Kristin E. Karlsson, Rickard Eriksson, Mats O. Karlsson and Joakim Nyberg
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: Traditionally the most commonly used model for survival analysis is the Cox proportional hazard (Cox PH) model [1]. The Cox PH model is a semi-parametric model that makes a parametric assumption regarding the effect of the covariates on the hazard function, but makes no assumption about the shape of the hazard function (i.e. the baseline hazard). This can be interpreted as it is the influence of the covariates that are of interest, not the nature of the hazard function as such.
As time to event models are increasingly used within the field of pharmacometrics it would be convenient to be able to add the Cox PH model to the battery of survival models to be tested in NONMEM and to be used within other procedures such as scm or sse in PsN [2]. Hence, the objective of this work was to implement the Cox PH model in NONMEM® [3].
Methods: The log partial likelihood for the Cox PH model with the Breslow approximation [4] is defined as: log L=∑jN{Siβ- dj*log[∑kЄRj exp(Xkβ)]} where N is the last observation in the data set, Rj is the set of subjects at risk at time j, Xk is the covariate vector for subject k, β is the covariate coefficient vector, Si=∑iЄDj Xi, where Dj is the set of subjects having an event at time j and dj is the number of events at time j. This definition of the likelihood, which allows for tied event times, was implemented in NONMEM®. As the risk set at time j (Rj) consists of the included subjects with event/censoring time ≥ j the data set was sorted according to decreasing event times, enabling the inner sums to be calculated on the fly. The implementation was tested on a data set which was simulated based on the survival model presented in Bruno et al [5]. Four covariates were tested: baseline tumor size, tumor size at week 6 (as time constant), number of metastases and ECOG status. The result from NONMEM® was compared to a Cox PH model run in R.
Results: The estimated covariate coefficients in NONMEM were: 0.00428, 0.792, 0.518, 0.189, and the same in R (with the Breslow option): 0.00429, 0.792, 0.519, and 0.189. The standard errors of the estimates were also equal between the software.
Conclusions: The likelihood for the Cox PH model with the Breslow approximation was successfully implemented in NONMEM® with the presumption that the subjects in the dataset are ordered by decreasing event times.
References:
[1] Cox D.R. Regression Models and Life-Tables. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 34, No. 2. (1972), pp.187-220.
[2] Keizer RJ, Karlsson MO, Hooker A. Modeling and Simulation Workbench for NONMEM: Tutorial on Pirana, PsN, and Xpose. CPT Pharmacometrics Syst Pharmacol. 2013 Jun 26;2:e50.
[3] Beal, S., Sheiner, L.B., Boeckmann, A., & Bauer, R.J., NONMEM User’s Guides. Introduction to NONMEM 7. Icon Development Solutions (2014).
[4] Breslow N. Covariance analysis of censored survival data. Biometrics. 1974 Mar;30(1):89–99.
[5] R Bruno et al. Simulations to Assess Phase II Noninferiority Trials of Different Doses of Capecitabine in Combination With Docetaxel for Metastatic Breast Cancer. CPT Pharmacometrics Syst Pharmacol. 2012:26;1:e19
Reference: PAGE 23 (2014) Abstr 3160 [www.page-meeting.org/?abstract=3160]
Poster: Methodology - Covariate/Variability Models