Implementation of event time distribution as a random effect in time-to-event analysis
Carolina Llanos-Paez1, Mats O. Karlsson1
1. Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Introduction: Different distributions of event times considered in time-to-event (TTE) analysis are well documented. These distributions include the exponential, Weibull, Gompertz, among others [1]. It has been reported that event times can be generated following any of those distributions by using a random variable with uniform distribution [2]. Including this random variable on the distribution of the event time may help to perform TTE analysis using the Full Random Effect Modelling (FREM) approach [3], and therefore take into account correlation between covariates without affecting parameter estimates as well as handle potential missing covariates.
Objectives: To perform TTE analysis adding a random effect on the event time to take into account correlation between covariates by using the FREM approach.
Methods: Single event times according to an exponential distribution (constant hazard) were repeatedly (n=100) simulated for 1,000 subjects and thereafter used to perform TTE analysis using: i) the random variable (RV) approach [2] and ii) the standard (STD) parametric modelling approach. Estimation properties of the two approaches were compared at different proportions of censored data (0%, 20%, 50% and 80%) using both constant hazard and Weibull models. Further estimation of covariate effects was performed using the RV model with the FREM approach (RV_FREM) and the STD model. Comparison between models were assessed in terms of their objective function value (OFV), parameter estimates and using the stochastic simulation and estimation (sse) tool to obtain the relative standard error (RSE) of estimates. In addition, a real data set (that was used previously to perform a published TTE analysis [4]) was used in this study to assess the significance of different covariates (treatment-group with digoxin (n=105) or placebo-group (n=112), sex, age, body weight and creatinine clearance) on the time of conversion to sinus rhythm in patients with acute atrial fibrillation using the FREM approach. All analysis and data simulation were performed using NONMEM® software v.7.4.3 with an Intel® FORTRAN compiler and PsN version 4.8.9.
Results: The same OFV and similar parameter estimates were obtained for the RV and STD model. The hazard estimate [RSE%] for a constant hazard for the RV/STD model was: 0.93/0.93 h-1 [0.34%/0.32%] (0% censored patients), 0.94/0.94 h-1 [0.37%/0.37%] (20% censored subjects), 0.96/0.95 h-1 [0.49%/0.50%] (50% censored subjects), and 0.98/0.98 h-1 [0.69%/0.70%] (80% censored subjects). The scale and shape parameter estimates for a Weibull distribution for the RV and STD model were similarly close. When a covariate effect was considered in both models, STD/RV_FREM, the ∆OFV (base model vs. model with covariate effect included) for the i) constant hazard was: 658/1,093 (0% censored subjects); 761/666 (20% censored subjects); 674/516 (50% censored subjects); 385/307 (80% censored subjects); ii) Weibull distribution was: 945/1,092 (0% censored subjects); 947/662 (20% censored subjects); 761/502 (50% censored subjects); 408/279 (80% censored subjects). Parameter estimates for the STD/RV models when using the real data set for the constant hazard was a hazard of 0.04/0.04 h-1; and for the Weibull distribution were a scale of 0.03/0.03 h-1 and shape parameter of 0.67/0.66. The ∆OFV (base model vs. model with all covariates included at the same time) for the STD/RV_FREM considering a constant hazard was 26/8 and considering a Weibull distribution was 21/8.
Conclusion: A random variable affecting the event time was incorporated in a TTE analysis using both simulated and real data sets. This allows the use of the FREM approach to perform TTE analysis. It may also allow other extension to standard TTE analysis, such as additional diagnostics, possibility to explore different distributions of the event time. Results from the real data set using the RV approach are comparable to the original model. While the present results are encouraging, further analysis is warranted, especially in the way of handling the censored data when using this additional random effect.
References:
[1] Bradburn MJ et al. Br J Cancer (2003) 89: 232–238
[2] Bender R et al. Statist. Med. (2005) 24:1713–1723
[3] Karlsson MO. PAGE 21 (2012) Abstr 2455
[4] Hennig et al. PAGE 18 (2009) Abstr 1504