Multiple test procedures for identifying the minimum effective and maximum safe doses of a drug

We address the problem of determining the therapeutic window of a drug by finding its minimum effective and maximum safe doses (MINED and MAXSD), The MINED is the lowest dose that exceeds the mean efficacy of the zero dose by a specified threshold, and the MAXSD is the highest dose that does not exceed the mean toxicity of the zero dose by a specified threshold. Step-down multiple test procedures are proposed to identify the MINED and MAXSD assuming a bivariate normal model. These procedures control the type I familywise error probability of declaring any ineffective dose as effective or any unsafe dose as safe at a prespecified level alpha. A new multivariate t-distribution is introduced whose critical points are required to implement the exact normal theory procedures. Because these critical points depend on the unknown correlation coefficient between the efficacy and safety variables, the Bonferroni method is proposed as an alternative, which amounts to separately testing for efficacy and safety, each at type I familywise error rate of alpha/2. The bootstrap versions of the exact normal theory procedures provide an approximate way to jointly test for efficacy and safety without the knowledge of the correlation coefficient, as well as to relax the bivariate normality assumption. The Bonferroni and bootstrap procedures are compared in a simulation study. It is shown that significant power gains are achieved by jointly testing for both efficacy and safety using bootstrap procedures, Coded data from an arthritis drug trial are analyzed to illustrate the procedures.