A Flexible Choice of Critical Constants for the Improved Hybrid Hochberg-Hommel Procedure

In Gou et al. (Biometrika 101(4), 899–911 2014), we proposed an improved hybrid Hochberg-Hommel (HH₀) step-up multiple test procedure. The procedure used a fixed, specified set of critical constants. In this follow-up note, we show that an entire class of critical constants can be derived by relaxing one of the assumptions in that paper. This allows a choice of more powerful critical constants. These critical constants do not control the type I error under positive dependence among the test statistics as shown by simulations. Therefore, they should be used only under independence and negative dependence when not only they control the type I error but also yield higher power. If positive dependence is expected, as is common in practice, then the HH₀ procedure should be used since it is robust with respect to type I error control under dependence.