Rationale for the update algorithm of the graphical approach to sequentially rejective multiple test procedures.

The graphical approach by Bretz et al. is a convenient tool to construct, visualize and perform multiple test procedures that are tailored to structured families of hypotheses while controlling the familywise error rate. A critical step is to update the transition weights following a pre-specified algorithm. In their original publication, however, the authors did not provide a detailed rationale for the update formula. This paper closes the gap and provides three alternative arguments for the update of the transition weights of the graphical approach. It is a legacy of the first author, based on an unpublished technical report from 2014, and after his untimely death reconstructed by the other two authors as a tribute to Willi Maurer's collaboration with Andy Grieve and contributions to biostatistics over many years.