In fitting the Cox regression model by maximizing the partial likelihood, the estimate of an explanatory variable X
will be infinite if the value of X
at each uncensored failure time is the largest of all the values of X
in the risk set at that time (Tsiatis, 1981; Bryson and Johnson, 1981). You can exploit this information to artificially create a data set that has the condition of monotone likelihood for the
Cox regression. The following DATA step modifies the Myeloma
data in Example 73.1 to create a new explanatory variable, Contrived
, which has the value 1 if the observed time is less than or equal to 65 and has the value 0 otherwise. The phenomenon of
monotone likelihood will be demonstrated in the new data set Myeloma2
.
data Myeloma2; set Myeloma; Contrived= (Time <= 65); run;
For illustration purposes, consider a Cox model with three explanatory variables, one of which is the variable Contrived
. The following statements invoke PROC PHREG to perform the Cox regression. The IPRINT option is specified in the MODEL statement
to print the iteration history of the optimization.
proc phreg data=Myeloma2; model Time*Vstatus(0)=LOGbun HGB Contrived / itprint; run;
The symptom of monotonity is demonstrated in Output 73.4.1. The log likelihood converges to the value of –136.56 while the coefficient for Contrived
diverges. Although the NewtonRaphson optimization process did not fail, it is obvious that convergence is questionable.
A close examination of the standard errors in the "Analysis of Maximum Likelihood Estimates" table reveals a very large value
for the coefficient of Contrived
. This is very typical of a diverged estimate.
Output 73.4.1: Monotone Likelihood Behavior Displayed
Maximum Likelihood Iteration History  

Iter  Ridge  Log Likelihood  LogBUN  HGB  Contrived 
0  0  154.8579914384  0.0000000000  0.000000000  0.000000000 
1  0  140.6934052686  1.9948819671  0.084318519  1.466331269 
2  0  137.7841629416  1.6794678962  0.109067888  2.778361123 
3  0  136.9711897754  1.7140611684  0.111564202  3.938095086 
4  0  136.7078932606  1.7181735043  0.112273248  5.003053568 
5  0  136.6164264879  1.7187547532  0.112369756  6.027435769 
6  0  136.5835200895  1.7188294108  0.112382079  7.036444978 
7  0  136.5715152788  1.7188392687  0.112383700  8.039763533 
8  0  136.5671126045  1.7188405904  0.112383917  9.040984886 
9  0  136.5654947987  1.7188407687  0.112383947  10.041434266 
10  0  136.5648998913  1.7188407928  0.112383950  11.041599592 
11  0  136.5646810709  1.7188407960  0.112383951  12.041660414 
12  0  136.5646005760  1.7188407965  0.112383951  13.041682789 
13  0  136.5645709642  1.7188407965  0.112383951  14.041691020 
14  0  136.5645600707  1.7188407965  0.112383951  15.041694049 
15  0  136.5645560632  1.7188407965  0.112383951  16.041695162 
16  0  136.5645545889  1.7188407965  0.112383951  17.041695572 
Next, the Firth correction was applied as shown in the following statements. Also, the profilelikelihood confidence limits for the hazard ratios are requested by using the RISKLIMITS=PL option.
proc phreg data=Myeloma2; model Time*Vstatus(0)=LogBUN HGB Contrived / firth risklimits=pl itprint; run;
PROC PHREG uses the penalized likelihood maximum to obtain a finite estimate for the coefficient of Contrived
(Output 73.4.2). The much preferred profilelikelihood confidence limits, as shown in (Heinze and Schemper, 2001), are also displayed.
Output 73.4.2: Convergence Obtained with the Firth Correction
Maximum Likelihood Iteration History  

Iter  Ridge  Log Likelihood  LogBUN  HGB  Contrived 
0  0  150.7361197494  0.0000000000  0.000000000  0.0000000000 
1  0  136.9933949142  2.0262484120  0.086519138  1.4338859318 
2  0  134.5796594364  1.6770836974  0.109172604  2.6221444778 
3  0  134.1572923217  1.7163408994  0.111166227  3.4458043289 
4  0  134.1229607193  1.7209210332  0.112007726  3.7923555412 
5  0  134.1228364805  1.7219588214  0.112178328  3.8174197804 
6  0  134.1228355256  1.7220081673  0.112187764  3.8151642206 
Analysis of Maximum Likelihood Estimates  

Parameter  DF  Parameter Estimate 
Standard Error 
ChiSquare  Pr > ChiSq  Hazard Ratio 
95% Hazard Ratio Profile Likelihood Confidence Limits 

LogBUN  1  1.72201  0.58379  8.7008  0.0032  5.596  1.761  17.231 
HGB  1  0.11219  0.06059  3.4279  0.0641  0.894  0.794  1.007 
Contrived  1  3.81516  1.55812  5.9955  0.0143  45.384  5.406  6005.404 