  When simultaneously testing multiple hypotheses, the usual approach in the
context of confirmatory clinical trials is to control the familywise error rate
(FWER), which bounds the probability of making at least one false rejection. In
many trial settings, these hypotheses will additionally have a hierarchical
structure that reflects the relative importance and links between different
clinical objectives. The graphical approach of Bretz et al. (2009) is a
flexible and easily communicable way of controlling the FWER while respecting
complex trial objectives and multiple structured hypotheses. However, the FWER
can be a very stringent criterion that leads to procedures with low power, and
may not be appropriate in exploratory trial settings. This motivates
controlling generalised error rates, particularly when the number of hypotheses
tested is no longer small. We consider the generalised familywise error rate
(k-FWER), which is the probability of making k or more false rejections, as
well as the tail probability of the false discovery proportion (FDP), which is
the probability that the proportion of false rejections is greater than some
threshold. We also consider asymptotic control of the false discovery rate
(FDR), which is the expectation of the FDP. In this paper, we show how to
control these generalised error rates when using the graphical approach and its
extensions. We demonstrate the utility of the resulting graphical procedures on
three clinical trial case studies.
