Approximate Bayesian methods in cure survival models: coupling P-splines with Laplace approximations for fast inference

Standard Bayesian methods for time-to-event data rely on Markov chain Monte Carlo (MCMC) to sample from posterior distributions and perform statistical inference. When confronted with the increasing sophistication of survival models to cope with applied challenges, the MCMC toolbox exhibits a spectrum of practical issues such as slow mixing samplers, potential high posterior correlation between parameters and a strong computational burden. In an attempt to overcome the drawbacks inherent to MCMC sampling, an approximate Bayesian inference methodology has recently been proposed by Rue et al. [1] that delivers accurate posterior approximations at a fast computational speed. We extend the INLA methodology in a Cox proportional hazards model where the baseline log-hazard is specified as a linear combination of cubic B-splines whose coefficients are assigned a multivariate normal prior. This is motivated by the generic idea underlying penalized B-splines by Eilers and Marx [2] and their Bayesian adaptation following Lang and Brezger [3]. Simulation results suggest that our approximate inference method is a promising alternative to MCMC in Bayesian P-spline models. The computational speed is increased by a factor of at least twenty as compared to classic MCMC methods without losing on precision and accuracy. Furthermore, approximate pointwise credible intervals for the conditional survival function can be obtained in a streamlined way. We conclude the presentation with an extension of the methodology to cure survival models where an unknown proportion of unidentified subjects are not at risk for the monitored event. [1] Rue, H., Martino, S., & Chopin, N. (2009). Journal of the Royal Statistical Society, 71(2), 319-392. [2] Eilers, P.H., & Marx, B.D. (1996). Statistical Science, 11(2), 89-102. [3] Lang, S., & Brezger, A. (2004). Journal of Computational and Graphical Statistics, 13(1), 183-212