Bayesian regularization for flexible baseline hazard functions in Cox survival models

Abstract

Fully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric approaches generally provide monotone estimations. Semi‐parametric choices allow for more flexible patterns but they can suffer from overfitting and instability. Regularization methods through prior distributions with correlated structures usually give reasonable answers to these types of situations. We discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards specified by a mixture of piecewise constant functions and by a cubic B‐spline function. For those “semi‐parametric” proposals, different prior scenarios ranging from prior independence to particular correlated structures are discussed in a real study with microvirulence data and in an extensive simulation scenario that includes different data sample and time axis partition sizes in order to capture risk variations. The posterior distribution of the parameters was approximated using Markov chain Monte Carlo methods. Model selection was performed in accordance with the deviance information criteria and the log pseudo‐marginal likelihood. The results obtained reveal that, in general, Cox models present great robustness in covariate effects and survival estimates independent of the baseline hazard specification. In relation to the “semi‐parametric” baseline hazard specification, the B‐splines hazard function is less dependent on the regularization process than the piecewise specification because it demands a smaller time axis partition to estimate a similar behavior of the risk.

Approximate Bayesian inference for mixture cure models

“…They provide a great flexibility to the modeling by allowing different patterns and multimodalities but some care is needed when working with them to avoid overfitting. To this effect, the elicitation of prior distributions is a relevant issue in the Bayesian approach to regularization 16 .…”

Bayesian survival analysis with BUGS

“…However, these specifications give restricted shapes which do not allow the presence of irregular behaviors. In addition, it is also possible to specify more flexible hazard shapes that allow for multimodal patterns by means of piecewise constant functions or spline functions, among other proposals 49 . Theoretical and methodological aspects of different flexible approaches to define the baseline hazard function can be found in several specific references such as Ibrahim et al, 8 Lin et al, 50 Mitra and Müller, 51 Bogaerts et al, 12 and Lázaro et al 49 The BUGS manual 43 shows a very flexible model implementation, using a Poisson approach, that allows the hazard to change at every observed event time.…”

“…Similarly, there is a wide range of approaches to define marginal prior distributions for h 0 parameters, from prior independence to prior correlation. Correlated scenarios account for shape restrictions and also avoid overfitting and strong irregularities in the estimation process 49 …”

“…Correlated scenarios account for shape restrictions and also avoid overfitting and strong irregularities in the estimation process. 49 The likelihood function of this specific model is not implemented in JAGS, so the "zeros trick" approach using a Poisson distribution has been used to specify it indirectly. 43,56…”

Bayesian survival analysis with BUGS

“…Regularisation methods try to modify the estimation procedures to give reasonable answers to these types of situations. Bayesian reasoning usually accounts for regularisation through prior distributions (Lázaro et al, 2020).…”