Simulating survival data when one subgroup lacks information

In this paper, we aim to show the process of simulating survival data when the distribution of the overall population and one subgroup (called "positive subgroup") as well as the proportion of the subgroup is known, while the distribution of the other subgroup (called "negative subgroup") is unknown. We propose a combination method which generates survival data of the positive subgroup and negative subgroup, respectively, and survival data of the overall population are the combination of the two subgroups. The parameters of the overall population and the positive subgroup need to satisfy certain constraints, otherwise the parameters may lead to contradictions. From simulation, we show that our proposed combination method can reflect the correlation between the test statistics of overall population and positive subgroup, which makes the simulated data more realistic and the results of simulation more reliable. Moreover, for a multiplicity control in trial design, the combination method can help to determine the $\alpha$ splitting strategy between primary endpoints, and is helpful in designs of clinical trials as shown in three applications.