# Example 9.14: confidence intervals for logistic regression models

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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.

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