Procedures for identifying and modeling time-to-event data in the presence of non--proportionality
For both randomized clinical trials and prospective cohort studies, the Cox regression model is a powerful tool for evaluating the effect of a treatment or an explanatory variable on time-to-event outcome. This method assumes proportional hazards over time. Systematic approaches to efficiently evaluate non-proportionality and to model data in the presence of non-proportionality are investigated. Six graphical methods are assessed to verify the proportional hazards assumption based on characteristics of the survival function, cumulative hazard, or the feature of residuals. Their performances are empirically evaluated with simulations by checking their ability to be consistent and sensitive in detecting proportionality or non-proportionality. Two-sample data are generated in three scenarios of proportional hazards and five types of alternatives (that is, non-proportionality). The usefulness of these graphical assessment methods depends on the event rate and type of non-proportionality. Three numerical (statistical testing) methods are compared via simulation studies to investigate the proportional hazards assumption. In evaluating data for proportionality versus non-proportionality, the goal is to test a non-zero slope in a regression of the variable or its residuals on a specific function of time, or a Kolmogorov-type supremum test. Our simulation results show that statistical test performance is affected by the number of events, event rate, and degree of divergence of non-proportionality for a given hazards scenario. Determining which test will be used in practice depends on the specific situation under investigation. Both graphical and numerical approaches have benefits and costs, but they are complementary to each other. Several approaches to model and summarize non-proportionality data are presented, including non-parametric measurements and testing, semi-parametric models, and a parametric approach. Some illustrative examples using simulated data and real data are also presented. In summary, we present a systemic approach using both graphical and numerical methods to identify non-proportionality, and to provide numerous modeling strategies when proportionality is violated in time-to-event data.