A common problem that is encountered in medical applications is the overall homogeneity of survival distributions when two survival curves cross each other. A survey demonstrated that under this condition, which was an obvious violation of the assumption of proportional hazard rates, the log-rank test was still used in 70% of studies. Several statistical methods have been proposed to solve this problem. However, in many applications, it is difficult to specify the types of survival differences and choose an appropriate method prior to analysis. Thus, we conducted an extensive series of Monte Carlo simulations to investigate the power and type I error rate of these procedures under various patterns of crossing survival curves with different censoring rates and distribution parameters. Our objective was to evaluate the strengths and weaknesses of tests in different situations and for various censoring rates and to recommend an appropriate test that will not fail for a wide range of applications. Simulation studies demonstrated that adaptive Neyman’s smooth tests and the two-stage procedure offer higher power and greater stability than other methods when the survival distributions cross at early, middle or late times. Even for proportional hazards, both methods maintain acceptable power compared with the log-rank test. In terms of the type I error rate, Renyi and Cramér—von Mises tests are relatively conservative, whereas the statistics of the Lin-Xu test exhibit apparent inflation as the censoring rate increases. Other tests produce results close to the nominal 0.05 level. In conclusion, adaptive Neyman’s smooth tests and the two-stage procedure are found to be the most stable and feasible approaches for a variety of situations and censoring rates. Therefore, they are applicable to a wider spectrum of alternatives compared with other tests.

In clinical and epidemiologic studies, hazard ratios are often applied to compare treatment effects between 2 groups for survival data. For competing-risks data, the corresponding quantities of interest are cause-specific hazard ratios and subdistribution hazard ratios. However, they both have some limitations related to model assumptions and clinical interpretation. Therefore, we recommend restricted mean time lost (RMTL) as an alternative measure that is easy to interpret in a competing-risks framework. Based on the difference in RMTL (RMTLd), we propose a new estimator, hypothetical test, and sample-size formula. Simulation results show that estimation of the RMTLd is accurate and that the RMTLd test has robust statistical performance (both type I error and statistical power). The results of 3 example analyses also verify the performance of the RMTLd test. From the perspectives of clinical interpretation, application conditions, and statistical performance, we recommend that the RMTLd be reported along with the hazard ratio in analyses of competing-risks data and that the RMTLd even be regarded as the primary outcome when the proportional hazards assumption fails.

Objectives: This study explored the prognostic factors and developed a prediction model for Chinese-American (CA) cervical cancer (CC) patients. We compared two alternative models (the restricted mean survival time (RMST) model and the proportional baselines landmark supermodel (PBLS model, producing dynamic prediction)) versus the Cox proportional hazards model in the context of time-varying effects. Setting and data sources: A total of 713 CA women with CC and available covariates (age at diagnosis, International Federation of Gynecology and Obstetrics (FIGO) stage, lymph node metastasis and radiation) from the Surveillance, Epidemiology and End Results database were included. Design: We applied the Cox proportional hazards model to analyse the all-cause mortality with the proportional hazards assumption. Additionally, we applied two alternative models to analyse covariates with time-varying effects. The performances of the models were compared using the C-index for discrimination and the shrinkage slope for calibration. Results: Older patients had a worse survival rate than younger patients. Advanced FIGO stage patients showed a relatively poor survival rate and low life expectancy. Lymph node metastasis was an unfavourable prognostic factor in our models. Age at diagnosis, FIGO stage and lymph node metastasis represented time-varying effects from the PBLS model. Additionally, radiation showed no impact on survival in any model. Dynamic prediction presented a better performance for 5-year dynamic death rates than did the Cox proportional hazards model. Conclusions: With the time-varying effects, the RMST model was suggested to explore diagnosis factors, and the PBLS model was recommended to predict a patient’s w -year dynamic death rate.

Background: Under competing risks, the commonly used sub-distribution hazard ratio (SHR) is not easy to interpret clinically and is valid only under the proportional sub-distribution hazard (SDH) assumption. This paper introduces an alternative statistical measure: the restricted mean time lost (RMTL). Methods: First, the definition and estimation methods of the measures are introduced. Second, based on the differences in RMTLs, a basic difference test (Diff) and a supremum difference test (sDiff) are constructed. Then, the corresponding sample size estimation method is proposed. The statistical properties of the methods and the estimated sample size are evaluated using Monte Carlo simulations, and these methods are also applied to two real examples. Results: The simulation results show that sDiff performs well and has relatively high test efficiency in most situations. Regarding sample size calculation, sDiff exhibits good performance in various situations. The methods are illustrated using two examples. Conclusions: RMTL can meaningfully summarize treatment effects for clinical decision making, which can then be reported with the SDH ratio for competing risks data. The proposed sDiff test and the two calculated sample size formulas have wide applicability and can be considered in real data analysis and trial design.

In clinical or epidemiological follow‐up studies, methods based on time scale indicators such as the restricted mean survival time (RMST) have been developed to some extent. Compared with traditional hazard rate indicator system methods, the RMST is easier to interpret and does not require the proportional hazard assumption. To date, regression models based on the RMST are indirect or direct models of the RMST and baseline covariates. However, time‐dependent covariates are becoming increasingly common in follow‐up studies. Based on the inverse probability of censoring weighting (IPCW) method, we developed a regression model of the RMST and time‐dependent covariates. Through Monte Carlo simulation, we verified the estimation performance of the regression parameters of the proposed model. Compared with the time‐dependent Cox model and the fixed (baseline) covariate RMST model, the time‐dependent RMST model has a better prediction ability. Finally, an example of heart transplantation was used to verify the above conclusions.