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
http://www.ats.ucla.edu/stat/sas/webbooks/reg/default.htm.

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 <.0001 Score 58.6092 3 <.0001 Wald 39.8751 3 <.0001 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq 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 Effect Estimate Confidence Limits female 4.404 1.832 10.584 read 1.109 1.054 1.167 science 1.099 1.036 1.167 Association of Predicted Probabilities and Observed Responses Percent Concordant 85.6 Somers' D 0.714 Percent Discordant 14.2 Gamma 0.715 Percent Tied 0.2 Tau-a 0.279 Pairs 7791 c 0.857

### Model Information

Model Information Data Set^{a}WORK.LOGIT Response Variable^{b}honcomp Number of Response Levels^{c}2 Model^{d}binary logit Optimization Technique^{e}Fisher's scoring Number of Observations Read^{f}200 Number of Observations Used^{f}200 Response Profile Ordered Total Value^{g}honcomp^{g}Frequency^{h}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 Status^{j}Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion^{k}Only^{l}Covariates^{m}AIC 233.289 168.236 SC 236.587 181.430 -2 Log L 231.289 160.236 Testing Global Null Hypothesis: BETA=0 Test^{n}Chi-Square^{o}DF^{o}Pr > ChiSq^{o}Likelihood Ratio 71.0525 3 <.0001 Score 58.6092 3 <.0001 Wald 39.8751 3 <.0001

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*'s are the frequency values of the

_{i}*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 against the null hypothesis that at least one of the predictors'
regression coefficient is not 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 Parameter^{p}DF^{q}Estimate^{r}Error^{s}Chi-Square^{t}Pr > ChiSq^{t}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 Effect^{u}Estimate^{v}Confidence Limits^{w}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

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 Concordant^{x}85.6 Somers' D^{bb}0.714 Percent Discordant^{y}14.2 Gamma^{cc}0.715 Percent Tied^{z}0.2 Tau-a^{dd}0.279 Pairs^{aa}7791 c^{ee}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.