3.2. Confidence Intervals

It’s standard practice in social science journals to report only point estimates and hypothesis tests for the coefficients. Most statisticians, on the other hand, hold that confidence intervals give a better picture of the sampling variability in the estimates.

Conventional confidence intervals for logit regression coefficients are easily computed by hand. For an approximate 95% confidence interval around a coefficient, simply add and subtract the standard error multiplied by 1.96 (2.00 is close enough for most purposes). But you can save yourself the trouble by asking LOGISTIC or GENMOD to do it for you. In LOGISTIC, the option in the MODEL statement for conventional (Wald) confidence intervals is CLPARM=WALD. In GENMOD, the corresponding MODEL option is WALDCI. The default is a 95% confidence interval. To change that to a 90% interval, put the option ALPHA=.10 in the MODEL statement.

There’s another method, called profile likelihood confidence intervals, that may produce better approximations, especially in smaller samples. This method involves an iterative evaluation of the likelihood function and produces intervals that are not generally symmetric around the coefficient estimate. GENMOD computes profile likelihood confidence intervals with the MODEL option LRCI (likelihood ratio confidence intervals). In LOGISTIC, the model option is CLPARM=PL (for profile likelihood). The profile likelihood method is computationally intensive so you may want to use it sparingly for large samples.

Here’s an example of both kinds of confidence intervals using LOGISTIC with the death-penalty data set:

PROC LOGISTIC DATA=my.penalty DES;
  MODEL death = blackd whitvic culp / CLPARM=BOTH;
RUN;

Besides the usual results, we get the numbers shown in Output 3.1. The intervals produced by the two methods are very similar but not identical.

Output 3.1. Confidence Intervals Produced by LOGISTIC

Parameter Estimates and 95% Confidence
Intervals

                                 Wald
                           Confidence Limits
            Parameter
Variable     Estimate       Lower       Upper

INTERCPT      -5.2182     -7.0407     -3.3957
BLACKD         1.6360      0.4689      2.8031
WHITVIC        0.8477     -0.2427      1.9381
CULP           1.2709      0.8863      1.6555

                           Profile Likelihood
                           Confidence Limits
            Parameter
Variable     Estimate       Lower       Upper

INTERCPT      -5.2182     -7.2485     -3.5710
BLACKD         1.6360      0.5289      2.8890
WHITVIC        0.8477     -0.2190      1.9811
CULP           1.2709      0.9187      1.6953

The output from GENMOD is sufficiently different that it’s worth examining as well. The program is:

PROC GENMOD DATA=my.penalty;
  MODEL death = blackd whitvic culp / D=B WALDCI LRCI;
RUN;

In the top half of Output 3.2, we see the Wald confidence intervals, labeled “Normal” because they are based on the normal distribution. Unfortunately, variable names aren’t given so you have to remember that PRM1 (parameter 1) is the intercept, PRM2 is the coefficient for BLACKD, and so on.

In the lower portion of the output, we find the profile likelihood results. The first column of numbers is the one to pay attention to. The remaining columns can usually be ignored, but here’s a brief explanation. When GENMOD is evaluating the profile likelihood function, it varies not only the parameter of interest but also the other parameters in the model. The additional columns tell us what values the other parameters took on when the parameter of interest was at its lower (or upper) limit.

Output 3.2. Confidence Intervals Produced by GENMOD

           Normal Confidence Intervals For Parameters

                  Two-Sided Confidence Coefficient: 0.9500
                  Parameter      Confidence Limits


                  PRM1          Lower       -7.0408
                  PRM1          Upper       -3.3956
                  PRM2          Lower        0.4689
                  PRM2          Upper        2.8031
                  PRM3          Lower       -0.2427
                  PRM3          Upper        1.9381
                  PRM4          Lower        0.8863
                  PRM4          Upper        1.6555


         Likelihood Ratio Based Confidence Intervals For Parameters

                  Two-Sided Confidence Coefficient: 0.9500
Param     Confidence Limits             Parameter Values
                               PRM1       PRM2       PRM3       PRM4

PRM1   Lower      -7.2485    -7.2485     2.6226     1.5941     1.6140
PRM1   Upper      -3.5710    -3.5710     0.8248     0.1857     1.0038
PRM2   Lower       0.5289    -4.0304     0.5289     0.4032     1.1420
PRM2   Upper       2.8890    -6.8385     2.8890     1.3096     1.5126
PRM3   Lower      -0.2190    -4.2409     1.1922    -0.2190     1.2389
PRM3   Upper       1.9811    -6.6011     2.1640     1.9811     1.4047
PRM4   Lower       0.9187    -3.9489     1.1767     0.6457     0.9187
PRM4   Upper       1.6953    -6.8361     2.3064     1.1261     1.6953

We’ve seen how to get confidence intervals for the β parameters in a logistic regression. What about confidence intervals for the odds ratios? LOGISTIC can compute them for you. In the MODEL statement, the CLODDS=WALD option requests the conventional Wald confidence intervals, and the CLODDS=PL option requests the profile likelihood intervals. Output 3.3 shows the results for the model just estimated. The “Unit” column indicates how much each independent variable is incremented to produce the estimated odds ratio. The default is 1 unit. For the variable CULP, each 1-point increase on the culpability scale multiplies the odds of a death sentence by 3.564.

Output 3.3. Odds Ratio Confidence Intervals in LOGISTIC

  Conditional Odds Ratios and 95% Confidence Intervals

                                     Profile Likelihood
                                      Confidence Limits
                            Odds
Variable        Unit       Ratio       Lower       Upper

BLACKD        1.0000       5.135       1.697      17.976
WHITVIC       1.0000       2.334       0.803       7.251
CULP          1.0000       3.564       2.506       5.448


  Conditional Odds Ratios and 95% Confidence Intervals

                                            Wald
                                      Confidence Limits
                            Odds
Variable        Unit       Ratio       Lower       Upper

BLACKD        1.0000       5.135       1.598      16.495
WHITVIC       1.0000       2.334       0.784       6.945
CULP          1.0000       3.564       2.426       5.236

If you want odds ratios for different increments, you can easily calculate them by hand. If O is the odds ratio for a 1-unit increment, O^k is the odds ratio for a k-unit increment. If that’s too much trouble, you can use the UNITS statement to produce “customized” odds ratios. For example, to get the odds ratio for a 2-unit increase in CULP, include the following statement in the LOGISTIC procedure:

UNITS culp=2 / DEFAULT=1;

The DEFAULT option tells SAS to print odds ratios and their confidence intervals for a one-unit increase in each of the other variables in the model.

Because GENMOD doesn’t compute odds ratios, it won’t produce confidence intervals for them. However, they’re easy to get with a hand calculator. The first step is to get confidence intervals for the original parameters. Let β be a parameter estimate, and let U and L be the upper and lower confidence limits for this parameter. The odds ratio estimate is e^β. The upper and lower odds ratio limits are e^U and e^L. This works for either conventional confidence intervals or profile likelihood confidence intervals. If you want confidence intervals for the transformation 100(e^β–1), discussed in Section 2.9, just substitute the upper and lower limits for β in this formula.