SAS Data Analysis Examples
Probit Regression

Probit regression, also called a probit model, is used to model dichotomous or binary outcome variables. In the probit model, the inverse standard normal distribution of the probability is modeled as a linear combination of the predictors.

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 binary.sas7bdat. You can store this anywhere you like, but our the example syntax assumes it has been stored in c:\data. This data set has a binary response (outcome, dependent) variable called admit. 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 the data.
proc means data="c:\data\binary";
  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:\data\binary";
  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 Probit Model

There are multiple ways to run a probit model in SAS, this page uses proc logistic with link=probit on the model statement. Alternative methods not shown on this page include using proc probit, or proc genmod. The advantage of running the model using proc logistic is that it is easier to specify the ordering of the categories than it is in proc probit. One possible advantage of using proc probit is that it will produce graphs that may help you interpret and explain the model.

Below we run the probit regression model using proc logistic. 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 parm=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?. The model statement specifies that we are modeling the outcome admit as a function of the predictor variables gre, gpa, and rank. The link=probit option fits a probit model rather than the default logit model.

proc logistic data=data.binary descending;
  class rank / param=ref ;
  model admit = gre gpa rank /link=probit;
run;

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

                       The LOGISTIC Procedure

                         Model Information

Data Set                      DATA.BINARY          Written by SAS
Response Variable             ADMIT
Number of Response Levels     2
Model                         binary probit
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        470.413
       SC              505.968        494.362
       -2 Log L        499.977        458.413


        Testing Global Null Hypothesis: BETA=0

Test                 Chi-Square       DF     Pr > ChiSq

Likelihood Ratio        41.5633        5         <.0001
Score                   40.1603        5         <.0001
Wald                    38.6596        5         <.0001


               Type 3 Analysis of Effects

                               Wald
       Effect      DF    Chi-Square    Pr > ChiSq

       GRE          1        4.4767        0.0344
       GPA          1        5.8685        0.0154
       RANK         3       21.3611        <.0001
              Analysis of Maximum Likelihood Estimates

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

Intercept       1     -3.3225      0.6633       25.0872        <.0001
GRE             1     0.00138    0.000650        4.4767        0.0344
GPA             1      0.4777      0.1972        5.8685        0.0154
RANK      1     1      0.9359      0.2453       14.5606        0.0001
RANK      2     1      0.5205      0.2109        6.0904        0.0136
RANK      3     1      0.1237      0.2240        0.3053        0.5806
Association of Predicted Probabilities and Observed Responses

      Percent Concordant     69.1    Somers' D    0.385
      Percent Discordant     30.6    Gamma        0.387
      Percent Tied            0.4    Tau-a        0.167
      Pairs                 34671    c            0.693

The table above gives information about the relationship between the predicted probabilities from our model, and the actual outcomes in our data.

The output shown above 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. 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 uses the 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 (i.e. 0 1 -1) that describes the desired comparison. 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 the contrast statement, see our FAQ page How can I create contrasts with proc logistic?.

proc logistic data=data.binary descending;
  class rank / param=ref ;
  model admit = gre gpa rank /link=probit;
  contrast 'rank 2 vs. 3' rank 0 1 -1 / estimate=parm;
run;


                       Contrast Rows Estimation and Testing Results

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

rank 2 vs. 3 PARM        1   0.3967   0.1681   0.05   0.0673   0.7261     5.5725     0.0182

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 (shown above). Under the heading Contrast Test Results we see the label for the contrast (rank 2 vs 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 0.3967, indicating that having attended an undergraduate institution with a rank of 2, versus an institution with a rank of 3, increases the z-score by 0.4.

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 word intercept followed by a 1 indicates that the intercept for the model is to be included in estimate.

proc logistic data=data.binary descending;
  class rank / param=ref ;
  model admit = gre gpa rank /link=probit; 	
  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=800        1        0.2452        0.6205


                           Contrast Rows Estimation and Testing Results

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

 gre=200   PROB         1    0.1821    0.0746    0.05    0.0720    0.3615     10.3375      0.0013
 gre=300   PROB         1    0.2206    0.0662    0.05    0.1136    0.3698     11.8864      0.0006
 gre=400   PROB         1    0.2635    0.0552    0.05    0.1676    0.3816     14.0168      0.0002
 gre=500   PROB         1    0.3103    0.0442    0.05    0.2296    0.4014     15.6564      <.0001
 gre=600   PROB         1    0.3604    0.0396    0.05    0.2861    0.4404     11.4014      0.0007
 gre=700   PROB         1    0.4130    0.0480    0.05    0.3222    0.5087      3.1785      0.0746
 gre=800   PROB         1    0.4672    0.0661    0.05    0.3415    0.5963      0.2452      0.6205

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 in the output. 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. & Lemeshow, S. (2000). Applied Logistic Regression (Second Edition). New York: John Wiley & Sons, Inc.

Long, J. Scott (1997). Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage Publications.

See Also

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