Statistics and Its Interface

Volume 11 (2018)

Number 4

Prior specifications to handle the monotone likelihood problem in the Cox regression model

Pages: 687 – 698

DOI: https://dx.doi.org/10.4310/SII.2018.v11.n4.a12

Authors

Frederico M. Almeida (Departamento de Estatística, ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

Enrico A. Colosimo (Departamento de Estatística, ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

Vinícius D. Mayrink (Departamento de Estatística, ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

Abstract

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.

Keywords

Firth correction, MCMC, proportional hazards, partial likelihood, survival analysis

2010 Mathematics Subject Classification

62N02

Received 5 October 2017

Published 19 September 2018