Avoiding infinite estimates of time-dependent effects in small-sample survival studies

We address the phenomenon of monotone likelihood in Cox regression with time-dependent effects. Monotone likelihood occurs in the fitting process of a Cox model if at least one parameter estimate diverges to +/- infinity. We show that the probability of monotone likelihood is increased by the inclusion of time-dependent effects, particularly in small samples with several unbalanced and highly predictive covariates, and with a high percentage of censoring. Firth's bias reduction procedure was shown to provide an ideal solution to monotone likelihood. Here we extend his idea to Cox regression with time-dependent effects. By penalized maximum likelihood estimation, finite hazard ratio estimates of constant and time-dependent effects can be obtained. Penalized likelihood ratio tests and profile penalized likelihood confidence intervals are proposed as tools for inference. A Monte Carlo study of Cox regression with time-dependent effects confirms advantages of Firth-corrected (FC) over standard Cox analysis in terms of average bias and median absolute deviation. We also compare the FC and standard Cox approaches by means of analyses of two studies with time-dependent effects. An SAS macro and an R package for FC Cox regression with time-varying covariates and time-dependent effects are available at: http://www.muw.ac.at/msi/biometrie/programs.