Validity and efficiency of approximation methods for tied survival times in Cox regression.

Survival-time studies sometimes do not yield distinct failure times. Several methods have been proposed to handle the resulting ties. The goal of this paper is to compare these methods. Simulations were conducted, in which failure times were generated for a two-sample problem with an exponential hazard, a constant hazard ratio, and no censoring. Failure times were grouped to produce heavy, moderate, and light ties, corresponding to a mean of 10.0, 5.0, and 2.5 failures per interval. Cox proportional hazards models were fit using each of three approximations for handling ties with each interval size for sample sizes of n = 25, 50, 250, and 500 in each group. The Breslow (1974, Biometrics 30, 89-99) approximation tends to underestimate the true beta, while the Kalbfleisch-Prentice (1973, Biometrika 60, 267-279) approximation tends to overestimate beta. As the ties become heavier, the bias of these approximations increases. The Efron (1977, Journal of the American Statistical Association 72, 557-565) approximation performs far better than the other two, particularly with moderate or heavy ties; even with n = 25 in each group, the bias is under 2%, and for sample sizes larger than 50 per group, it is less than 1%. Except for the heaviest ties in the smallest sample, confidence interval coverage for all three estimators fell in the range of 94-96%. However, the tail probabilities were asymmetric with the Breslow and Kalbfleisch-Prentice formulas; using the Efron approximation, they were closer to the nominal 2.5%. Although the Breslow approximation is the default in many standard software packages, the Efron method for handling ties is to be preferred, particularly when the sample size is small either from the outset or due to heavy censoring.