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Multinomial Logistic Regression

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.

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.

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 d:\data\hsbdemo 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 d:\data\hsbdemo.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 = 200proc 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 -------------------------------------------------------------------

- Multinomial logistic regression: the focus of this page.
- Multinomial probit regression: similar to multinomial logistic regression but with independent normal error terms.
- Multiple-group discriminant function analysis: A multivariate method for multinomial outcome variables
- Multiple logistic regression analyses, one for each pair of outcomes: One problem with this approach is that each analysis is potentially run on a different sample. The other problem is that without constraining the logistic models, we can end up with the probability of choosing all possible outcome categories greater than 1.
- Collapsing number of categories to two and then doing a logistic regression: This approach suffers from loss of information and changes the original research questions to very different ones.
- Ordinal logistic regression: If the outcome variable is truly ordered and if it also satisfies the assumption of proportional odds, then switching to ordinal logistic regression will make the model more parsimonious.
- Alternative-specific multinomial probit regression: allows different error structures therefore allows to relax the independence of irrelevant alternatives (IIA, see below "Things to Consider") assumption. This requires that the data structure be choice-specific.
- Nested logit model: also relaxes the IIA assumption, also requires the data structure be choice-specific.

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**.

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 d:\data\hsbdemo 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 408.193 359.963Testing 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

- In the output above, the likelihood ratio chi-square of
48.23 with a p-value < 0.0001 tells us that our model as a whole fits
significantly better than an empty model (i.e., a model with no
predictors)
- Several model fit measures such as the AIC are listed under Model Fit Statistics

- 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:\[ln\left(\frac{P(prog=general)}{P(prog=academic)}\right) = b_{10} + b_{11}(ses=2) + b_{12}(ses=3) + b_{13}write\] \[ln\left(\frac{P(prog=vocation)}{P(prog=academic)}\right) = b_{20} + b_{21}(ses=2) + b_{22}(ses=3) + b_{23}write\] where \(b\)'s are the regression coefficients.
- A one-unit increase in the variable
**write**is associated with a .058 decrease in the relative log odds of being in general program vs. academic program . - A one-unit increase in the variable
**write**is associated with a .1136 decrease in the relative log odds of being in vocation program vs. academic program. - The relative log odds of being in general program vs. in academic program will
decrease by 1.163 if moving from the lowest level of
**ses**(**ses**==1) to the highest level of**ses**(**ses**==3). - The overall effects of
**ses**and**write**are listed under "Type 3 Analysis of Effects", and both are significant. - The ratio of the probability of choosing one outcome category over the
probability of choosing the baseline category is often referred to as relative risk
(and it is also sometimes referred to as odds as we have just used to described the
regression parameters above). Relative risk can be obtained by
exponentiating the linear equations above, yielding regression coefficients that
are relative risk ratios for a unit change in the predictor variable.
In the case of two categories, relative risk ratios are equivalent to
odds ratios, which are listed in the output as well.
- The odds ratio for a one-unit increase in the variable
**write**is .944 (exp(-.0579) from the regression coefficients above the odds ratios) for being in general program vs. academic program. - The odds ratio of switching from
**ses**= 1 to 3 is .313 for being in general program vs. academic program. In other words, the expected risk of staying in the general program is lower for subjects who are high in**ses**.

- The odds ratio for a one-unit increase in the variable

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 0.1785; for the general program (program type 1),
the probability is 0.7009.

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.

- The Independence of Irrelevant Alternatives (IIA) assumption: Roughly, the IIA assumption means that adding or deleting alternative outcome categories does not affect the odds among the remaining outcomes.
- Diagnostics and model fit: Unlike logistic regression where there are many statistics for performing model diagnostics, it is not as straightforward to do diagnostics with multinomial logistic regression models. Some model fit statistics are listed in the output.
- Pseudo-R-Squared: The R-squared offered in the output is basically the change in terms of log-likelihood from the intercept-only model to the current model. It does not convey the same information as the R-square for linear regression, even though it is still "the higher, the better".
- Sample size: Multinomial regression uses a maximum likelihood estimation method. Therefore, it requires a large sample size. It also uses multiple equations. Therefore, it requires an even larger sample size than ordinal or binary logistic regression.
- Complete or quasi-complete separation: Complete separation implies that only one value of a predictor variable is associated with only one value of the response variable. You can tell from the output of the regression coefficients that something is wrong. You can then do a two-way tabulation of the outcome variable with the problematic variable to confirm this and then rerun the model without the problematic variable.
- Empty cells or small cells: You should check for empty or small cells 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.
- Sometimes observations are clustered into groups (e.g., people within families, students within classrooms). In such cases, you may want to see our page on non-independence within clusters.

- Hosmer, D. and Lemeshow, S. (2000) Applied Logistic Regression (Second Edition). New York: John Wiley & Sons, Inc..
- Agresti, A. (1996) An Introduction to Categorical Data Analysis. New York: John Wiley & Sons, Inc.