Adjusted score functions for monotone likelihood in the Cox regression model

Standard inference procedures for the Cox model involve maximizing the partial likelihood function. Monotone partial likelihood is an issue that frequently happens in the analysis of health science studies. Monotone likelihood mainly occurs in samples with substantial censoring of survival times and is associated with categorical covariates. In particular, and more frequently, it usually happens when one level of a categorical covariate has just experienced censoring times. In order to overcome this problem, Heinze and Schemper proposed an adjusted partial likelihood score function obtained by suitably adapting the general approach of Firth for mean bias reduction. The procedure is effective in preventing infinite estimates. As an alternative solution, we propose an approach based on the adjusted score function recently suggested by Kenne Pagui et al for median bias reduction. This procedure also solves the infinite estimate problem and has an additional advantage of being invariant under componentwise reparameterizations. This latter fact is fundamental under Cox model since hazards ratio interpretation is obtained by exponentiating parameter estimates. Numerical studies of the proposed method suggest better inference properties than those of the mean bias reduction. A real-data application related to a melanoma skin dataset is used as illustration for a comparison basis of the methods.