Approximately exact inference for the common odds ratio in several 2 x 2 tables

The conditional maximum likelihood estimator (MLE) of the common odds ratio in a sequence of independent 2 x 2 tables is known to be superior to the M antel-Haenszel estimator in terms of asymptotic efficiency and has the furt her advantage that its exact distribution is known. However, a long-standin g barrier to the widespread use of this estimator has been computational in tractability; in particular, the calculation of significance levels, confid ence sets, and power based on the exact distribution requires fast and effi cient algorithms. An important class of such algorithms form the basis of S tatXact, a software package able to solve various aspects of the exact infe rence problem for a sequence of several 2 x 2 tables in real time. We provi de an alternative methodology by developing several useful saddlepoint appr oximations to the exact distribution of the conditional RILE. The approxima tions are derived from an interesting representation for hypergeometric ran dom variables as a sum of independent Bernoulli random variables and provid e fast, accurate calculations of power functions, p values, and confidence sets. The primary computational burden is in determining the roots of a cer tain polynomial, which need be done numerically only once for each table. C onsequently, the required computational effort is typically minimal; for ex ample, all of the examples herein were done using code written by the autho rs entirely in S-PLUS.