Graphical approaches for multiple comparison procedures using weighted Bonferroni, Simes, or parametric tests

Abstract

The confirmatory analysis of pre-specified multiple hypotheses has become common in pivotal clinical trials. In the recent past multiple test procedures have been developed that reflect the relative importance of different study objectives, such as fixed sequence, fallback, and gatekeeping procedures. In addition, graphical approaches have been proposed that facilitate the visualization and communication of Bonferroni-based closed test procedures for common multiple test problems, such as comparing several treatments with a control, assessing the benefit of a new drug for more than one endpoint, combined non-inferiority and superiority testing, or testing a treatment at different dose levels in an overall and a subpopulation. In this paper, we focus on extended graphical approaches by dissociating the underlying weighting strategy from the employed test procedure. This allows one to first derive suitable weighting strategies that reflect the given study objectives and subsequently apply appropriate test procedures, such as weighted Bonferroni tests, weighted parametric tests accounting for the correlation between the test statistics, or weighted Simes tests. We illustrate the extended graphical approaches with several examples. In addition, we describe briefly the gMCP package in R, which implements some of the methods described in this paper.

Step-Down Parametric Procedures for Testing Correlated Endpoints in a Group-Sequential Trial

“…For m ≥ 3, the monotonicity condition (Bretz et al 2011) does not hold for positive correlations, although it holds for the cases of independent or negative correlations.…”

“…In situations where the correlations between endpoints are known for some but not for all, Bretz et al (2011) provided a solution, which requires using a submatrix of those fully known correlations for solving Equation (3). It can be shown that their solution preserves consonancy in a consonant parametric procedure.…”

“…For an example, consider the testing of three correlated endpoints H 1 , H 2 , and H 3 , with ρ 12 and ρ 23 known while ρ 13 unknown. The solution of Bretz et al (2011) uses either ρ 12 or ρ 23 , presumably the larger one, but not both. An alternative solution is to use the partially parametric procedure of Seneta and Chen (2005), which uses all known pairwise correlations in boundary calculations, thus, can be more powerful.…”

“…where w i (J) > 0 is the weight for H i and ξ J ≥ 1 is the solution to the following equation (Bretz et al 2011). Note that the probability is evaluated under the global (intersection) null hypothesis:…”

Analyzing noninferiority trials: it is time for advantage deficit assessment – an observational study of published noninferiority trials

“…[7][8][9][10] Evaluation of safety along with efficacy in superiority trials has been around for a long time, and solid methodologies have been developed for simultaneous comparisons and multiple endpoint testing. [11][12][13][14][15][16][17][18] Adapting these to NI trials, several approaches for multiple comparison procedures allowing simultaneous testing of noninferiority and superiority of the multiple endpoints in NI trials are available. These include a mixture of graphical approaches, alpha allocation, and hierarchical gatekeeping, eg, Bretz et al 12,13 Bauer et al, 11 Bristol, 18 Nishikawa et al 17 and Guilbaud.…”