Cancer Survival Estimates Due to Non-Uniform Loss to Follow-Up and Non-Proportional Hazards

Background: Cancer survival depends on loss to follow-up (LFU) and non-proportional hazards (non-PH). If LFU is high, survival will be over-estimated. If hazard is non-PH, rank tests will provide biased inference and Cox-model will provide biased hazard-ratio. We assessed the bias due to LFU and non-PH factor in cancer survival and provided alternate methods for unbiased inference and hazard-ratio. Materials and Methods: Kaplan-Meier survival were plotted using a realistic breast cancer (BC) data-set, with >40%, 5-year LFU and compared it using another BC data-set with <15%, 5-year LFU to assess the bias in survival due to high LFU. Age at diagnosis of the latter data set was used to illustrate the bias due to a non-PH factor. Log-rank test was employed to assess the bias in p-value and Cox-model was used to assess the bias in hazard-ratio for the non-PH factor. Schoenfeld statistic was used to test the non-PH of age. For the non-PH factor, we employed Renyi statistic for inference and time dependent Cox-model for hazard-ratio. Results: Five-year BC survival was 69% (SE: 1.1%) vs. 90% (SE: 0.7%) for data with low vs. high LFU respectively. Age (<45, 46-54 & >54 years) was a non-PH factor (p-value: 0.036). However, survival by age was significant (log-rank p-value: 0.026), but not significant using Renyi statistic (p=0.067). Hazard ratio (HR) for age using Cox-model was 1.012 (95%CI: 1.004 -1.019) and the same using time-dependent Cox-model was in the other direction (HR: 0.997; 95% CI: 0.997- 0.998). Conclusion: Over-estimated survival was observed for cancer with high LFU. Log-rank statistic and Cox-model provided biased results for non-PH factor. For data with non-PH factors, Renyi statistic and time dependent Cox-model can be used as alternate methods to obtain unbiased inference and estimates.