Statistical methods and models in the analysis of time to event data

摘要

Time to event data are common in medical research, where outcomes are the time until the occurrence of an event of interest that are subject to censoring. The main goal for the analysis of time to event data is to investigate the effects of covariates on the hazard for an event of interest and estimate the survival probability. The article by Zhou ( ) provides a nice overview on building a clinical prediction model on time to event data and assessing the prediction accuracy of a clinical prediction model, describing the model construction and model validation using R functions ( ). The article introduces the Cox proportional hazards model ( ) as a clinical prediction model for time to event data, which is the most widely used statistical model in studies of time to event data to estimate the effects of covariates on the hazard for an event of interest, and describes visualizing the survival probability for a patient with specific values of covariates using nomogram with R. The article also introduces R code to compute the concordance index (i.e., C-index) and draw a calibration plot to evaluate the discriminatory and calibration accuracy of a Cox regression model. However, the limitation of the article is that the authors did not describe about checking the proportional hazards assumption. In practice, the proportional hazards assumption may not be plausible for some applications. Thus, it is necessary to assess the proportionality assumption since a violation of the assumption may affect statistical inference. The most commonly used approaches for checking the assumption are to use a test based on the Schoenfeld residuals ( ), and draw the Schoenfeld residuals plot ( ) and the curves ( ), where is the Kaplan-Meier estimator ( ). A test based on the Schoenfeld residuals can be conducted using the function cox.zph() in survival package in R, which provides a two-sided P value for the test under the null hypothesis of proportionality. If a P value is less than the prespecified significance level, it indicates a violation of the proportionality assumption. The Schoenfeld residuals plot can be drawn using the function cox.zph() in R. When a covariate is continuous, if a plot of Schoenfeld residuals versus times for the covariate shows patterns other than a constant, it indicates that there is a departure from the proportionality assumption for the covariate. As another graphical check, a plot of versus log(time) can be used to investigate the assumption. If the curves for each of the levels of a covariate are parallel, it indicates that the proportional hazards assumption holds for the covariate. When the proportional hazards assumption does not hold for some covariates, other regression models may be used to estimate the effects of covariates on the hazard function for an event of interest. For example, an additive hazards model ( - ) or accelerated failure time model ( - ) can be used as alternatives to the proportional hazards model.