Logistic regression, also called a logit model, is used to model dichotomous
outcome variables. In the logit model the log odds of the outcome is modeled as a linear
combination of the predictor variables.

Please note: The purpose of this page is to show how to use various data analysis commands.
It does not cover all aspects of the research process which researchers are expected to do. In
particular, it does not cover data cleaning and checking, verification of assumptions, model
diagnostics and potential follow-up analyses.

Examples

Example 1:  Suppose that we are interested in the factors
that influence whether a political candidate wins an election.  The
outcome (response) variable is binary (0/1);  win or lose.
The predictor variables of interest are the amount of money spent on the campaign, the
amount of time spent campaigning negatively, and whether the candidate is an
incumbent.
Example 2:  A researcher is interested in how variables, such as GRE (Graduate Record Exam scores),
GPA (grade
point average) and prestige of the undergraduate institution, effect admission into graduate
school. The outcome variable, admit/don’t admit, is binary.

Description of the data

For our data analysis below, we are going to expand on Example 2 about getting
into graduate school.  We have generated hypothetical data, which can be
obtained from our website by clicking on https://stats.idre.ucla.edu/wp-content/uploads/2016/02/binary.sas7bdat.
You can store this anywhere you like, but the syntax below assumes it has been
stored in the directory c:data.
This data set has a binary response (outcome, dependent) variable called admit,
which is equal to 1 if the individual was admitted to graduate school, and 0
otherwise. There are three
predictor variables: gre, gpa, and rank. We will treat the variables
gre and gpa as continuous.
The variable rank takes on the values 1 through 4. Institutions with a rank of 1 have the highest
prestige, while those with a rank of 4 have the lowest. We start out by looking
at some descriptive statistics.

proc means data="c:databinary";
  var gre gpa;
run;
                              The MEANS Procedure

Variable      N            Mean         Std Dev         Minimum         Maximum
-------------------------------------------------------------------------------
GRE         400     587.7000000     115.5165364     220.0000000     800.0000000
GPA         400       3.3899000       0.3805668       2.2600000       4.0000000
-------------------------------------------------------------------------------


proc freq data="c:databinary";
  tables rank admit admit*rank;
run;

                   The FREQ Procedure

                                 Cumulative    Cumulative
RANK    Frequency     Percent     Frequency      Percent
----------------------------------------------------------
   1          61       15.25            61        15.25
   2         151       37.75           212        53.00
   3         121       30.25           333        83.25
   4          67       16.75           400       100.00
   
                                  Cumulative    Cumulative
ADMIT    Frequency     Percent     Frequency      Percent
----------------------------------------------------------
   0         273       68.25           273        68.25
   1         127       31.75           400       100.00



                 Table of ADMIT by RANK

ADMIT     RANK

Frequency|
Percent  |
Row Pct  |
Col Pct  |       1|       2|       3|       4|  Total
---------+--------+--------+--------+--------+-
       0 |     28 |     97 |     93 |     55 |    273
         |   7.00 |  24.25 |  23.25 |  13.75 |  68.25
         |  10.26 |  35.53 |  34.07 |  20.15 |
         |  45.90 |  64.24 |  76.86 |  82.09 |
---------+--------+--------+--------+--------+-
       1 |     33 |     54 |     28 |     12 |    127
         |   8.25 |  13.50 |   7.00 |   3.00 |  31.75
         |  25.98 |  42.52 |  22.05 |   9.45 |
         |  54.10 |  35.76 |  23.14 |  17.91 |
---------+--------+--------+--------+--------+-
Total          61      151      121       67      400
            15.25    37.75    30.25    16.75   100.00

Analysis methods you might consider

Below is a list of some analysis methods you may have encountered.
Some of the methods listed are quite reasonable while others have either
fallen out of favor or have limitations.

Using the logit model

Below we run the logistic regression model. To model 1s rather than 0s, we
use the descending option. We do this because by default, proc logistic models
0s rather than 1s, in this case that would mean predicting the probability of
not getting into graduate school (admit=0) versus getting in (admit=1).
Mathematically, the models are equivalent, but conceptually, it probably makes
more sense to model the probability of getting into graduate school versus not getting in. The class statement tells SAS that rank is a
categorical variable. The param=ref option after the slash requests dummy
coding, rather than the default effects coding, for the levels of rank.
For more information on dummy versus effects coding in proc logistic, see
our FAQ page: In PROC LOGISTIC why aren’t the coefficients consistent with the odds ratios?.

proc logistic data="c:databinary" descending;
  class rank / param=ref ;
  model admit = gre gpa rank;
run;

The output from proc logistic is broken into several sections each of which is discussed below.

The LOGISTIC Procedure

                          Model Information

Data Set                      c:databinary     Written by SAS
Response Variable             ADMIT
Number of Response Levels     2
Model                         binary logit
Optimization Technique        Fisher's scoring


Number of Observations Read         400
Number of Observations Used         400


          Response Profile

 Ordered                      Total
   Value        ADMIT     Frequency

       1            1           127
       2            0           273

Probability modeled is ADMIT=1.


      Class Level Information

Class     Value     Design Variables

RANK      1          1      0      0
          2          0      1      0
          3          0      0      1
          4          0      0      0


                    Model Convergence Status

         Convergence criterion (GCONV=1E-8) satisfied.

         Model Fit Statistics

                             Intercept
              Intercept            and
Criterion          Only     Covariates

AIC             501.977        470.517
SC              505.968        494.466
-2 Log L        499.977        458.517
                
                
        Testing Global Null Hypothesis: BETA=0

Test                 Chi-Square       DF     Pr > ChiSq

Likelihood Ratio        41.4590        5         <.0001
Score                   40.1603        5         <.0001
Wald                    36.1390        5         <.0001 

Type 3 Analysis of Effects 

                        Wald 
Effect       DF   Chi-Square    Pr > ChiSq
GRE          1        4.2842        0.0385
GPA          1        5.8714        0.0154
RANK         3       20.8949        0.0001

              Analysis of Maximum Likelihood Estimates

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

Intercept       1     -5.5414      1.1381       23.7081        <.0001
GRE             1     0.00226     0.00109        4.2842        0.0385
GPA             1      0.8040      0.3318        5.8714        0.0154
RANK      1     1      1.5514      0.4178       13.7870        0.0002
RANK      2     1      0.8760      0.3667        5.7056        0.0169
RANK      3     1      0.2112      0.3929        0.2891        0.5908

             Odds Ratio Estimates

                  Point          95% Wald
Effect         Estimate      Confidence Limits

GRE               1.002       1.000       1.004
GPA               2.235       1.166       4.282
RANK 1 vs 4       4.718       2.080      10.701
RANK 2 vs 4       2.401       1.170       4.927
RANK 3 vs 4       1.235       0.572       2.668


Association of Predicted Probabilities and Observed Responses

Percent Concordant     69.3    Somers' D    0.386
Percent Discordant     30.7    Gamma        0.386
Percent Tied            0.0    Tau-a        0.168
Pairs                 34671    c            0.693

The output gives a test for the overall effect of rank, as well as coefficients
that describe the difference between the reference group (rank=4) and each of the other
three groups. We can also test for differences between the other levels of rank. For example,
we might want to test for a difference in coefficients for rank=2 and rank=3, that is, to
compare the odds of admission for students who attended a university with a rank of
2, to students who attended a
university with a rank of 3. We can test this type of hypothesis by adding a contrast
statement to the code for proc logistic. The syntax shown below is the
same as that shown above, except that it includes a contrast statement.
Following the word contrast, is the label that will appear in the output,
enclosed in single quotes (i.e., ‘rank 2 vs. rank 3’). This is followed by
the name of the variable we wish to test hypotheses about (i.e., rank),
and a vector that describes the desired comparison (i.e., 0 1 -1). In this
case the value computed is the difference between the coefficients for rank=2 and rank=3.
After the slash (i.e., / ) we use the estimate = parm option to
request that the estimate be the difference in coefficients. For
more information on use of the contrast statement, see our FAQ page:
How can I create contrasts with proc logistic?.

proc logistic data="c:databinary" descending;
  class rank / param=ref ;
  model admit = gre gpa rank;
  contrast 'rank 2 vs 3' rank 0 1 -1 / estimate=parm;
run;
             Contrast Test Results

                             Wald
Contrast         DF    Chi-Square    Pr > ChiSq

rank 2 vs 3       1        5.5052        0.0190


                      Contrast Estimation and Testing Results by Row

                                   Standard                                Wald
Contrast    Type      Row Estimate    Error  Alpha Confidence Limits Chi-Square Pr > ChiSq

rank 2 vs 3 PARM        1   0.6648   0.2833   0.05   0.1095   1.2200     5.5052     0.0190

Because the models are the same, most of the output produced by the above proc logistic
command is the same as before. The only difference is the additional output produced by the contrast
statement. Under the heading Contrast Test Results we see the label for the
contrast (rank 2 versus 3) along with its degrees of freedom, Wald chi-square
statistic, and p-value. Based on the p-value in this table we know that the coefficient for
rank=2 is significantly different from the coefficient for rank=3.
The second table, shows more detailed information, including the actual
estimate of the difference (under Estimate), it’s standard error, confidence
limits, test statistic, and p-value. We can see that the estimated
difference was 0.6648, indicating that having attended an undergraduate
institution with a rank of 2,
versus an institution with a rank of 3, increases the log odds of admission
by 0.67.

You can also use predicted probabilities to help you understand the model.
The contrast statement can be used to estimate predicted probabilities by
specifying estimate=prob. In the syntax below we use multiple contrast
statements to estimate the predicted probability of admission as gre
changes from 200 to 800 (in increments of 100). When estimating the predicted
probabilities we hold gpa constant at
3.39 (its mean), and rank at 2. The
term intercept followed by a 1 indicates that the intercept for
the model is to be included in estimate.

proc logistic data="c:databinary" descending;
  class rank / param=ref ;
  model admit = gre gpa rank;
  contrast 'gre=200' intercept 1 gre 200 gpa 3.3899 rank 0 1 0  / estimate=prob;
  contrast 'gre=300' intercept 1 gre 300 gpa 3.3899 rank 0 1 0  / estimate=prob;
  contrast 'gre=400' intercept 1 gre 400 gpa 3.3899 rank 0 1 0  / estimate=prob;
  contrast 'gre=500' intercept 1 gre 500 gpa 3.3899 rank 0 1 0  / estimate=prob;
  contrast 'gre=600' intercept 1 gre 600 gpa 3.3899 rank 0 1 0  / estimate=prob;
  contrast 'gre=700' intercept 1 gre 700 gpa 3.3899 rank 0 1 0  / estimate=prob;
  contrast 'gre=800' intercept 1 gre 800 gpa 3.3899 rank 0 1 0  / estimate=prob;
run;
           Contrast Test Results

                          Wald
Contrast      DF    Chi-Square    Pr > ChiSq

gre=200        1        9.7752        0.0018
gre=300        1       11.2483        0.0008
gre=400        1       13.3231        0.0003
gre=500        1       15.0984        0.0001
gre=600        1       11.2291        0.0008
gre=700        1        3.0769        0.0794
gre=800        1        0.2175        0.6409


                         Contrast Estimation and Testing Results by Row

                                    Standard                                    Wald
Contrast  Type       Row  Estimate     Error   Alpha   Confidence Limits  Chi-Square  Pr > ChiSq

gre=200   PROB         1    0.1844    0.0715    0.05    0.0817    0.3648      9.7752      0.0018
gre=300   PROB         1    0.2209    0.0647    0.05    0.1195    0.3719     11.2483      0.0008
gre=400   PROB         1    0.2623    0.0548    0.05    0.1695    0.3825     13.3231      0.0003
gre=500   PROB         1    0.3084    0.0443    0.05    0.2288    0.4013     15.0984      0.0001
gre=600   PROB         1    0.3587    0.0399    0.05    0.2847    0.4400     11.2291      0.0008
gre=700   PROB         1    0.4122    0.0490    0.05    0.3206    0.5104      3.0769      0.0794
gre=800   PROB         1    0.4680    0.0685    0.05    0.3391    0.6013      0.2175      0.6409

As with the previous example, we have omitted most of the proc logistic output, because it is the same as
before. The predicted probabilities are included in the column labeled Estimate in the second table
shown above. Looking at the estimates, we can see that the predicted
probability of being admitted is only 0.18 if one’s gre score is 200,
but increases to 0.47 if one’s gre score is 800, holding gpa at its mean
(3.39), and rank at 2.

Things to consider

References

Hosmer, D. and Lemeshow, S. (2000). Applied Logistic Regression (Second Edition).

New York: John Wiley and Sons, Inc.

Long, J. Scott (1997). Regression Models for Categorical and Limited Dependent Variables.

Thousand Oaks, CA: Sage Publications.

See also