(W-136) Gompertz Cure Rate Survival Models with Stan and Brms

Andrew Tredennick, Ph.D. – Senior Scientist, Statistics, Metrum Research Group; Tim Waterhouse, Ph.D. – Principal Scientist, Statistics, Metrum Research Group

Objectives: In drug safety data it is common for a proportion of the population to never experience a specified adverse event. Standard parametric survival models assume the survival curve approaches zero and fail to characterize this aspect of safety data. A Gompertz distribution with a negative scale parameter allows the survival curve to have a non-zero asymptote and may better characterize survival data where a proportion of the population never experiences the event of interest1. The brms package in R enables Bayesian modeling with Stan^2,3. The Gompertz distribution is not natively supported in brms or Stan, and a custom probability distribution was implemented in the software to enable cure rate model fitting and inference.

Methods: The following Gompertz distribution functions were defined as custom Stan functions: log probability density, log cumulative distribution, log complementary cumulative distribution, and random generation function. The Stan functions and the log likelihood of the survival model were used to define the Gompertz model as a custom family in brms. Data were simulated from a mixture cure rate model with a constant hazard and were used to demonstrate implementation of the custom brms family. With the custom brms family in place, standard brms tooling was used for model fitting and inference.

Results: The Gompertz distribution better characterized the simulated survival curve than an exponential, lognormal, or Weibull distribution. The leave-one-out expected log pointwise predictive density (ELPD) model criterion identified the Gompertz model as the most favorable model, and visual predictive checks showed that the asymptotic assumption of the standard survival models led to an overprediction of adverse events in the latter part of the simulated study. The Gompertz model did not suffer from such a misspecification and characterized the data well.

Conclusions: The Gompertz distribution characterized the simulated survival data, allowing for a fraction of the patient population to never experience the adverse event. Implementation of custom distributions in brms and Stan can be extended to other models not natively supported in either software.

Citations: Citations:
[1] Gieser, P. W., M. N. Chang, P. V. Rao, J. J. Shuster, and J. Pullen. 1998. “Modelling Cure Rates Using the Gompertz Model with Covariate Information.” Statistics in Medicine 17 (8): 831–39.
[2] Bürkner P (2017). “brms: An R Package for Bayesian Multilevel Models Using Stan.” Journal of Statistical Software, 80(1), 1–28. doi:10.18637/jss.v080.i01.
[3] Stan Development Team. 2019. Stan Modeling Language Users Guide and Reference Manual, 2.26. https://mc-stan.org.