In applications of clinical trials, the hypotheses to be tested often exhibit a hierarchical structure, and are usually hierarchically ordered based on their importance, clinical relevance, or dose concentration, etc. Thus, they are tested in a pre-defined fixed sequence. In some more complex cases, the hypotheses to be tested are hierarchically grouped into several families, and thus the families are tested in a sequential order. Although such problems of structured multiple testing have received much attention and several popular FWER controlling procedures, such as conventional fixed sequence procedure, fallback procedure and gatekeeping procedure have been introduced, not much progress has been made yet advancing their theory and methods.
This research contributes to the development of theory and methods of multiple testing problems under structured hypotheses in the following aspects. First, a class of generalized fixed sequence procedures is introduced for testing a single family of hypotheses, which allow each hypothesis to be tested even though some early hypotheses in the sequence are not rejected. A condition of a given generalized fixed sequence procedure can strongly control the FWER under arbitrary dependence is proposed. Based on the condition, three special generalized fixed sequence procedures controlling the FWER are developed. Through extensive simulation studies, the advantages of proposed procedures are shown over the existing FWER controlling procedures in terms of the FWER control and power. When the pairwise joint distributions of the true null p-values are known, these procedures can be improved further by incorporating such pairwise correlation information while maintaining the control of the FWER.
Secondly, a family-based graphical approach is proposed to construct general stepwise multilevel family-based procedures for testing multiple hierarchically ordered families of hypotheses. The resulting procedures can be elegantly represented by directed acyclic graphs. Though some examples, it is shown that the proposed family- based graphical approach can present the testing strategy simpler and more efficiently than the existing hypothesis - based graphical approach.
Thirdly, a Bonferroni-based gatekeeping procedure with retesting option for testing hierarchically ordered families of hypotheses is proposed. By this procedure, each family of hypotheses is repeatedly tested using Bonferroni procedure with updated local critical values. It is proved that the proposed procedure can strongly control the global FWER under arbitrary dependence.
Lastly, a multilevel partial hierarchical procedure in dealing with the problem of testing multiple families of hypotheses with partially ordered hierarchical structure is introduced. One hypothesis in current family is of interest only if some hypotheses in the previous families satisfy certain conditions. It is shown that the proposed procedure can control the FWER strongly at level α under the assumption that p-values of hypotheses from different families are independent.