Senior Principal Scientist Amgen Inc., United States
Disclosure(s):
Po-Wei Chen, PhD: No relevant disclosure to display
Objectives: The Cox proportional hazard model had been used extensively to evaluate time-to-event (TTE) endpoints in drug development. The model assumes that event times are unique and distinct across subjects. When tied times occur, correction methods can be used to minimize bias in parameter estimation. Efron’s correction method is considered a balanced choice between the accuracy of estimate (over Breslow method) and computational intensity (over Exact method) and is used as the default option in R, however, its implementation in NONMEM has not been developed. This method introduces Efron correction in NONMEM for evaluating the effects of time-variant and time-invariant covariates on TTE endpoints in Cox regression.
Methods: To implement the Cox Regression with Efron Correction in NONMEM, simulation datasets with 500 subjects were created, consisting of a time-variant covariate, a time-invariant covariate, and a TTE response variable. The event times were generated based on a bathtubs-like baseline hazard function and prespecified covariates’ effects on the baseline hazard for the events. Censoring times were generated from a constant hazard. The time records were obtained from either the event time or censoring time, whichever occurred first. The tied-time records were created by rounding up the times to the closest day, week, or month, mimicking scenarios when the time records are collected by day, week, or month. To implement Cox regression in NONMEM, the datasets were rearranged such that the subjects were organized by event/censoring times in descending order. For tied-time records, subjects with tied censoring times arranged before subjects with tied event times. The partial likelihood calculation for Efron correction employed a do-while loop and two variables defined in abbreviation block to facilitate calculations for the numerators and denominators of the partial likelihoods across the events over time. Lastly, model parameters were estimated in NONMEM and R for validation and comparison.
Results: Tie-time records of 305, 444, and 484 were generated when rounding up the event/censoring times to the nearest day, week, and month, respectively, over 195 distinct days, 56 distinct weeks, and 16 distinct months. Estimated coefficients and the corresponding standard errors for the effects of time-invariant and time-variant covariates were found to be similar between NONMEM and R across all cases, when applying Efron correction in the Cox regression analyses (differences < 0.2%). This indicates that the newly proposed workflow in NONMEM is comparable to R and can be effectively adopted for TTE analysis with tie-records.
Conclusions: The developed NONMEM workflow successfully implements Efron correction for tied-time records in Cox regression. Compared to traditional statistical software, this workflow can be extended to incorporate complex nonlinear time-variant covariates and pharmacokinetic and pharmacodynamic models with ordinary differential equations for TTE analysis, conveniently within NONMEM in the field of pharmacometrics.
Citations: [1] Efron, B. The Efficiency of Cox's Likelihood Function for Censored Data, Journal of the American Statistical Association, vol. 72, no. 359, 1977, pp. 557–565. [2] Zhang, Z. et al. “Time-varying covariates and coefficients in Cox regression models.” Annals of translational medicine vol. 6,7 (2018): 121. doi:10.21037/atm.2018.02.12 [3] Karlsson, KE. et al. “Estimating a Cox proportional hazard model in NONMEM”. PAGE - Population Approach Group Europe. (2024) https://www.page-meeting.org/pdf_assets/4156-Poster_PAGE%20_2014_COXPH_Kristin_Karlsson.pdf
