This page shows an example of logistic regression with footnotes
explaining the output. The data were collected on 200 high school students, with
measurements on various tests, including science, math, reading and social
studies. The response variable is high writing test score (honcomp),
where a writing score greater than or equal to 60 is considered high, and less
than 60 considered low; from which we explore its relationship with
gender (female), reading test score (read), and science test score
(science). The dataset used in this page can be downloaded from
SAS Web Books Regression with SAS.

data logit;
 set "c:temphsb2";
 honcomp = (write >= 60);
run;

proc logistic data= logit descending; 
 model honcomp = female read science;
run;

The LOGISTIC Procedure

              Model Information
Data Set                      WORK.LOGIT
Response Variable             honcomp
Number of Response Levels     2
Model                         binary logit
Optimization Technique        Fisher's scoring

Number of Observations Read         200
Number of Observations Used         200

          Response Profile
 Ordered                      Total
   Value      honcomp     Frequency
       1            1            53
       2            0           147
Probability modeled is honcomp=1.

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

         Model Fit Statistics
                             Intercept
              Intercept            and
Criterion          Only     Covariates
AIC             233.289        168.236
SC              236.587        181.430
-2 Log L        231.289        160.236

        Testing Global Null Hypothesis: BETA=0
Test                 Chi-Square       DF     Pr > ChiSq
Likelihood Ratio        71.0525        3          ChiSq
Intercept     1    -12.7772      1.9759       41.8176

Model Information

            Model Information
Data Seta                      WORK.LOGIT
Response Variableb             honcomp
Number of Response Levelsc     2
Modeld                         binary logit
Optimization Techniquee        Fisher's scoring

Number of Observations Readf         200
Number of Observations Usedf         200

          Response Profile
 Ordered                      Total
   Valueg     honcompg    Frequencyh
       1            1            53
       2            0           147
Probability modeled is honcomp=1.i

a. Data Set – This the data set used in this procedure.

b. Response Variable – This is the response variable in the logistic
regression.

c. Number of Response Levels – This is the number of levels our
response variable has.

d. Model – This is the type of regression model that was fit to our
data. The term logit and logistic are exchangeable.

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).

f. Number of Observations Read and Number of Observations Used –
This is the number of observations read and the number of observation used in
the analysis. The Number of Observations Used may be less than the
Number of Observations Read if there are missing values for any variables
in the equation. By default, SAS does a listwise deletion of incomplete cases.

g. Ordered Value and honcomp – Ordered value refers to
how SAS orders/models the levels of the dependent variable. When
we specified the descending option in the procedure statement, SAS treats
the levels of honcomp in a descending order (high to low), such that when
the logit regression coefficients are estimated, a positive coefficient
corresponds to a positive relationship for high write status, and a negative
coefficient has a negative relationship with high write status. Special
attention needs to be placed on the ordered value since it can lead to erroneous
interpretation. By default SAS models the 0s, whereas most other statistics
packages model the 1s. The descending option is necessary so that SAS
models the 1’s.

h. Total Frequency – This is the frequency distribution of the
response variable. Our response variable has 53 observations with a high write
score and 147 with a low write score.

i. Probability modeled is honcomp=1 – This is a note informing which
level of the response variable we are modeling. See superscript g for further
discussion of the descending option and its influence on which level of
the response variable is being modeled.

Model Fit Statistics

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

         Model Fit Statistics
                             Intercept
              Intercept            and
Criterionk         Onlyl    Covariatesm
AIC             233.289        168.236
SC              236.587        181.430
-2 Log L        231.289        160.236

        Testing Global Null Hypothesis: BETA=0
Testn                Chi-Squareo       DFo    Pr > ChiSqo
Likelihood Ratio        71.0525        3

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

k. Criterion – Underneath are various measurements used to assess the
model fit. The first two, Akaike Information Criterion (AIC) and Schwarz
Criterion (SC) are deviants 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.

    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 dependent variable and s is the number of predictors in the
model. AIC is used for the comparison of nonnested models on the same
sample. Ultimately, the model with the smallest AIC is
considered the best, although the AIC value itself is not meaningful.

    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 and the value itself is not meaningful..

    -2 Log L – This is negative two times the log-likelihood. The
-2 Log L is used in hypothesis tests for nested models and the value in
itself is not meaningful.

l. Intercept Only – This column refers to the respective criterion
statistics with no predictors in the model, i.e., just the response variable.

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

n. Test – These are three asymptotically equivalent Chi-Square tests.
They test the null hypothesis that all of the predictors’
regression coefficients are simultaneously equal to zero in the model. The difference between
them are where on the log-likelihood function they are evaluated. For further
discussion, see Categorical
Data Analysis, Second Edition, by Alan Agresti (pages 11-13).

    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) = 231.289-160.236 = 71.05, where L(null
model) refers to the Intercept Only model and L(fitted model)
refers to the Intercept and Covariates model.

    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.

    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.

o. Chi-Square, DF and Pr > ChiSq – These are the Chi-Square
test statistic, Degrees of Freedom (DF) and associated p-value (PR>ChiSq)
corresponding to the specific test 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. The DF defines
the distribution of the Chi-Square test statistics and is defined by the number
of predictors in the model. Typically,  PR>ChiSq is compared to a
specified alpha level, our willingness to accept a type I error, which is
often 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.

Analysis of Maximum Likelihood Estimates

             Analysis of Maximum Likelihood Estimates

                               Standard          Wald
Parameterp   DFq    Estimater      Errors   Chi-Squaret   Pr > ChiSqt
Intercept     1    -12.7772      1.9759       41.8176        <.0001
female        1      1.4825      0.4474       10.9799        0.0009
read          1      0.1035      0.0258       16.1467        <.0001
science       1      0.0948      0.0305        9.6883        0.0019

    
           Odds Ratio Estimates

              Point                95% Wald
Effectu     Estimatev      Confidence Limitsw

female        4.404       1.832      10.584
read          1.109       1.054       1.167
science       1.099       1.036       1.167

p. Parameter – Underneath are the predictor variables in the model and
the intercept.

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

r. Estimate – These are the binary logit regression estimates for the
Parameters in the model. The logistic regression model models the log
odds of a positive response (probability modeled is honcomp=1) as a linear
combination the predictor variables. This is written as
log[ p / (1-p) ] = b0 + b1*female + b2*read + b3 *science,

where p is the probability that honcomp is 1. For our model, we have,

log[ p / (1-p) ] = -12.78 + 1.48*female + 0.10*read +
0.09*science.

We can interpret the parameter estimates as follows: for a one
unit change in the predictor variable, the difference in log-odds for a positive
outcome is expected to change by the respective coefficient, given the other
variables in the model are held constant.

Intercept – This is the logistic regression
estimate when all variables in the model are evaluated at zero. For males (the
variable female evaluated at zero) with zero read and science
test scores, the log-odds for high write score is -12.777. Note that evaluating
read and science at zero is out of the range of plausible test
scores. If the test scores were mean-centered, the intercept would have a
natural interpretation: the expected log-odds for high write score for males
with an average read and science test score.

female – This is the estimated logistic regression
coefficient comparing females to males, given the other variables are held
constant in the model. The difference in log-odds is expected to be 1.4825 units
higher for females compared to males, while holding the other variables constant
in the model.

read – This is the estimate logistic regression
coefficient for a one unit change in read score, given the other
variables in the model are held constant. If a student were to increase her
read score by one point, her difference in log-odds for high write score is
expected to increase by 0.10 unit, given the other variables in the model are
held constant.

science – This is the estimate logistic regression
coefficient for a one unit change in science score, given the other
variables in the model are held constant. If a student were to increase her
science score by one point, the difference in log-odds for high write score
is expected to increase by 0.095 unit, given the other variables in the model
are held constant.

s. Standard Error – These are the standard errors of the individual
regression coefficients. They are used in both the 95% Wald Confidence Limits,
superscript w, and the Chi-Square test statistic, superscript t.

t. Chi-Square and Pr > ChiSq – These are the test statistics
and p-values, respectively, testing the null hypothesis that an individual
predictor’s regression coefficient is zero, given the other predictor variables are in the model. The Chi-Square test statistic is the squared
ratio of the Estimate to the Standard Error of the respective
predictor. The Chi-Square value follows a central Chi-Square
distribution with degrees of freedom given by DF, which is used to test
against the alternative hypothesis that the Estimate is not equal to
zero. The probability that a particular Chi-Square test statistic is as
extreme as, or more so, than what has been observed under the null hypothesis is
defined by Pr>ChiSq.

u. Effect – Underneath are the predictor variables that are
interpreted in terms of odds ratios.

v. Point Estimate – Underneath are the odds ratio corresponding to
Effect. The odds ratio is obtained by exponentiating the Estimate,
exp[Estimate]. The difference in the log of
two odds is equal to the log of the ratio of these two odds. The log of the ratio
of two odds is the log odds ratio. Hence, the interpretation of Estimate–the
coefficient was interpreted as the difference in log-odds–could also be
done in terms of log-odds ratio. When the Estimate is exponentiated, the
log-odds ratio becomes the odds ratio.
We can interpret the odds ratio as follows: for a one
unit change in the predictor variable, the odds ratio for a positive
outcome is expected to change by the respective coefficient, given the other
variables in the model are held constant.

w. 95% Wald Confidence Limits – This is the Wald Confidence Interval
(CI) of an individual odds ratio, given the other
predictors are in the model. For a given predictor variable with a level of 95%
confidence, we’d say that we are 95% confident that upon repeated trials, 95% of
the CI’s would include the “true” population odds ratio. The CI is equivalent to the Chi-Square test statistic: if the CI includes
one, we’d fail to reject the null hypothesis that a particular regression
coefficient equals zero and the odds ratio equals one, given the other predictors are in the model. An advantage
of a CI is that it is illustrative; it provides information on where the “true”
parameter may lie and the precision of the point estimate for the odds ratio.

Association of Predicted Probabilities and Observed Responses

Association of Predicted Probabilities and Observed Responses
Percent Concordantx     85.6    Somers' Dbb    0.714
Percent Discordanty     14.2    Gammacc        0.715
Percent Tiedz            0.2    Tau-add        0.279
Pairsaa                 7791    cee            0.857

x. Percent Concordant – A pair of observations with different observed
responses is said to be concordant if the observation with the lower ordered
response value (honcomp = 0) has a lower predicted mean score than the observation with the
higher ordered response value (honcomp = 1). See Pairs, superscript aa,
for what defines a pair.

y. 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.

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

aa. Pairs – This is the total number of distinct pairs in which one case
has an observed outcome different from the other member of the pair. In the Response Profile table in the Model Information section above, we see that there are 53 observations with honcomp=1 and 147 observations with honcomp=0. Thus the total number of pairs with different outcomes is 53*147=7791.

bb. 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, 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:
(85.6-14.2)/100 = 0.714.

cc. 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.

dd. Tau-a – Kendall’s Tau-a is a modification of Somer’s D that takes
into the account the difference between the number of possible paired
observations and the number of paired observations with a 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.

ee. c – c is equivalent to the well known measure ROC. c
ranges from 0.5 to 1, where 0.5 corresponds to the model randomly predicting the
response, and a 1 corresponds to the model perfectly discriminating the
response.