Example 9.14: confidence intervals for logistic regression models

Recently a student asked about the difference between confint() and confint.default() functions, both available in the MASS library to calculate confidence intervals from logistic regression models. The following example demonstrates that they yield different results.

R

ds = read.csv("http://www.math.smith.edu/r/data/help.csv")
library(MASS)
glmmod = glm(homeless ~ age + female, binomial, data=ds)

> summary(glmmod)
Call:
glm(formula = homeless ~ age + female, family = binomial, data = ds)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.3600  -1.1231  -0.9185   1.2020   1.5466  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)  
(Intercept) -0.89262    0.45366  -1.968   0.0491 *
age          0.02386    0.01242   1.921   0.0548 .
female      -0.49198    0.22822  -2.156   0.0311 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 625.28  on 452  degrees of freedom
Residual deviance: 617.19  on 450  degrees of freedom
AIC: 623.19

Number of Fisher Scoring iterations: 4

> exp(confint(glmmod))
Waiting for profiling to be done...
                2.5 %    97.5 %
(Intercept) 0.1669932 0.9920023
age         0.9996431 1.0496390
female      0.3885283 0.9522567
> library(MASS)
> exp(confint.default(glmmod))
                2.5 %    97.5 %
(Intercept) 0.1683396 0.9965331
age         0.9995114 1.0493877
female      0.3909104 0.9563045

Why are they different? Which one is correct?

SAS

Fortunately the detailed documentation in SAS can help resolve this. The logistic procedure (section 4.1.1) offers the clodds option to the model statement. Setting this option to both produces two sets of CL, based on the Wald test and on the profile-likelihood approach. (Venzon, D. J. and Moolgavkar, S. H. (1988), “A Method for Computing Profile-Likelihood Based Confidence Intervals,” Applied Statistics, 37, 87–94.)

ods output cloddswald = waldcl cloddspl = plcl;
proc logistic data = "c:\book\help.sas7bdat"  plots=none;
class female (param=ref ref='0');
model homeless(event='1') = age female / clodds = both;
run;

 Odds Ratio Estimates and Profile-Likelihood Confidence Intervals

 Effect                Unit     Estimate     95% Confidence Limits

 AGE                 1.0000        1.024        1.000        1.050
 FEMALE 1 vs 0       1.0000        0.611        0.389        0.952


        Odds Ratio Estimates and Wald Confidence Intervals

 Effect                Unit     Estimate     95% Confidence Limits

 AGE                 1.0000        1.024        1.000        1.049
 FEMALE 1 vs 0       1.0000        0.611        0.391        0.956

Unfortunately, the default precision of the printout isn't quite sufficient to identify whether this distinction aligns with the differences seen in the two R methods. We get around this by using the ODS system to save the output as data sets (section A.7.1). Then we can print the data sets, removing the default rounding formats to find all of the available precision.

title "Wald CL";
proc print data=waldcl; format _all_; run;
title "PL CL";
proc print data=plcl; format _all_; run;

                              Wald CL  
                                     Odds
   Obs    Effect           Unit    RatioEst    LowerCL    UpperCL

    1     AGE                1      1.02415    0.99951    1.04939
    2     FEMALE 1 vs 0      1      0.61143    0.39092    0.95633
 
  
                               PL CL                            
                                     Odds
   Obs    Effect           Unit    RatioEst    LowerCL    UpperCL

    1     AGE                1      1.02415    0.99964    1.04964
    2     FEMALE 1 vs 0      1      0.61143    0.38853    0.95226

With this added precision, we can see that the confint.default() function in the MASS library generates the Wald confidence limits, while the confint() function produces the profile-likelihood limits. This also explains the confint() comment "Waiting for profiling to be done..." Thus neither CI from the MASS library is incorrect, though the profile-likelihood method is thought to be superior, especially for small sample sizes. Little practical difference is seen here.