Approximate Monte Carlo conditional inference in exponential families

This article presents an algorithm for approximate frequentist conditional inference on two or more parameters for any regression model in the General ized Linear Model (GLIM) family. We thereby extend highly accurate inferenc e beyond the cases of logistic regression and contingency tables implimente d in commercially available software. The method makes use of the double sa ddlepoint approximations of Skovgaard (1987, Journal of Applied Probability 24, 875-887) and Jensen (1992, Biometrika 79, 693-703) to the conditional cumulative distribution function of a sufficient statistic given the remain ing sufficient statistics. This approximation is then used in conjunction w ith noniterative Monte Carlo methods to generate a sample from a distributi on that approximates the joint distribution of the sufficient statistics as sociated with the parameters of interest conditional on the observed values of the sufficient statistics associated with the nuisance parameters. This algorithm is an alternate approach to that presented by Kolassa and Tanner (1994, Journal of the American Statistical Association 89, 697-702), in wh ich a Markov chain is generated whose equilibrium distribution under certai n regularity conditions approximates the joint distribution of interest. In Kolassa and Tanner (1994), the Gibbs sampler was used in conjunction with these univariate conditional distribution function approximations. The meth od of this paper does not require the construction and simulation of a Mark ov chain, thus avoiding the need to develop regularity conditions under whi ch the algorithm converges and the need for the data analyst to check conve rgence of the particular chain. Examples involving logistic and truncated P oisson regression are presented.