A note on familywise error rate for a primary and secondary endpoint

Hung et al. (2007) considered the problem of controlling the type I error rate for a primary and secondary endpoint in a clinical trial using a gatekeeping approach in which the secondary endpoint is tested only if the primary endpoint crosses its monitoring boundary. They considered a two-look trial and showed by simulation that the naive method of testing the secondary endpoint at full level α at the time the primary endpoint reaches statistical significance does not control the familywise error rate at level α. Tamhane et al. (2010) derived analytic expressions for familywise error rate and power and confirmed the inflated error rate of the naive approach. Nonetheless, many people mistakenly believe that the closure principle can be used to prove that the naive procedure controls the familywise error rate. The purpose of this note is to explain in greater detail why there is a problem with the naive approach and show that the degree of alpha inflation can be as high as that of unadjusted monitoring of a single endpoint.