(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 H_oi^E: µ_i £ µ₀, i = 1, ..., k. Here µ₀ and µ₁, ..., µ_k denote the parameter under the control (µ₀) and the k treatments, respectively. For safety the shifted one sided null hypotheses H_oi^S: 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 H_oi^E is rejected and is considered to be safe if H_oi^S is rejected. If both H_oi^E and H_oi^S are rejected it is considered to be effective and safe.

By considering only the k sub-families (H_oi^E, H_oi^E È H_oi^S), 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 (H_oi^E, i = 1, ..., k) and (H_oi^S, 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.