A Note on the Use of Confidence Limits following Rejection of a Null Hypothesis

The equivalence of some tests of hypothesis and confidence limits is well known.When, however, the confidence limits are computed only after rejection of a null hypothesis, the usual unconditional confidence limits are no longer valid.This refers to a strict two-stage inference procedure: first test the hypothesis of interest and if the test results in a rejection decision, then proceed with estimating the relevant parameter.Under such a situation, confidence limits should be computed conditionally on the specified outcome of the test under which estimation proceeds.Conditional confidence sets will be longer than unconditional confidence sets and may even contain values of the parameter previously rejected by the test of hypothesis.Conditional confidence limits for the mean of a normal population with known variance are used to illustrate these results.In many applications, these results indicate that conditional estimation is probably not good practice.