A solution to the problem of monotone likelihood in Cox regression

The phenomenon of monotone likelihood is observed in the fitting process of a Cox model if the likelihood converges to a finite value while at least o ne parameter estimate diverges to +/-infinity. Monotone likelihood primaril y occurs in small samples with substantial censoring of survival times and several highly predictive covariates. Previous options to deal with monoton e likelihood have been unsatisfactory. The solution we suggest is an adapta tion of a procedure by Firth (1993, Biometrika 80, 27-38) originally develo ped to reduce the bias of maximum likelihood estimates. This procedure prod uces finite parameter estimates by means of penalized maximum likelihood es timation. Corresponding Wald-type tests and confidence intervals are availa ble, but it is shown that penalized likelihood ratio tests and profile pena lized likelihood confidence intervals are often preferable. An empirical st udy of the suggested procedures confirms satisfactory performance of both e stimation and inference. The advantage of the procedure over previous optio ns of analysis is finally exemplified in the analysis of a breast cancer st udy.