Version info: Code for this page was tested in SPSS 20.

Multinomial logistic regression is used to model 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.

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.

get data "D:datahsbdemo.sav".

crosstabs
  /tables=prog by ses
  /statistics=chisq 
  /cells=count.

sort cases by prog.
split file by prog.
descriptives var = write 
/statistics = mean stddev.
split file off.

Analysis methods you might consider

Using the multinomial logit model

Below we use the nomreg command to estimate a multinomial logistic
regression model. We specify the baseline comparison group to be the academic
group using (base=2).

nomreg prog (base = 2) by ses with write
/print = lrt cps mfi parameter summary.

$$lnleft(frac{P(prog=general)}{P(prog=academic)}right) = b_{10} + b_{11}(ses=1) + b_{12}(ses=2) + b_{13}write$$
$$lnleft(frac{P(prog=vocation)}{P(prog=academic)}right) = b_{20} + b_{21}(ses=1) + b_{22}(ses=2) + b_{23}write$$
where (b)’s are the regression coefficients.

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).  Thus, exponentiating the linear
equations above
yields relative risks. Regression coefficients represent the change
in log relative risk (log odds) per unit change in the predictor.
Exponentiating regression coefficients will therefore yield relative
risk ratios.  SPSS
includes relative risk
ratios in the output, under the column “Exp(B)”.

Tests for the overall effect of ses and write are outputted by
the nomreg command. Below we see that the effects are statistically
significant.

You can also use predicted probabilities to help you understand the model. You can calculate predicted probabilities using the
SPSS matrix command.
Below we calculate the predicted probability of choosing each program type at each level of
ses, holding
write at its means.

Matrix.
* intercept1 intercept2 pared public gpa.
* these coefficients are taken from the output.
compute b_gen = {1.689354 ; -0.057928 ; 1.162832 ; 0.629541}.
compute b_voc = {4.235530 ; -0.113603 ; 0.982670 ; 1.274063}.
* overall design matrix including means of public and gpa.
compute x = {{1 ; 1; 1}, make(3, 1, 52.775), {1, 0; 0, 1; 0, 0}}.
compute lp_gen = exp(x * b_gen).
compute lp_voc = exp(x * b_voc).
compute lp_aca = {1; 1; 1}.
compute p_gen = lp_gen/(lp_aca + lp_gen + lp_voc).
compute p_voc = lp_voc/(lp_aca + lp_gen + lp_voc).
compute p_aca = lp_aca/(lp_aca + lp_gen + lp_voc).
compute p = {p_gen, p_aca, p_voc}.
print p /title 'Predicted Probabilities for Outcomes 1 2 3 for ses 1 2 3 at mean of write'.
End Matrix.

Run MATRIX procedure:

Predicted Probabilities for Outcomes 1 2 3 for ses 1 2 3 at mean of write
   .3581989665   .4396824687   .2021185647
   .2283388262   .4777491509   .2939120229
   .1784967500   .7009009604   .1206022896

------ END MATRIX -----

Column 1 contains the predicted probabilities for prog = general, where ses equals 1, 2 and 3 on each successive row. Columns 2 and 3 are the same for prog = academic and prog = vocational, respectively. We can also calculate predicted probabilities as we vary write from 30 to 70, when ses = 1.

Matrix.
* intercept1 intercept2 pared public gpa.
* these coefficients are taken from the output.
compute b_gen = {1.689354 ; -0.057928 ; 1.162832 ; 0.629541}.
compute b_voc = {4.235530 ; -0.113603 ; 0.982670 ; 1.274063}.
* overall design matrix including means of public and gpa.
compute x = {make(5,1,1), {30; 40; 50; 60; 70}, make(5,1,1), make(5,1,0)}.
compute lp_gen = exp(x * b_gen).
compute lp_voc = exp(x * b_voc).
compute lp_aca = {1; 1; 1; 1; 1}.
compute p_gen = lp_gen/(lp_aca + lp_gen + lp_voc).
compute p_voc = lp_voc/(lp_aca + lp_gen + lp_voc).
compute p_aca = lp_aca/(lp_aca + lp_gen + lp_voc).
compute p = {p_gen, p_aca, p_voc}.
print p /title 'Predicted Probabilities for Outcomes 1 2 3 for write 30 40 50 60 70 at ses=1'.
End Matrix.

Run MATRIX procedure:

Predicted Probabilities for Outcomes 1 2 3 for write 30 40 50 60 70 at ses=1
   .2999966732   .0984378501   .6015654767
   .3656613530   .2141424912   .4201961559
   .3698577661   .3865775582   .2435646757
   .3083735022   .5752505689   .1163759289
   .2199925775   .7324300249   .0475773976

------ END MATRIX -----

Column 1 contains the predicted probabilities for prog = general, where write equals 30, 40, 50, 60 and 70 for rows 1 through 5, respectively. Columns 2 and 3 are the same for prog = academic and prog = vocational, respectively.

Things to consider

See also

References