Version info: Code for this page was tested in
SAS 9.3.

Multinomial logistic regression is for modeling nominal
outcome variables, in which the log odds of the outcomes are 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 of multinomial logistic regression

Example 1. People’s occupational choices might be influenced
by their parents’ occupations and their own education level. We can study the
relationship of one’s occupation choice with education level and father’s
occupation.  The occupational choices will be the outcome variable which
consists of categories of occupations.

Example 2. A biologist may be interested in food choices that alligators make. Adult alligators might have
difference preference than young ones. The outcome variable here will be the
types of food, and the predictor variables might be the length of the alligators
and other environmental variables.

Example 3. Entering high school students make program choices among general program,
vocational program and academic program. Their choice might be modeled using
their writing score and their social economic status.

Description of the data

For our data analysis example, we will expand the third example using the
hsbdemo data set. You can download the data
here .

proc contents data = "c:hsbdemo";
run;

The CONTENTS Procedure

Data Set Name        c:datahsbdemo                          Observations          200
Member Type          DATA                                     Variables             13
Engine               V9                                       Indexes               0
Created              Thursday, August 29, 2013 09:42:59 AM    Observation Length    40
Last Modified        Thursday, August 29, 2013 09:42:59 AM    Deleted Observations  0
Protection                                                    Compressed            NO
Data Set Type                                                 Sorted                YES
Label                Written by SAS

Data Representation  WINDOWS_64
Encoding             wlatin1  Western (Windows)


         Engine/Host Dependent Information

Data Set Page Size          4096
Number of Data Set Pages    3
First Data Page             1
Max Obs per Page            101
Obs in First Data Page      42
Number of Data Set Repairs  0
Filename                    c:datahsbdemo.sas7bdat
Release Created             9.0301M1
Host Created                X64_7PRO


     Alphabetic List of Variables and Attributes

 #    Variable    Type    Len    Label

12    AWARDS      Num       3
13    CID         Num       3
 2    FEMALE      Num       3
11    HONORS      Num       3    honores eng
 1    ID          Num       4
 8    MATH        Num       3    math score
 5    PROG        Num       3    type of program
 6    READ        Num       3    reading score
 4    SCHTYP      Num       3    type of school
 9    SCIENCE     Num       3    science score
 3    SES         Num       3
10    SOCST       Num       3    social studies score
 7    WRITE       Num       3    writing score


 Sort Information

Sortedby       PROG
Validated      YES
Character Set  ANSI

The data set contains variables on 200 students. The outcome variable is prog, program type. The predictor variables
are social economic status, ses,  a three-level categorical variable
and writing score, write, a continuous variable. Let’s start with
getting some descriptive statistics of the
variables of interest.

proc freq data = "c:hsbdemo";
tables prog*ses / chisq norow nocol nofreq;
run;

The FREQ Procedure

Table of PROG by SES

PROG(type of program)     SES

Percent |       1|       2|       3|  Total
--------+--------+--------+--------+
      1 |   8.00 |  10.00 |   4.50 |  22.50
--------+--------+--------+--------+
      2 |   9.50 |  22.00 |  21.00 |  52.50
--------+--------+--------+--------+
      3 |   6.00 |  15.50 |   3.50 |  25.00
--------+--------+--------+--------+
Total         47       95       58      200
           23.50    47.50    29.00   100.00


Statistics for Table of PROG by SES

Statistic                     DF       Value      Prob
------------------------------------------------------
Chi-Square                     4     16.6044    0.0023
Likelihood Ratio Chi-Square    4     16.7830    0.0021
Mantel-Haenszel Chi-Square     1      0.0598    0.8068
Phi Coefficient                       0.2881
Contingency Coefficient               0.2769
Cramer's V                            0.2037

Sample Size = 200

proc sort data = "c:hsbdemo";
by prog;
run;

proc means data = "c:hsbdemo";
var write;
by prog;
run;

type of program=1

The MEANS Procedure

              Analysis Variable : WRITE writing score

  N            Mean         Std Dev         Minimum         Maximum
-------------------------------------------------------------------
 45      51.3333333       9.3977754      31.0000000      67.0000000
-------------------------------------------------------------------


type of program=2

              Analysis Variable : WRITE writing score

  N            Mean         Std Dev         Minimum         Maximum
-------------------------------------------------------------------
105      56.2571429       7.9433433      33.0000000      67.0000000
-------------------------------------------------------------------


type of program=3

              Analysis Variable : WRITE writing score

  N            Mean         Std Dev         Minimum         Maximum
-------------------------------------------------------------------
 50      46.7600000       9.3187544      31.0000000      67.0000000
-------------------------------------------------------------------

Analysis methods you might consider

Multinomial logistic regression

Below we use proc logistic to estimate a multinomial logistic
regression model. The outcome prog and the predictor ses are both
categorical variables and should be indicated as such on the class statement. We
can specify the baseline category for prog using (ref = “2”) and
the reference group for ses using (ref = “1”). The param=ref option
on
the class statement tells SAS to use dummy coding rather than effect coding
for the variable ses. Note that the levels of prog are defined as:

1=general

2=academic (reference group)

3=vocational

proc logistic data = "c:hsbdemo";
class prog (ref = "2") ses (ref = "1") / param = ref;
model prog = ses write / link = glogit;
run;
	The LOGISTIC Procedure

                         Model Information

Data Set                      c:datahsbdemo       Written by SAS
Response Variable             PROG                  type of program
Number of Response Levels     3
Model                         generalized logit
Optimization Technique        Newton-Raphson


Number of Observations Read         200
Number of Observations Used         200


          Response Profile

 Ordered                      Total
   Value         PROG     Frequency

       1            1            45
       2            2           105
       3            3            50

Logits modeled use PROG=2 as the reference category.


   Class Level Information

                      Design
Class     Value     Variables

SES       1          0      0
          2          1      0
          3          0      1


                    Model Convergence Status

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


         Model Fit Statistics

                             Intercept
              Intercept            and
Criterion          Only     Covariates

AIC             412.193        375.963
SC              418.790        402.350
-2 Log L        408.193        359.963

        Testing Global Null Hypothesis: BETA=0

Test                 Chi-Square       DF     Pr > ChiSq

Likelihood Ratio        48.2299        6         <.0001
Score                   45.1588        6         <.0001
Wald                    37.2946        6         <.0001


        Type 3 Analysis of Effects

                        Wald
Effect      DF    Chi-Square    Pr > ChiSq

SES          4       10.8162        0.0287
WRITE        2       26.4633        <.0001


                  Analysis of Maximum Likelihood Estimates

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

Intercept      1        1      2.8522      1.1664        5.9790        0.0145
Intercept      3        1      5.2182      1.1635       20.1128        <.0001
SES       2    1        1     -0.5333      0.4437        1.4444        0.2294
SES       2    3        1      0.2914      0.4764        0.3742        0.5407
SES       3    1        1     -1.1628      0.5142        5.1137        0.0237
SES       3    3        1     -0.9827      0.5956        2.7224        0.0989
WRITE          1        1     -0.0579      0.0214        7.3200        0.0068
WRITE          3        1     -0.1136      0.0222       26.1392        <.0001


                  Odds Ratio Estimates

                           Point          95% Wald
Effect          PROG    Estimate      Confidence Limits

SES   2 vs 1    1          0.587       0.246       1.400
SES   2 vs 1    3          1.338       0.526       3.404
SES   3 vs 1    1          0.313       0.114       0.856
SES   3 vs 1    3          0.374       0.116       1.203
WRITE           1          0.944       0.905       0.984
WRITE           3          0.893       0.855       0.932

Two models are tested in this multinomial regression, one comparing
membership to general versus academic program and one comparing membership to
vocational versus academic program. They correspond to the two equations below:

$$lnleft(frac{P(prog=general)}{P(prog=academic)}right) = b_{10} + b_{11}(ses=2) + b_{12}(ses=3) + b_{13}write$$
$$lnleft(frac{P(prog=vocation)}{P(prog=academic)}right) = b_{20} + b_{21}(ses=2) + b_{22}(ses=3) + b_{23}write$$

where (b)s are the regression coefficients.

Using the test statement, we can also test specific hypotheses within
or even across logits, such as if the effect of ses=3 in
predicting general versus academic equals the effect of ses = 3 in
predicting vocational versus academic.  Use of the test statement requires the
unique names SAS assigns each parameter in the model.  The option outest
on the proc logistic statement produces an output dataset with
the parameter names and values.  We can get these names by printing them,
and we transpose them to be more readable.  The noobs option on the proc print
statement suppresses observation numbers, since they are meaningless in the parameter dataset.

proc logistic data = "c:hsbdemo" outest = mlogit_param;
class prog (ref = "2") ses (ref = "1") / param = ref;
model prog = ses write / link = glogit;
run;proc transpose data = mlogit_param;
run;
proc print noobs;
run;
_NAME_         _LABEL_                      PROG

Intercept_1    Intercept: PROG=1           2.852
Intercept_3    Intercept: PROG=3           5.218
SES2_1         SES 2: PROG=1              -0.533
SES2_3         SES 2: PROG=3               0.291
SES3_1         SES 3: PROG=1              -1.163
SES3_3         SES 3: PROG=3              -0.983
WRITE_1        writing score: PROG=1      -0.058
WRITE_3        writing score: PROG=3      -0.114
_LNLIKE_       Model Log Likelihood     -179.982

Here we see the same parameters as in the output above, but with their unique SAS-given names.
We are interested in testing whether  SES3_general is equal to SES3_vocational,
which we can now do with the test statement.  The code preceding the “:”
on the test statement is a label identifying the test in the output, and it must
conform to SAS variable-naming rules (i.e., 32 characters in length or less, letters,
numerals, and underscore).

proc logistic data = "c:hsbdemo" outest = mlogit_param;
class prog (ref = "2") ses (ref = "1") / param = ref;
model prog = ses write / link = glogit;
SES3_general_vs_SES3_vocational: test SES3_1 - SES3_3;
run;
		
		***SOME OUTPUT OMITTED***

                 Linear Hypotheses Testing Results

                                          Wald
 Label                              Chi-Square      DF    Pr > ChiSq

 SES3_general_vs_SES3_vocational        0.0772       1        0.7811

The effect of ses=3 for predicting general versus academic is not different from the effect of
ses=3 for predicting vocational versus academic.

You can also use predicted probabilities to help you understand the model.
You can calculate predicted probabilities using the lsmeans statement and
the ilink option. For multinomial data, lsmeans requires glm
rather than reference (dummy) coding, even though they are essentially
the same, so be sure to respecify the coding on the class statement.
However, glm coding only allows the last category to be the reference
group (prog = vocational and ses = 3)and will ignore any other
reference group specifications.   Below we use lsmeans to
calculate the predicted probability of choosing program type academic or general at each level
of ses, holding write at its means.

proc logistic data = "c:hsbdemo" outest = mlogit_param;
class prog ses / param = glm;
model prog = ses write / link = glogit;
lsmeans ses / e ilink cl;
run;

***SOME OUTPUT OMITTED***
                         Coefficients for SES Least Squares Means

                 type of
Parameter        program    SES      Row1      Row2      Row3      Row4      Row5      Row6

Intercept        1                      1         1         1
Intercept        2                                                    1         1         1
SES 1            1          1           1
SES 1            2          1                                         1
SES 2            1          2                     1
SES 2            2          2                                                   1
SES 3            1          3                               1
SES 3            2          3                                                             1
writing score    1                 52.775    52.775    52.775
writing score    2                                               52.775    52.775    52.775

***SOME OUTPUT OMITTED***

                   SES Least Squares Means

                              Standard
type of                       Error of       Lower       Upper
program    SES        Mean        Mean        Mean        Mean

1          1        0.3582     0.07264      0.2158      0.5006
1          2        0.2283     0.04512      0.1399      0.3168
1          3        0.1785     0.05405     0.07256      0.2844
2          1        0.4397     0.07799      0.2868      0.5925
2          2        0.4777     0.05526      0.3694      0.5861
2          3        0.7009     0.06630      0.5709      0.8309

The predicted probabilities are in the “Mean” column.  Thus, for ses
= 3 and write = 52.775, we see that the probability of being the academic
program (program type 2) is 0.7009; for the general program (program type 1),
the probability is 0.1785.

To obtain predicted probabilities for the program type vocational, we can reverse the ordering of the categories
using the descending option on the proc logistic statement.
This will make academic the reference group for prog and 3 the reference
group for ses.

proc logistic data = "c:hsbdemo" outest = mlogit_param descending;
class prog ses / param = glm;
model prog = ses write / link = glogit;
lsmeans ses / e ilink cl;
run;


***SOME OUTPUT OMITTED***
                        Coefficients for SES Least Squares Means

                 type of
Parameter        program    SES      Row1      Row2      Row3      Row4      Row5      Row6

Intercept        3                      1         1         1
Intercept        2                                                    1         1         1
SES 1            3          1           1
SES 1            2          1                                         1
SES 2            3          2                     1
SES 2            2          2                                                   1
SES 3            3          3                               1
SES 3            2          3                                                             1
writing score    3                 52.775    52.775    52.775
writing score    2                                               52.775    52.775    52.775

***SOME OUTPUT OMITTED***

                   SES Least Squares Means

                              Standard
type of                       Error of       Lower       Upper
program    SES        Mean        Mean        Mean        Mean

3          1        0.2021     0.05996     0.08459      0.3197
3          2        0.2939     0.05036      0.1952      0.3926
3          3        0.1206     0.04643     0.02960      0.2116
2          1        0.4397     0.07799      0.2868      0.5925
2          2        0.4777     0.05526      0.3694      0.5861
2          3        0.7009     0.06630      0.5709      0.8309

Here we see the probability of being in the vocational program when ses = 3 and
write = 52.775 is 0.1206, which is what we would have expected since (1 –
0.7009 – 0.1785) = 0.1206, where 0.7009 and 0.1785 are the probabilities of
being in the academic and general programs under the same conditions.

Things to consider

See Also

References