Cox Regression Using Firth's Correction: 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 one parameter estimate diverges to (plus or minus) infinity. Monotone likelihood primarily occurs in small samples with substantial censoring of survival times and several highly predictive covariates. Statistical software packages for Cox regression using the maximum likelihood method cannot appropriately deal with this problem. A new procedure, based on Firth`s bias correction method (Firth, 1993), to solve the problem has been proposed by Heinze and Schemper (2001). It has been shown that unlike the standard maximum likelihood method, this method always leads to finite parameter estimates. Heinze and Dunkler (2008) applied the Firth correction to estimate time-dependent effects with sparse survival data and showed that again the correction helps in improving point and interval estimation.
We developed a SAS macro and an R package to make this method available from within one of these widely used statistical software packages (Heinze, 1999; Heinze, 2006). Our programs are also capable of performing interval estimation based on profile penalized log likelihood (PPL) and of plotting the PPL function as was suggested by Heinze and Schemper (2001).
The programs are capable of estimating time-dependent effects and optionally use the counting process formulation of the Cox model (similarly to SAS/PROC PHREG or R/survival/coxph). The SAS macro was developed using the SAS System for Windows 9.1. The core routines of this program reside in a dynamic link library (DLL). Therefore, its use is limited to Windows XP and compatible platforms (Windows NT, 95/98, 2000).
The R package is available here as Windows binary distribution (.zip) or as a source package (.tar.gz). It can also be downloaded from CRAN for use with any platform.
Please note that users of SAS version 9.2 can apply Firth's correction by specifying the option FIRTH in the model statement of PROC PHREG. Profile penalized likelihood confidence intervals can be computed by specifying RL=PL in combination with the FIRTH option. However, PROC PHREG does not provide corresponding p-values from penalized likelihood ratio tests (as does our fc06 macro). As in our macro, the PROC PHREG's FIRTH option cannot be combined with other tie corrections than the BRESLOW method.
Heinze, G., Dunkler, D. (2008): "Avoiding infinite estimates of time-dependent effects in small-sample survival studies", Statistics in Medicine 27, 6455 - 6469
Heinze, G. (2006): "FC06: A SAS macro for Cox regression with Firth's penalization", Technical Report 1/2006, Section for Clinical Biometrics, CeMSIIS, Medical University of Vienna
Heinze, G., Ploner, M. (2002): "SAS and SPLUS programs to perform Cox regression without convergence problems", Computer Methods and Programs in Biomedicine 67, 217 - 223
Ploner, M., Heinze, G. (2001): "An SPLUS library to perform Cox regression without convergence problems", Technical Report 1/2001, Section for Clinical Biometrics, CeMSIIS, University of Vienna
Heinze, G., Schemper, M. (2001): "A Solution to the Problem of Monotone Likelihood in Cox Regression", Biometrics 57 (1), 114 - 119
Heinze, G. (1999): "The application of Firth's procedure to Cox and logistic regression", Technical Report 10/1999, Section for Clinical Biometrics, CeMSIIS, Medical University of Vienna
Firth, D. (1993): "Bias reduction of maximum likelihood estimates", Biometrika 80, 27 - 38