The log-rank test is commonly used to test the equivalence of two survival distributions under right censoring. Jung et al. (2005) proposed a modified log-rank test for noninferiority trials and its corresponding sample size calculation. In this article, we extend the use of the modified log-rank test for clinical trials with various types of nonconventional study objectives and propose its sample size calculation under general null and alternative hypotheses. The proposed formula is so flexible that we can specify any survival distributions and accrual pattern. The proposed methods are illustrated with designing real clinical trials. Through simulations, the modified log-rank test and the derived formula for sample size calculation are shown to have satisfactory small sample performance.