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:\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.
- Logistic regression, the focus of this page.
- Probit regression. Probit analysis will produce results similar tologistic regression. The choice of probit versus logit depends largely onindividual preferences.
- OLS regression. When used with a binary response variable, this model is knownas a linear probability model and can be used as a way todescribe conditional probabilities. However, the errors (i.e., residuals)
from the linear probability model violate the homoskedasticity andnormality of errors assumptions of OLSregression, resulting in invalid standard errors and hypothesis tests. Fora more thorough discussion of these and other problems with the linearprobability model, see Long (1997, p. 38-40). - Two-group discriminant function analysis. A multivariate method for
dichotomous outcome variables. - Hotelling’s T2. The 0/1 outcome is turned into thegrouping variable, and the former predictors are turned into outcomevariables. This will produce an overall test of significance but will notgive individual coefficients for each variable, and it is unclear the extentto which each “predictor” is adjusted for the impact of the other”predictors.”
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:\data\binary" 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:\data\binary 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.
- The first part of the above output tells us the file being analyzed (c:\data\binary) and the number of
observations used. We see that all 400 observations in our data set were
used in the analysis (fewer observations would have been used if any of our
variables had missing values). - We also see that SAS is modeling admit using a binary logit model and
that the probability that of admit = 1 is being modeled. (If we omitted the descending option, SAS would model
admit being 0 and our results would be completely reversed.)
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
- The portion of the output labeled Model Fit Statistics describes and tests the overall fit of the model.
The -2 Log L (499.977) can be used in comparisons of nested models, but we
won’t show an example of that here. - In the next section of output, the likelihood ratio chi-square of 41.4590 with a p-value of 0.0001
tells us that our model as a whole fits significantly better than an empty
model. The Score and Wald tests are asymptotically equivalent tests of the
same hypothesis tested by the likelihood ratio test, not surprisingly, these
tests also indicate that the model is statistically significant. - The section labeled Type 3 Analysis of Effects, shows the hypothesis
tests for each of the variables in the model individually. The chi-square
test statistics and associated p-values shown in the table indicate that
each of the three variables in the model significantly improve the model fit. For gre and gpa, this test duplicates the test of the
coefficients shown below. However, for class variables (e.g., rank), this table gives the
multiple degree of freedom test for the overall effect of the variable.
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
- The above table shows the coefficients (labeled Estimate), their
standard errors (error), the Wald Chi-Square statistic, and associated
p-values. The coefficients for gre, and gpa are statistically
significant, as are the terms for rank=1 and rank=2 (versus
the omitted category rank=4). The logistic
regression coefficients give the change in the log odds of the outcome for a
one unit increase in the predictor variable.- For every one unit change in
gre, the log odds of admission (versus non-admission) increases by 0.002. - For a one unit increase in gpa, the log odds of being admitted to graduate school increases by 0.804.
- The coefficients for the categories of rank have a slightly different interpretation. For example,
having attended an undergraduate institution with a rank of 1,
versus an institution with a rank of 4, increases the log odds of
admission by 1.55.
- For every one unit change in
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 first table above gives the coefficients as odds ratios. An odds
ratio is the exponentiated coefficient, and can be interpreted as the multiplicative
change in the odds for a one unit change in the predictor variable. For example,
for a one unit
increase in gpa, the odds of being admitted to graduate school (versus not being admitted)
increase by a factor of 2.24. For more information on interpreting odds ratios
see our FAQ page: How do I interpret odds ratios
in logistic regression?
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:\data\binary" 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:\data\binary" 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
- Empty cells or small cells: You should check for empty or smallcells by doing a crosstab between categorical predictors and the outcome
variable. If a cell has very few cases (a small cell), the model may
become unstable or it might not run at all. - Separation or quasi-separation (also called perfect prediction): A
condition in which the outcome does not vary at some levels of the
independent variables. See our page
FAQ: What is complete or quasi-complete separation in logistic/probit
regression and how do we deal with them? for information on models with
perfect prediction. - Sample size: Both logit and probit models require more cases than
OLS regression because they use maximum likelihood estimation techniques. It
is sometimes possible to estimate models for binary outcomes in datasets
with only a small number of cases using exact logistic regression (available
with the exact option in proc logistic). For more information see our data analysis example for
exact logistic regression. It is also important to keep in mind that
when the outcome is rare, even if the overall dataset is large, it can be
difficult to estimate a logit model. - Pseudo-R-squared: Many different measures of psuedo-R-squared
exist. They all attempt to provide information similar to that provided by
R-squared in OLS regression; however, none of them can be interpreted
exactly as R-squared in OLS regression is interpreted. For a discussion of
various pseudo-R-squareds see Long and Freese (2006) or our FAQ page
What are
pseudo R-squareds? - Diagnostics: The diagnostics for logistic regression are different
from those for OLS regression. For a discussion of model diagnostics for
logistic regression, see Hosmer and Lemeshow (2000, Chapter 5). Note that
diagnostics done for logistic regression are similar to those done for probit regression. - By default, proc logistic models the probability of the lower
valued category (0 if your variable is coded 0/1), rather than the higher
valued category.
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
- How do I interpret odds ratios in
logistic regression? - Why are my logistic results
reversed? - SAS Annotated Output:
proc logistic - SAS Seminar: Logistic
Regression in SAS - AS Textbook Examples:
Applied Logistic Regression (Second
Edition) by David Hosmer and Stanley Lemeshow - A Tutorial on Logistic Regression
(PDF)
by Ying So, from SUGI Proceedings, 1995, courtesy of
SAS). - Some Issues in Using PROC LOGISTIC
for Binary Logistic Regression (PDF)
by David C. Schlotzhauer, courtesy of SAS). - Logistic Regression Examples Using the SAS System by SAS Institute
- Logistic Regression Using the SAS System: Theory and Application by
Paul D. Allison