• Censored data;
  • Cure model;
  • Improper distribution;
  • Long-term survivors;
  • Proportional hazards model;
  • Semiparametric model;
  • Survival data;
  • Two-sample test

Summary. In the two-sample comparison of survival times with long-term survivors, the overall difference between the two distributions reflects differences occurring in early follow-up for susceptible subjects and in long-term follow-up for nonsusceptible subjects. In this setting, we propose statistics for testing (i) no overall, (ii) no short-term, and (iii) no long-term difference between the two distributions to be compared. The statistics are derived as follows. A semiparametric model is defined that characterizes a short-term effect and a long-term effect. By approximating this model about no difference in early survival, a time-dependent proportional hazards model is obtained. The statistics are obtained from this working model. The asymptotic distributions of the statistics for testing no overall or no short-term effects are ascertained, while that of the statistic for testing no long-term effect is valid only when the short-term effect is small. Simulation studies investigate the power properties of the proposed tests for different configurations. The results show the interesting behavior of the proposed tests for situations where a short-term effect is expected. An example investigating the impact of progesterone receptors status on local tumor relapse for patients with early breast cancer illustrates the use of the proposed tests.