Counting tables using the double-saddlepoint approximation

A double-saddlepoint approximation is proposed for the number of contingency tables with counts satisfying certain linear constraints. Computation of the approximation involves fitting a generalized linear model for geometric responses which can be accomplished almost instantaneously using the iterated weighted least squares algorithm. The approximation is far superior to other analytical approximations that have been proposed, and is shown to be highly accurate in a range of examples, including some for which analytical approximations were previously unavailable. A similar approximation is proposed for tables consisting of only zeros and ones based on a logistic regression model. A higher order adjustment to the basic double saddlepoint further improves the accuracy of the approximation in almost all cases. Computer code for implementing methods described in the article is provided as supplemental material.