The Geometric Generalized Birnbaum–Saunders model with long-Term Survivors

Background: Survival rates are important to show the progress of the disease and the effect of treatments. The estimation of survival probabilities especially in presence of highly censored data is challenging. In this study, Fuzzy Product Limit Estimator (FPLE) is introduced to mitigate the challenge. Methods: In a longitudinal study, data of 173 CRC patients were analyzed. To estimate survival probabilities, mean and median survival time, Fuzzy Product Limit Estimator (FPLE), a data-driven method, was applied to the data. It provides a smooth survival probability curve and the continuation of the survival curve is not a concern in the case while the largest observed time is censored. Results: One-year survival rate for CRC patients was estimated to be 83% using FPLE and KM methods. The five-year survival rate was estimated to be 37% and 52% by the FPLE and KM methods, respectively. The largest observed time in data (71.96 months) was censored, so the survival rate after 71.96 months was not estimable by the KM method. But 10-year and 20-year survival rates were estimated by FPLE as 0.21 and 0.09. The mean (median) survival time was estimated 45.97 (65) and 82.69 (41.70) months by KM and FPLE methods, respectively. Conclusion: In presence of highly censored survival data, the FPLE method provides acceptable estimates of CRC patients' survival rate. Also, the continuation of the survival curve was estimated after the largest observed time. The smaller estimates by the FPLE at 5-year could be considered as warning that the actual survival rate is lower than that reported by the KM method.