Improving on exact tests by approximate conditioning

The often conservative nature, for discrete data, of so-called exact tests seems usually the result of unnecessarily precise conditioning. We consider avoiding this by conditioning only approximately on the sufficient statistics for nuisance parameters. Modest relaxation of conditioning results in small loss in terms of the rationale for conditional inference, but can greatly reduce the difficulties caused by discreteness. Exact calculation of p-values based on approximate conditioning is possible, but unattractive both in terms of the amount of calculation involved and in requiring explicit specification of the extent to which conditioning is to be relaxed. It is shown that there is a highly accurate, easily computed and very natural asymptotic approximation that avoids these difficulties.