Analysis of lifetime data has a long history in the landscape of statistical literature. Over the past three decades or so, survival models have witnessed a notable facelift of the underlying techniques used to analyze time-to-event data with a myriad of extensions and refinements aiming at a better understanding of potentially complex survival patterns. Among these refinements, cure models are of particular interest as they leave room for the existence of a cure fraction, i.e. a fraction of long-term survivors for which the event of interest will never occur, no matter how far we look in the future horizon. The relatively recent technological breakthroughs in medicine, especially in cancer research, have led to the indisputable biological evidence of a ‘‘cure’’ phenomenon that should not be ignored from a statistical modeling perspective to avoid biased results.  

This article focuses on mixture cure models where the population under study is divided into a group of subjects who will experience the event of interest over some finite time horizon and another group of cured subjects who will never experience the event irrespective of the duration of follow-up. Within the Bayesian framework, the authors develop a new methodology for fast and flexible inference based on Laplace approximations and penalized B-splines. In the so-called Laplacian-P-splines mixture cure (LPSMC) model, the cure proportion is fitted by a logistic regression approach and the survival function of susceptible subjects is assumed to follow a Cox proportional hazards model, where the baseline hazard is approximated by P-splines. The proposed method has the advantage of being completely sampling-free as opposed to traditional Markov chain Monte Carlo (MCMC) algorithms that are much more costly to implement due to their iterative scheme to sample from posterior distributions. In addition, the Laplace approximation to the conditional posterior of latent variables is based on analytical formulas for the gradient and Hessian of the log-likelihood, resulting in a substantial speed-up in approximating posterior distributions. 

The authors also propose a fully stochastic algorithm based on a Metropolis-Langevin-within-Gibbs sampler (MLWG) for inference in the mixture cure model. As such, the end user has the opportunity to choose between a totally sampling-free approach (as provided by LPSMC) that is extremely fast and a more classic stochastic sampler. The algorithms are brought to the public via an intuitive software package coded in the R language. The paper measures the statistical performance and computational efficiency of LPSMC in different simulation settings and illustrates its use on three applications involving real survival data. Results show that LPSMC is an appealing alternative to MCMC for approximate Bayesian inference in standard mixture cure models.