Confidence band and confidence interval of restricted mean survival time of length-biased data

The restricted mean survival time (RMST) is an important measure in clinical studies, particularly when samples are length-biased. It provides valuable insights into survival over a specific period, with the area under the survival function playing a crucial role in this evaluation. When the data are length-biased, traditional parametric and classic methods for analyzing RMST are not appropriate. To overcome this challenge, nonparametric and semi-parametric methods, such as the empirical likelihood (EL) and adjusted empirical likelihood (AEL) methods, are applicable. We introduce the EL and AEL methods for RMST with length-biased right-censoring. We have demonstrated that the limiting distribution of the empirical log-likelihood ratio follows a scaled chi-square distribution. We have also shown that the likelihood ratio has weakly converged to a mean-zero Gaussian process, thereby constructing a confidence band. Our simulation section compared the confidence intervals and bands obtained from the normal approximation (NA) method, the EL, and the AEL approaches. Our results indicate that the EL and AEL methods provide better coverage probability than the NA method. In addition, we have presented two real-world data applications using datasets from the Channing House and Stanford heart transplant patients. This demonstration serves to illustrate the effectiveness of our proposed method in a practical setting.