Example 8.15: Firth logistic regression

In logistic regression, when the outcome has low (or high) prevalence, or when there are several interacted categorical predictors, it can happen that for some combination of the predictors, all the observations have the same event status. A similar event occurs when continuous covariates predict the outcome too perfectly.

This phenomenon, known as "separation" (including complete and quasi-complete separation) will cause problems fitting the model. Sometimes the only symptom of separation will be extremely large standard errors, while at other times the software may report an error or a warning.

One approach to handling this sort of problem is exact logistic regression, which we discuss in section 4.1.2. But exact logistic regression is complex and may require prohibitive computational resources. Another option is to use a Bayesian approach. Here we show how to use a penalized likelihood method originally proposed by Firth (1993 Biometrika 80:27-38) and described fully in this setting by Georg Heinze (2002 Statistics in Medicine 21:2409-2419 and 2006 25:4216-4226). A nice summary of the method is shown on a web page that Heinze maintains. In later entries we'll consider the Bayesian and exact approaches.

Update: see bottom of the post.

SAS
In SAS, the corrected estimates can be found using the firth option to the model statement in proc logistic. We'll set up the problem in the simple setting of a 2x2 table with an empty cell. Here, we simply output three observations with three combinations of predictor and outcome, along with a weight variable which contains the case counts in each cell of the table

data testfirth;
   pred=1; outcome=1; weight=20; output;
   pred=0; outcome=1; weight=20; output;
   pred=0; outcome=0; weight=200; output;
run;

In the proc logistic code, we use the weight statement, available in many procedures, to suggest how many times each observation is to be replicated before the analysis. This approach can save a lot of space.

proc logistic data = testfirth;
  class outcome pred (param=ref ref='0');
  model outcome(event='1') = pred / cl firth;
  weight weight;
run;

Without the firth option, the parameter estimate is 19.7 with a standard error of 1349. In contrast, here is the result of the above code.

            Analysis of Maximum Likelihood Estimates

                              Standard         Wald
Parameter     DF   Estimate      Error   Chi-Square   Pr > ChiSq

Intercept      1    -2.2804     0.2324      96.2774       <.0001
pred      1    1     5.9939     1.4850      16.2926       <.0001

Note here that these no suggestion in this part of the output that the Firth method was employed. That appears only at the very top of the voluminous output.

R
In R, we can use Heinze's logistf package, which includes the logistf() function. We'll make the same table as in SAS by constructing two vectors of length 240 using the c() and rep() functions.

pred = c(rep(1,20),rep(0,220))
outcome = c(rep(1,40),rep(0,200))
lr1 = glm(outcome ~ pred, binomial)
>summary(lr1)

Call:
glm(formula = outcome ~ pred, family = binomial)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-0.4366  -0.4366  -0.4366  -0.4366   2.1899  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept)   -2.3026     0.2345  -9.818   <2e-16 ***
pred          20.8687  1458.5064   0.014    0.989    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 216.27  on 239  degrees of freedom
Residual deviance: 134.04  on 238  degrees of freedom
AIC: 138.04

Number of Fisher Scoring iterations: 17

Note that the estimate differs slightly from what SAS reports. Here's a more plausible answer.

>library(logistf)
>lr2 = logistf(outcome ~ pred)
>summary(lr2)

logistf(formula = outcome ~ pred)

Model fitted by Penalized ML
Confidence intervals and p-values by Profile Likelihood 

                 coef  se(coef) lower 0.95 upper 0.95 Chisq p
(Intercept) -2.280389 0.2324057  -2.765427  -1.851695   Inf 0
pred         5.993961 1.4850029   3.947048  10.852893   Inf 0

Likelihood ratio test=78.95473 on 1 df, p=0, n=240
Wald test = 16.29196 on 1 df, p = 5.429388e-05

Covariance-Matrix:
            [,1]        [,2]
[1,]  0.05401242 -0.05401242
[2,] -0.05401242  2.20523358

Here, the estimates are nearly identical to SAS, but the standard errors differ.


Update:
Georg Heinze, author of the logistf() function, suggests the following two items.

First, in SAS, the model statement option clparm=pl will generate profile penalized likelihood confidence intervals, which should be similar to those from logistf(), It certainly makes sense to use confidence limits that more closely reflect the fitting method.

Second, in R, there is a weight option in both glm() and in logistf() that is similar to the weight statement in SAS. For example, the data used above could have been input and run as:

pred = c(1,0,0)
outcome = c(1,1,0)
weight=c(20,20,200)
lr1 = glm(outcome ~ pred, binomial, weights=weight)
lr2 = logistf(outcome ~ pred,  weights=weight)