Empirical Comparison of the Breslow Estimator and the Kalbfleisch Prentice Estimator for Survival Functions

When analyzing time-to-event data in a non-parametric setting without considering covariates, the Kaplan-Meier estimator is widely used to estimate the survival function. When considering covariates, the Cox proportional hazards model is widely used to account for covariates effects. In this setting, for the baseline survival function, the most commonly used approach is the Breslow method, which estimates the baseline survival function as an exponential function of the cumulative baseline hazard function. However, an unnatural and undesirable feature of the Breslow estimator is that, its estimated survival probability will never reaches zero even if the last observation is an event. In this article, we consider an less commonly used alternative, the Kalbfleisch Prentice estimator for the baseline survival function. It is the counterpart of the Kaplan-Meier estimator in a setting with covariates, and thus similarly as the Kaplan Meier estimator, it will reach zero if the last observation is an event. To evaluate the usefulness of the Kalbfleisch Prentice estimator and its relative performance comparing with the Breslow estimator, we conduct simulation studies across a range of conditions by varying the true survival time distribution, sample size, censoring rate and covariate values. We compare the performance of the two estimators regarding bias, mean squared error and relative mean squared error. In most situations in our study, the Kalbfleisch Prentice estimator results in less bias and smaller mean squared error than the Breslow estimator. Their differences are especially clear at the tail of the distribution. The implications of such differences in applications are discussed. We advocate the use of Kalbfleisch Prentice estimator in practice, and further research on its properties.