This page shows an example of probit regression analysis with footnotes
explaining the output in SAS. The data in this example were gathered on
undergraduates applying to graduate school and includes undergraduate GPAs, the
reputation of the school of the undergraduate (a topnotch indicator), the
students’ GRE score, and whether or not the student was admitted to graduate
school. Using this dataset ( https://stats.idre.ucla.edu/wp-content/uploads/2016/02/probit.sas7bdat ), we can predict admission to graduate school
using undergraduate GPA, GRE scores, and the reputation of the school of the
undergraduate.  Our outcome variable is binary, and we will use a probit
model.  Thus, our model will calculate a predicted probability of admission
based on our predictors. The probit model does so using the cumulative distribution
function of the standard
normal.

First, let us examine the dataset and our response variable. Our binary
outcome variable must be coded with zeros and ones, so we will include a
frequency of our outcome variable admit to check this.

data probit; 
  set "C:Dataprobit.sas7bdat"; 
run;

proc means data = probit;
  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 = probit;

  table topnotch admit;

run;

The FREQ Procedure

                                     Cumulative    Cumulative
TOPNOTCH    Frequency     Percent     Frequency      Percent
       0         335       83.75           335        83.75
       1          65       16.25           400       100.00

                                  Cumulative    Cumulative
ADMIT    Frequency     Percent     Frequency      Percent
    0         273       68.25           273        68.25
    1         127       31.75           400       100.00

We have now examined the data and the range of our predictors is acceptable
and our outcome variable is properly coded with zeroes and ones. To run a probit model in SAS, we will use proc logistic and specify probit as
our link function.  By default, SAS predicts the lowest value of the
outcome variable.  In this case, SAS would thus be predicting admit
= 0, or non-admission.  Because we are interested in predicting admission (admit
= 1), we have indicated that our model is to predict the “event” of admit
= 1.

proc logistic data = probit;
  model admit (event = '1') = gre topnotch gpa / link = probit;
run;

NOTE: It is also possible to run a probit regression in SAS using proc
probit, though it is more difficult to specify the predicted outcome as we
did with (event = ‘1’) using proc logistic. We can order the data
so that the predicted outcomes occur first in our dataset, then indicate order = data in
our proc probit function.

proc sort data = probit;
  by descending admit;
run;

proc probit data = probit order = data;
  class admit;
  model admit = gre topnotch gpa;
run;

The output below is from the proc logistic command.

Model Information
Data Set                      WORK.PROBIT
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            0           273
       2            1           127

Probability modeled is ADMIT=1.

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

Model Fit Statistics
                             Intercept
              Intercept            and
Criterion          Only     Covariates
AIC             501.977        485.887
SC              505.968        501.853
-2 Log L        499.977        477.887


Testing Global Null Hypothesis: BETA=0
Test                 Chi-Square       DF     Pr > ChiSq
Likelihood Ratio        22.0897        3         <.0001
Score                   21.5235        3         <.0001
Wald                    21.5263        3         <.0001

Analysis of Maximum Likelihood Estimates
                               Standard          Wald
Parameter    DF    Estimate       Error    Chi-Square    Pr > ChiSq
Intercept     1     -2.7978      0.6476       18.6630        <.0001
GRE           1     0.00152    0.000640        5.6661        0.0173
TOPNOTCH      1      0.2730      0.1803        2.2923        0.1300
GPA           1      0.4010      0.1948        4.2370        0.0396


Association of Predicted Probabilities and Observed Responses
Percent Concordant     63.9    Somers' D    0.283
Percent Discordant     35.6    Gamma        0.284
Percent Tied            0.5    Tau-a        0.123
Pairs                 34671    c            0.641

Model Information

Model Information
Data Seta                      WORK.PROBIT
Response Variableb             ADMIT
Number of Response Levelsc     2
Modeld                         binary probit
Optimization Techniquee        Fisher's scoring

a. Data Set – This is the SAS dataset analyzed with probit regression.

b. Response Variable – This is the outcome (a.k.a. dependent) variable in the probit
regression.

c. Number of Response Levels – This is the number of levels of the
dependent variable. Our dependent variable has two levels: 0 and 1.

d. Model – This is the model that SAS is fitting. Here, binary refers
to the outcome variable (the two levels of admit) and probit refers to
the distribution used in fitting the model.

e. Optimization Technique – This refers to the iterative method of
estimating the regression parameters. In SAS, the default is method is Fisher’s
scoring method, whereas in Stata, it is the Newton-Raphson algorithm. Both
techniques yield the same estimate for the regression coefficient; however, the
standard errors differ between the two methods. For further discussion, see

Regression Models for Categorical and Limited Dependent Variables by J.
Scott Long (page 56).

Response Profile

Response Profile
 Ordered                      Total
   Valuef        ADMITg    Frequencyh
       1            0           273
       2            1           127

Probability modeled is ADMIT=1.i

f. Ordered Value – This refers to how SAS
orders (e.g., models) the levels of the dependent variable, admit.

g. ADMIT – This lists the values in the outcome variable, admit.
We can see how these values are ordered by SAS by looking at the corresponding
ordered value (superscript f).

h. Total Frequency – This is the observed frequency distribution of
subjects in the dependent variable. Of our 400 subjects, 273 were not admitted (admit
= 0) and 127 wereadmitted(admit = 1).

i. Probability modeled is ADMIT = 1 – This indicates the value of our
outcome variable that is being modeled.  From this, we know to interpret
the predicted values from the probit model as the predicted probability of
admission (admit = 1).

Model Fit

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

Model Fit Statistics
                             Intercept
              Intercept            and
Criterionk         Onlyl    Covariatesm
AICn            501.977        485.887
SCo             505.968        501.853
-2 Log Lp       499.977        477.887

Testing Global Null Hypothesis: BETA=0
Testq                 Chi-Squarer       DFs     Pr > ChiSqt
Likelihood Ratiou        22.0897        3         <.0001
Scorev                   21.5235        3         <.0001
Waldw                    21.5263        3         <.0001

j. Model Convergence Status – This describes whether or not the maximum-likelihood
algorithm has converged and what kind of convergence criterion is used
for convergence. The default convergence criterion is the relative gradient
convergence criterion (GCONV), and the default precision is 10^-8.

k. Criterion – These are various measurements used to assess the model
fit. See superscripts n, o and p. The first two, Akaike Information Criterion (AIC)
and Schwarz Criterion (SC) are variants of negative two times the
Log-Likelihood (-2 Log L). AIC and SC penalize the
Log-Likelihood by the number of predictors in the model.

l. Intercept Only – This column refers to the respective Criterion
statistics with no predictors.

m. Intercept and Covariates – This column corresponds to the
respective Criterion statistics for the fitted model. A fitted model
includes all predictors and the intercept. We can compare the values
in this column with the criteria corresponding Intercept Only value to
assess model fit/significance.

n. AIC – This is the Akaike Information Criterion. It is calculated as
AIC = -2 Log L + 2((k-1) + s), where k is the number of
levels of the outcome variable and s is the number of predictors in the
model. AIC is used for the comparison of models from different samples or
nonnested models. Ultimately, the model with the smallest AIC is
considered the best.

o. SC – This is the Schwarz Criterion. It is defined as – 2 Log L + ((k-1)
+ s)*log(Σ f_i), where f_i‘s are the
frequency values of the i^th observation, and k and s
were defined previously. Like AIC, SC penalizes for the number of
predictors in the model and the smallest SC is most desirable.

p. -2 Log L – This is negative two times the log likelihood. The -2
Log L is used in hypothesis tests for nested models.

q. Test – These are three asymptotically equivalent Chi-Square tests.
They test against the null hypothesis that at least one of the predictors’
regression coefficient is not equal to zero in the model. The differences between
the three tests can be attributed to evaluating the log-likelihood function at
different points. For further
discussion, see Categorical
Data Analysis, Second Edition, by Alan Agresti (pages 11-13).

r. Chi-Square – This is the Chi-Square test statistic
corresponding to the specific test that all of the predictors are
simultaneously equal to zero.

s. DF – This is the number of degrees of freedom.  It determines
the distribution of the Chi-Square test statistics and is defined by the number
of predictors in the model. Our model includes three predictors, so DF =
3.

t. Pr > ChiSq – This is the probability the Chi-Square test statistic (or a more extreme test statistic) would be observed under the null hypothesis
that a particular predictor’s regression coefficient is zero, given that the
rest of the predictors are in the model. For a given alpha level, Pr >
ChiSq determines whether or not the null hypothesis
can be rejected. If Pr > ChiSq is less than alpha, then the null hypothesis can be rejected and the parameter
estimate is considered statistically significant at that alpha level.

u. Likelihood Ratio – This is the Likelihood Ratio (LR) Chi-Square
test that at least one of the predictors’ regression coefficient is not equal to
zero in the model. The LR Chi-Square statistic can be calculated by  -2 Log
L(null model) – 2 Log L(fitted model) = 499.977 – 477.887 = 22.0897, where
L(null model) refers to the Intercept Only model and L(fitted model)
refers to the Intercept and Covariates model.

v. Score – This is the Score Chi-Square Test that at least one of the
predictors’ regression coefficient is not equal to zero in the model.

w. Wald – This is the Wald Chi-Square Test that at least one of the
predictors’ regression coefficient is not equal to zero in the model.

Parameter Estimates

Analysis of Maximum Likelihood Estimates
                               Standard          Wald
Parameterx   DFy   Estimatez      Erroraa  Chi-Squarebb   Pr > ChiSqcc
Intercept     1     -2.7978      0.6476       18.6630        <.0001
GRE           1     0.00152    0.000640        5.6661        0.0173
TOPNOTCH      1      0.2730      0.1803        2.2923        0.1300
GPA           1      0.4010      0.1948        4.2370        0.0396


Association of Predicted Probabilities and Observed Responses
Percent Concordantdd     63.9    Somers' Dhh    0.283
Percent Discordantee     35.6    Gammaii        0.284
Percent Tiedff            0.5    Tau-ajj        0.123
Pairsgg                 34671    ckk            0.641

x. Parameter – These refer to the independent variables in the model
as well as intercepts (a.k.a. constants) for the adjacent levels of the
dependent variable.

y. DF – This column gives the degrees of freedom corresponding to the
Parameter. For each Parameter estimated in the model, one DF
is required, and the DF defines the Chi-Square distribution to test
whether the individual regression coefficient is zero given the other variables
are in the model.

z. Estimate – These are the regression coefficients. The predicted
probability of admission can be calculated using these coefficients.  For a
given record, the predicted probability of admission is

where F is the cumulative distribution function of the
standard normal. However, interpretation of the coefficients in probit
regression is not as straightforward as the interpretations of coefficients in
linear regression or logit regression.  The increase in probability
attributed to a one-unit increase in a given predictor is dependent both on the
values of the other predictors and the starting value of the given predictors.
For example, if we hold gre and topnotch constant at zero, the one
unit increase in gpa from 2 to 3 has a different effect than the one unit
increase from 3 to 4 (note that the probabilities do not change by a common
difference or common factor):

and the effects of these one unit increases are different if we
hold gre and topnotch constant at their respective means instead
of zero:

However, there are limited ways in which we can interpret the
individual regression coefficients.  A positive coefficient mean that an increase in the
predictor leads to an increase in the predicted probability.  A negative
coefficient means that an increase in the predictor leads to a decrease in the
predicted probability.

    Intercept – The constant term is -2.797884.  This
means that if all of the predictors (gre, topnotch and gpa) are evaluated at
zero, the predicted probability of admission is F(-2.797884) = 0.002571929. So,
as expected, the predicted probability of a student with a GRE score of zero and
a GPA of zero from a non-topnotch school has an extremely low predicted
probability of admission.

gre – The coefficient of gre is 0.0015244.
This means that an increase in GRE score increases the predicted probability of
admission.

topnotch – The coefficient of topnotch is
0.2730334. This means attending a top notch institution as an undergraduate
increases the predicted probability of admission.

gpa – The coefficient of gpa is 0.4009853.
This means that an increase in GPA increases the predicted probability of
admission.

aa. Standard Error – These are the standard errors of the individual
regression coefficients. They are used in the calculation of the Wald
Chi-Square test statistic, superscript bb.

bb. Wald Chi-Square – This is the Wald test statistic for the
hypothesis test that an individual predictor’s regression coefficient is zero
given the rest of the predictors are in the model. The Wald Chi-Square
test statistic is the squared ratio of the Estimate to the Standard
Error of the respective predictor. The probability that a particular Wald
Chi-Square test statistic is as extreme as, or more so, than what has been
observed under the null hypothesis is given by Pr > ChiSq.

cc. Pr > ChiSq – This is the p-value corresponding to the Wald
Chi-Square test statistic that all of the predictors are simultaneously
equal to zero. We are testing the probability (Pr > ChiSq) of observing
a Chi-Square statistic as extreme as, or more so, than the observed one
under the null hypothesis; the null hypothesis is that all of the regression
coefficients in the model are equal to zero. Typically,  Pr > ChiSq is
compared to a specified alpha level, our willingness to accept a type I error,
which is typically set at 0.05 or 0.01. The small p-value from the all three
tests would lead us to conclude that at least one of the regression
coefficients in the model is not equal to zero

    The Wald
Chi-Square test statistic for the Intercept is18.6630 with an
associated p-value <.0001. If we set our alpha level to 0.05, we would reject
the null hypothesis and conclude that the model intercept has been found to be
statistically different from zerogiven gre, topnotch and gpa
are in the model.

The Wald Chi-Square test statistic for the predictor gre is5.6661 with an associated p-value of 0.0173. If we set our alpha level to
0.05, we would reject the null hypothesis and conclude that the regression
coefficient for gre has been found to be statistically different from
zero in estimating gre given topnotch and gpa are in the
model.

The Wald Chi-Square test statistic for the predictor topnotch
is2.2923 with an associated p-value of 0.1300. If we set our alpha level
to 0.05, we would fail to reject the null hypothesis and conclude that the
regression coefficient for topnotch has not been found to be
statistically different from zero in estimating topnotch given gre
and gpa are in the model.

The Wald Chi-Square test statistic for the predictor gpa is4.2370 with an associated p-value of 0.0396. If we set our alpha level to
0.05, we would reject the null hypothesis and conclude that the regression
coefficient for gpa has been found to be statistically different from
zero in estimating gpa given topnotch and gpa are in the
model.

dd. Percent Concordant – A pair of observations with different
observed responses is said to be concordant if the observation with the lower
ordered response value has a lower predicted mean score than the observation
with the higher ordered response value.

ee. Percent Discordant – If the observation with the lower ordered
response value has a higher predicted mean score than the observation with the
higher ordered response value, then the pair is discordant.

ff. Percent Tied – If a pair of observations with different responses
is neither concordant nor discordant, it is a tie.

gg. Pairs – This is the total number of distinct pairs with one case
having a positive response (admit = 1) and the other having a negative
response (admit = 0). The total number of ways the 400 observations can
be paired up (excluding be matched up with themselves) is 400(399)/2 = 79,800.
Of the 79,800 possible pairings, 34,671 have different values on the response
variable and 79,800 – 34,671 = 45,129 have the same value on the response
variable.

hh. Somers’ D – Somer’s D is used to determine the strength and
direction of relation between pairs of variables. Its values range from -1.0
(all pairs disagree) to 1.0 (all pairs agree). It is defined as (n_c-n_d)/t
where n_c is the number of pairs that are concordant, and n_d
the number of pairs that are discordant, and t is the number of total number of
pairs with different responses. In our example, it equals the difference between
the percent concordant and the percent discordant divided by 100:
(63.9-35.6)/100 = 0.283.

ii. Gamma – The Goodman-Kruskal Gamma method does not penalize for
ties on either variable. Its values range from -1.0 (no association) to 1.0
(perfect association). Because it does not penalize for ties, its value will
generally be greater than the values for Somer’s D.

jj. Tau-a – Kendall’s Tau-a is a modification of Somer’s D to take
into the account the difference between the number of possible paired
observations and the number of paired observations with different response. It
is defined to be the ratio of the difference between the number of concordant
pairs and the number of discordant pairs to the number of possible pairs (2(n_c-n_d)/(N(N-1)).
Usually Tau-a is much smaller than Somer’s D since there would be many paired
observations with the same response.

kk. c – Another measure of rank correlation of ordinal variables. It
ranges from 0 to (no association) to 1 (perfect association). It is a variant of
Somer’s D index.