Methods of Handling Tied Events in the Cox Proportional Hazard Model

Abstract

The Cox proportional hazard model is one of the most common methods used in time to event data analysis. The model is based on several restrictive assumptions one of which concerns tied events, i.e. events with exactly the same survival time. If time were measured in a perfectly continuous scale, such cases would never occur. In real applications time is usually measured in a discrete manner which results in the existence of ties in most survival data. However, if this assumption is violated, it should not necessarily prevent analysis by using the Cox model. The current paper presents and compares five ways proposed for handling tied events. On the basis of the calculations performed it can be stated that exact expression and the discrete model give the best results in terms of fit statistics; however, they are the most time-consuming. Efron and Breslow approximations are much faster but result in worse model fit. In the case analysed the Efron method seems to be the best choice, taking into account differences in parameters estimates, fit statistics and calculation time. What is more, a simple method based on subtracting a tiny random value from each tied survival time performed surprisingly well; both in terms of parameter estimates compared with the exact expression, as well as fit statistics. In general, in the case of large datasets and/or a large number of ties, if estimation precision is not as important as estimation time, Breslow or - more preferably - Efron approximations might be used. However, if time is not limited, one should consider choosing an exact method or discrete model that can provide better fit statistics and more efficient parameter estimates.