THE COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN 2r FACTORIAL EXPERIMENTS AFTER PRELIMINARY HYPOTHESIS TESTING

We consider a 2^r factorial experiment with at least two replicates. Our aim is to find a confidence interval for θ, a specified linear combination of the regression parameters (for the model written as a regression, with factor levels coded as ?1 and 1). We suppose that preliminary hypothesis tests are carried out sequentially, beginning with the rth‐order interaction. After these preliminary hypothesis tests, a confidence interval for θ with nominal coverage 1 ?α is constructed under the assumption that the selected model had been given to us a priori. We describe a new efficient Monte Carlo method, which employs conditioning for variance reduction, for estimating the minimum coverage probability of the resulting confidence interval. The application of this method is demonstrated in the context of a 2³ factorial experiment with two replicates and a particular contrast θ of interest. The preliminary hypothesis tests consist of the following two‐step procedure. We first test the null hypothesis that the third‐order interaction is zero against the alternative hypothesis that it is non‐zero. If this null hypothesis is accepted, we assume that this interaction is zero and proceed to the second step; otherwise, we stop. In the second step, for each of the second‐order interactions we test the null hypothesis that the interaction is zero against the alternative hypothesis that it is non‐zero. If this null hypothesis is accepted, we assume that this interaction is zero. The resulting confidence interval, with nominal coverage probability 0.95, has a minimum coverage probability that is, to a good approximation, 0.464. This shows that this confidence interval is completely inadequate.