovgaard (1987), upon which the Gibbs-Skovgaard algorithm rests, will be developed. Higher-order terms in this expansion will be calculated. When applied to exponential families, the double saddlepoint approximation requires the existence of maximum likelihood estimators for the regression parameters associated with statistics conditioned on. An extension of these methods removing this requirement will be developed. The performance of this expansion will be compared against the performance of competing approximations, including the sequential saddlepoint approximation of Fraser, Reid, and Wong (1991). The accuracy of the distribution approximation achieved by using the Markov chain resulting from the Gibbs-Skovgaard algorithm will be assessed. Computer software that can be shared with other sophisticated users will be developed. A subsidiary aim of this work is to explore the use of saddlepoint methods in performing approximate maximum likelihood estimation in generalized linear models with random effects. Saddlepoint integration methods will be used to evaluate the integral defining the unconditional likelihood.
