Firth’s penalized likelihood approach is a method of addressing issues of separability, small sample sizes, and bias of the parameter estimates. This example performs some comparisons between results from using the FIRTH option to results from the usual unconditional, conditional, and exact logistic regression analyses. When the sample size is large enough, the unconditional estimates and the Firth penalizedlikelihood estimates should be nearly the same. These examples show that Firth’s penalized likelihood approach compares favorably with unconditional, conditional, and exact logistic regression; however, this is not an exhaustive analysis of Firth’s method. For more detailed analyses with separable data sets, see Heinze (2006, 1999) and Heinze and Schemper (2002).
A 22 table with one cell having zero frequency, where the rows of the table are the levels of a covariate while the columns are the levels of the response variable, is an example of a quasicompletely separated data set. The parameter estimate for the covariate under unconditional logistic regression will move off to infinity, although PROC LOGISTIC will stop the iterations at an earlier point in the process. An exact logistic regression is sometimes performed to determine the importance of the covariate in describing the variation in the data, but the medianunbiased parameter estimate, while finite, might not be near the true value, and one confidence limit (for this example, the upper) is always infinite.
The following DATA step produces 1000 different 22 tables, all following an underlying probability structure, with one cell having a near zero probability of being observed:
%let beta0=15; %let beta1=16; data one; keep sample X y pry; do sample=1 to 1000; do i=1 to 100; X=rantbl(987987,.4,.6)1; xb= &beta0 + X*&beta1; exb=exp(xb); pry= exb/(1+exb); cut= ranuni(393993); if (pry < cut) then y=1; else y=0; output; end; end; run;
The following statements perform the biascorrected and exact logistic regression on each of the 1000 different data sets, output the odds ratio tables by using the ODS OUTPUT statement, and compute various statistics across the data sets by using the MEANS procedure:
ods exclude all; proc logistic data=one; by sample; class X(param=ref); model y(event='1')=X / firth clodds=pl; ods output cloddspl=firth; run; proc logistic data=one exactonly; by sample; class X(param=ref); model y(event='1')=X; exact X / estimate=odds; ods output exactoddsratio=exact; run; ods select all; proc means data=firth; var LowerCL OddsRatioEst UpperCL; run; proc means data=exact; var LowerCL Estimate UpperCL; run;
The results of the PROC MEANS statements are summarized in Table 54.16. You can see that the odds ratios are all quite large; the confidence limits on every table suggest that the covariate X
is a significant factor in explaining the variability in the data.
Table 54.16: Odds Ratio Results
Method 
Mean Estimate 
Standard Error 
Minimum Lower CL 
Maximum Upper CL 

Firth 
231.59 
83.57 
10.40 
111317 
Exact 
152.02 
52.30 
11.16 

Casecontrol models contain an intercept term for every casecontrol pair in the data set. This means that there are a large number of parameters compared to the number of observations. Breslow and Day (1980) note that the estimates from unconditional logistic regression are biased with the corresponding odds ratios off by a power of 2 from the true value; conditional logistic regression was developed to remedy this.
The following DATA step produces 1000 casecontrol data sets, with pair
indicating the strata:
%let beta0=1; %let beta1=2; data one; do sample=1 to 1000; do pair=1 to 20; ran=ranuni(939393); a=3*ranuni(9384984)1; pdf0= pdf('NORMAL',a,.4,1); pdf1= pdf('NORMAL',a,1,1); pry0= pdf0/(pdf0+pdf1); pry1= 1pry0; xb= log(pry0/pry1); x= (xb&beta0*pair/100) / &beta1; y=0; output; x= (xb&beta0*pair/100) / &beta1; y=1; output; end; end; run;
Unconditional, conditional, exact, and Firthadjusted analyses are performed on the data sets, and the mean, minimum, and maximum odds ratios and the mean upper and lower limits for the odds ratios are displayed in Table 54.17. Caution: Due to the exact analyses, this program takes a long time and a lot of resources to run. You might want to reduce the number of samples generated.
ods exclude all; proc logistic data=one; by sample; class pair / param=ref; model y=x pair / clodds=pl; ods output cloddspl=oru; run; data oru; set oru; if Effect='x'; rename lowercl=lclu uppercl=uclu oddsratioest=orestu; run; proc logistic data=one; by sample; strata pair; model y=x / clodds=wald; ods output cloddswald=orc; run; data orc; set orc; if Effect='x'; rename lowercl=lclc uppercl=uclc oddsratioest=orestc; run; proc logistic data=one exactonly; by sample; strata pair; model y=x; exact x / estimate=both; ods output ExactOddsRatio=ore; run; proc logistic data=one; by sample; class pair / param=ref; model y=x pair / firth clodds=pl; ods output cloddspl=orf; run; data orf; set orf; if Effect='x'; rename lowercl=lclf uppercl=uclf oddsratioest=orestf; run; data all; merge oru orc ore orf; run; ods select all; proc means data=all; run;
You can see from Table 54.17 that the conditional, exact, and Firthadjusted results are all comparable, while the unconditional results are several orders of magnitude different.
Table 54.17: Odds Ratio Estimates
Method 
N 
Minimum 
Mean 
Maximum 

Unconditional 
1000 
0.00045 
112.09 
38038 
Conditional 
1000 
0.021 
4.20 
195 
Exact 
1000 
0.021 
4.20 
195 
Firth 
1000 
0.018 
4.89 
71 
Further examination of the data set all
shows that the differences between the square root of the unconditional odds ratio estimates and the conditional estimates
have mean –0.00019 and standard deviation 0.0008, verifying that the unconditional odds ratio is about the square of the conditional
odds ratio. The conditional and exact conditional odds ratios are also nearly equal, with their differences having mean 3E–7
and standard deviation 6E–6. The differences between the Firth and the conditional odds ratios can be large (mean 0.69, standard
deviation 5.40), but their relative differences, , have mean 0.20 with standard deviation 0.19, so the largest differences occur with the larger estimates.