Extending Sample Size Calculation for the Cox Proportional Hazards Model

In clinical trials with time-to-event endpoints, it is not uncommon to see a significant proportion of patients being cured (or long-term survivors), such as trials for the non-Hodgkin\u27s lymphoma disease. The popularly used sample size formula derived under the Cox proportional hazards (PH) model may not be proper to design a survival trial with a cure fraction, since the PH model assumption may be violated. To account for a cure fraction, the PH cure model is widely used in practice, where a PH model is used for survival times of uncured patients and a logistic distribution is used for the probability of patients being cured. As the first project of this dissertation, we develop a sample size formula based on the PH cure model by investigating the asymptotic distributions of the standard weighted log-rank statistics under the null and local alternative hypotheses. The derived sample size formula under the PH cure model is more flexible since it can be used to test the differences in the short-term survival and/or long-term survival. On the other hand, PH model with time-dependent variables (referred as extended PH model) has been widely used in medical and clinical studies. However, there are no methods available to determine the sample size when the effect of treatment or exposure can change with time. Therefore as the second project of this dissertation, we develop a sample size calculation method based on the extended PH model by investigating the asymptotic distributions of the standard weighted log-rank statistics under the null and local alternative hypotheses. The derived sample size formula is an extension of Schoenfeld\u27s sample size formula for the standard Cox PH model. In both projects, the impacts of accrual methods and durations of accrual and follow-up periods on sample size are investigated as numerical examples. The performance of the proposed formulae is evaluated by extensive simulation studies, and examples using data from clinical trial and cohort studies are given to illustrate the applications of our proposed methods