Statistics and Its Interface
Volume 11 (2018)
Prior specifications to handle the monotone likelihood problem in the Cox regression model
Pages: 687 – 698
The monotone likelihood is a phenomenon that may affect the fitting process of well-established regression models such as the Cox proportional hazards model. In short, the problem occurs when the likelihood converges to a finite value, while at least one parameter estimate diverges to $\pm$infinity. In survival analysis, monotone likelihood primarily appears in samples with substantial censored times and containing many categorical covariates; it is often observed when one level of a categorical covariate has not experienced any failure. A solution suggested in the literature (known as Firth correction) is an adaptation of a method originally created to reduce the bias of maximum likelihood estimates. The method leads to a finite estimate by means of a penalized maximum likelihood procedure. In this case, the penalty might be interpreted as a Jeffreys type of prior widely used in the Bayesian context; however, this approach has some drawbacks, especially biased estimators and high standard errors. The present paper explores other penalties for the partial likelihood function in the flavor of Bayesian prior distributions. A simulation study is developed, based on Monte Carlo replications and distinct sample sizes, to evaluate the impact of the suggested priors in terms of inference. Results show that a greater bias reduction can be achieved with respect to the Firth correction; however, this performance depends on the uncertainty level of the prior (vague priors do not manage well the monotone shape). A real application is also presented to illustrate the analysis using a melanoma skin data set.
Firth correction, MCMC, proportional hazards, partial likelihood, survival analysis
2010 Mathematics Subject Classification
This research is partially supported by grants from: Fundação de Amparo à Pesquisa de Minas Gerais (FAPEMIG), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).
Received 5 October 2017
Published 19 September 2018