P. Bauer, W.Brannath and M.Posch

(Department of Medical Statistics, University of Vienna, Austria)

We consider the situation where k treatments and a (zero) control are compared with respect to efficacy and safety. For efficacy the null hypotheses for the many one comparisons in terms of the paramenter of interest are defined as HoiE: i £ 0, i = 1, ..., k. Here 0 and 1, ..., k denote the parameter under the control (0) and the k treatments, respectively. For safety the shifted one sided null hypotheses HoiS: q i £ q 0 + d are investigated, where d is the prefixed safety margin for the corresponding parameter of interest.

A treatment i is considered to be effective if HoiE is rejected and is considered to be safe if HoiS is rejected. If both HoiE and HoiS are rejected it is considered to be effective and safe.

By considering only the k sub-families (HoiE, HoiE È HoiS), i = 1, ..., k, the multiple levels applied within the sub-families can be adjusted in a stepwise way. Within the sub-family a hierarchical procedure with a fixed sequence of testing is used. This multiple level- procedure can also be applied to the problem of simultaneously establishing superiority of a treatment to a (zero) control and D -equivalence to an active control and is more powerful than the procedure by Bauer et al. (1998). If order restrictions are assumed to hold among the parameters of interest a split strategy by applying adjusted multiple levels within the two sub-families (HoiE, i = 1, ..., k) and (HoiS, i = 1, ..., k) can be applied. If all treatments are found to be effective or all treatments are found to be safe this leads to an improvement of the Bonferroni-splitting.

A possible generalization to continuous families with the corresponding confidence intervals is given.

Bauer, P., Röhmel, J., Maurer, W. and Hothorn, L. (1998) Testing strategies in multi-dose experiments including active control. Statist. Med., 17, 2133-46.