This manuscript considers inference on a single parameter in a multivariate canonical exponential family, where the effect of nuisance parameters on the p-value is mitigated by conditioning on the event that the sufficient statistics associated with the nuisance parameters lie in a neighborhood about the observed value. This manuscript has three aims. First, we provide a method for approximating p-values using approximate conditioning that is more accurate than that presented by Pierce and Peters (Biometrika 86(1999) 265-277), at the price of greater computational difficulty. Second, we examine the sensitivity of approximate conditioning methods to the values of the nuisance parameters. Third, we describe a method for presenting a valid approximate-conditioning observed significance level accounting for this dependence on the nuisance parameters.
