gms | German Medical Science

Introduction: In survival analysis, events sometimes only start to occur after a certain delay since entry time and this delay period might vary for different treatments or groups. While parametric delay models like the three-parameter Weibull distribution might adequately describe this process the estimation of delay via standard maximum likelihood is severely biased in small samples. Different estimation methods have been proposed that are less biased, for instance the maximum product of spacings estimation (MPSE) method [1] being one of the first. Different modifications to the maximum likelihood estimation (MLE) have also been proposed [2], [3]. However, there has not been much debate about parametric significance tests in this situation. Building on the different estimation methods we assess various parametric significance tests for delay in a two group setting.

Methods: Based on the different estimation methods we have implemented likelihood ratio tests and bootstrap tests for delay. We consider the log-rank test as benchmark as this non-parametric test is frequently used in practice. A Monte-Carlo study sheds light on the finite-sample properties of these test methods, using the delayed exponential distribution and the delayed Weibull distribution as foundational data models for our simulations. To assess the effect of censoring we have applied random right-censoring to an increasing degree in the simulations.

Results: The Monte-Carlo study shows that the bootstrap test based on MPSE and the likelihood ratio test based on corrected MLE are more powerful than the log-rank test in many scenarios. The corrected MLE based test also handles right-censored observations well. To demonstrate the utility of the proposed parametric tests in practice we reanalyze data from a published brain cancer experiment [4]. We find statistically significant differences in delay that the log-rank test does not uncover. A freely available implementation of all proposed methods in the R-programming environment is provided at https://CRAN.R-project.org/package=incubate..

Discussion: We have shown that parametric significance tests have more power than the standard non-parametric log-rank test in many situations. These parametric tests allow to focus on a specific aspect of the survival and they have no problem when the survival curves of both groups cross, a situation where the hazard is not proportional and hence the power of the log-rank test is particularly low. However, parametric tests are sensitive to their model assumptions. As a remedy, goodness-of-fit tests allow to assess if the model assumptions underlying the parametric test are grossly violated.

Conclusion: We believe that for time-to-event data with delay the presented parametric tests deserve a place besides the standard non-parametric log-rank test as it is often more powerful to uncover specific differences in survival whenever the assumed parametric model is appropriate. The R-package (https://cran.r-project.org/web/packages/incubate/index.html) is readily available for this purpose.

The authors declare that they have no competing interests.

The authors declare that an ethics committee vote is not required.