## 1 Introduction

Multiple test procedures are often used in the analysis of clinical trials addressing multiple objectives, such as comparing several treatments with a control and assessing the benefit of a new drug for more than one endpoint. Several multiple test procedures have been developed in the recent past that allow one to map the relative importance of the different study objectives as well as their relation onto an appropriately tailored multiple test procedure.

A common strategy to reduce the degree of multiplicity is to group the hypotheses into primary and secondary objectives (O'Neill, 1997). Test procedures accounting for the inherent logical relationships include fixed sequence tests (Maurer et al., 1995; Westfall and Krishen, 2001), gatekeeping procedures (Bauer et al., 1998; Westfall and Krishen, 2001; Dmitrienko et al., 2003) and fallback procedures (Wiens, 2003; Huque and Alosh, 2008). Li and Mehrotra (2008) introduced a more general approach for adapting the significance level to test secondary hypotheses based on the finding for the primary hypotheses. Alosh and Huque (2009) introduced the notion of consistency when testing for an effect in the overall population and in a specific subgroup. The authors extended this consistency concept to other situations (Alosh and Huque, 2010), including how to address multiplicity issues of a composite endpoint and its components in clinical trials (Huque et al., 2011). Hung and Wang (2009, 2010) considered some controversial multiple test problems, with emphasis on regulatory applications, and pointed out illogical problems that may arise with recently developed multiple test procedures.

In this paper, we focus on graphical approaches which have been introduced independently by Bretz et al. (2009) and Burman et al. (2009). The key idea is to express the resulting multiple test procedures by directed, weighted graphs, where each node corresponds to an elementary hypothesis, together with a simple algorithm to generate such graphs while sequentially testing the individual hypotheses. Using graphical approaches, one can explore different test strategies together with the clinical team and thus tailor the multiple test procedure to the given study objectives. So far, the description of these graphical approaches has focused on Bonferroni-based test procedures. In this paper, we investigate extensions of the original ideas. In particular, we discuss in Section 2 how a separation between the weighting strategy and the test procedure facilitates the application of a graphical approach beyond Bonferroni-based test procedures. In Section 3, we illustrate these ideas with different test procedures. We start with a brief review of Bonferroni-based test procedures and subsequently describe parametric graphical approaches that account for the correlation between the test statistics as well as graphical approaches using the Simes test. In Section 4, we describe the gMCP package in R which implements some of the methods discussed in this paper and illustrate it with a clinical trial example using a truncated Holm procedure. Concluding remarks are given in Section 5.