Validity and Efficiency of Approximation Methods for Tied Survival Times in Cox Regression

Abstract

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.

Maximum likelihood estimation for tied survival data under Cox regression model via EM-algorithm

“…But it has larger biases than the EM for normal and exponential covariate distributions, especially for larger ties, similar to RB procedure. Our simulation study shows that Efron procedure has the best performance of the existing methods dealing tied survival data, which is consistent with the findings of Hertz- Picciotto and Rockhill (1997). Compared with the EM procedure, Efron procedure works better with binary and uniform covariate distributions.…”

“…These distributions account for some of the practical situations where covariates are binary, approximately uniformly distributed, symmetrical or skewed. Here we choose the binary covariate as .6931×Bernoulli (1, .7), which results in a hazard ratio of 2 in the two sample situation when β = 1, a scenario considered by Hertz-Picciotto and Rockhill (1997).…”

“…As far as we know, there have been very little study on the validity and efficiency of the various approximation methods for tied survival data under Cox regression model. A previous paper by Hertz-Picciotto and Rockhill (1997) examining this question with simulations considered only two sample case without censoring.…”

Time to pregnancy among Danish laboratory technicians who were a part of the National Birth Cohort

“…Validation studies of time to pregnancy have shown that the recall of time to pregnancy seems accurate even over longer time periods (12,14,17,18). We asked about time to pregnancy in four categories in order to assist the women and accepted a possible minimal loss in precision estimates of the fecundability ratio (27).…”

Fluoroquinolone Impact on Nasal Methicillin-Resistant and Methicillin-Sensitive Staphylococcus aureus Colonization Durations in Neurologic Long-Term-Care Facilities

“…Those associated with time to loss of colonization in univariate analyses (2-sided P Ͻ 0.20) were included in the multivariable models and maintained in the final models when their HR estimates remained statistically significantly associated (2-sided P Ͻ 0.05) with loss of colonization using a manual backward stepwise procedure, yielding the final adjusted models. All univariate and multivariable models were adjusted for center and fitted with the Efron method for handling ties (23). Interactions between the strain's methicillin resistance phenotype and all the covariates remaining in the final model of loss of S. aureus colonization were systematically evaluated.…”