The large sample coverage probability of confidence intervals in general regression models after a preliminary hypothesis test

We derive a computationally convenient formula for the large sample coverage probability of a confidence interval for a scalar parameter of interest following a preliminary hypothesis test that a specified vector parameter takes a given value in a general regression model. Previously, this large sample coverage probability could only be estimated by simulation. Our formula only requires the evaluation, by numerical integration, of either a double or triple integral, irrespective of the dimension of this specified vector parameter. We illustrate the application of this formula to a confidence interval for the log odds ratio of myocardial infarction when the exposure is recent oral contraceptive use, following a preliminary test that two specified interactions in a logistic regression model are zero. For this real-life data, we compare this large sample coverage probability with the actual coverage probability of this confidence interval, obtained by simulation.