Conditional rejection probabilities of student's t-test and design adaptations

For clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t-test and show that it can be non monotone in the interim mean of the data and non monotone in the non-centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t-test and give a numerical example. Additionally, the conditional rejection probability given the interim t-statistic is investigated. It does not depend on the unknown a and can be used in stochastic curtailment procedures.