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

Examples of ordered logistic regression

Example 1:  A marketing research firm wants to
investigate what factors
influence the size of soda (small, medium, large or extra large) that people
order at a fast-food chain.  These factors may include what type of
sandwich is ordered (burger or chicken), whether or not fries are also ordered,
and age of the consumer.  While the outcome variable, size of soda, is
obviously ordered, the difference between the various sizes is not consistent.
The differences are 10, 8, 12 ounces, respectively.

Example 2:  A researcher is interested in what factors influence medaling
in Olympic swimming.  Relevant predictors include at training hours, diet,
age, and popularity of swimming in the athlete’s home country.  The
researcher believes that the distance between gold and silver is larger than the
distance between silver and bronze.

Example 3:  A study looks at factors that influence the decision of
whether to apply to graduate school.  College juniors are asked if they are
unlikely, somewhat likely, or very likely to apply to graduate school.
Hence, our outcome variable has three categories.  Data on parental educational status, whether the undergraduate institution is
public or private, and current GPA is also collected.   The
researchers have reason to believe that the “distances” between these three
points are not equal.  For example, the “distance” between “unlikely” and
“somewhat likely” may be shorter than the distance between “somewhat likely” and
“very likely”.

Description of the data

For our data analysis below, we are going to expand on Example 3 about
applying to graduate school.  We have generated hypothetical data,
which can be downloaded: ologit.

This hypothetical data set has a three-level variable called apply
(coded 0, 1, 2), that we
will use as our response (i.e., outcome, dependent) variable.  We also have three
variables that we will use as predictors:  pared, which is a 0/1
variable indicating whether at least one parent has a graduate degree; public, which is a 0/1 variable where 1 indicates
that the undergraduate institution is a public university and 0 indicates that it is
a private university, and gpa, which is the student’s grade point average.

proc freq data = ologit;
tables apply;
tables pared;
tables public;
run;

APPLY	Frequency	Percent	Cumulative Frequency	Cumulative Percent
0	220	55.00	220	55.00
1	140	35.00	360	90.00
2	40	10.00	400	100.00

 

PARED	Frequency	Percent	Cumulative Frequency	Cumulative Percent
0	337	84.25	337	84.25
1	63	15.75	400	100.00

 

PUBLIC	Frequency	Percent	Cumulative Frequency	Cumulative Percent
0	343	85.75	343	85.75
1	57	14.25	400	100.00

proc means data = ologit;
var gpa;
run;

The MEANS Procedure

                      Analysis Variable : GPA

  N            Mean         Std Dev         Minimum         Maximum
-------------------------------------------------------------------
400       2.9989250       0.3979409       1.9000000       4.0000000
-------------------------------------------------------------------

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.

Ordered logistic regression

Before we run our ordinal logistic model, we will see if any cells (created
by the crosstab of our categorical and response variables) are empty or
extremely small.  If any are, we may have difficulty running our model.
We have used some options on the tables statements to clean up the output.
Perhaps the most important option is the missprint option; this will have
SAS include missing values as a category in the table.  Because we have no
missing values in this data set, this option is not really needed; we have
included it here only to show its use.

proc freq data = ologit;
tables apply*pared / nopercent norow nocol missprint;
tables apply*public / nopercent norow nocol missprint;
run;

The FREQ Procedure

Table of APPLY by PARED

APPLY     PARED

Frequency|       0|       1|  Total
---------+--------+--------+
       0 |    200 |     20 |    220
---------+--------+--------+
       1 |    110 |     30 |    140
---------+--------+--------+
       2 |     27 |     13 |     40
---------+--------+--------+
Total         337       63      400


Table of APPLY by PUBLIC

APPLY     PUBLIC

Frequency|       0|       1|  Total
---------+--------+--------+
       0 |    189 |     31 |    220
---------+--------+--------+
       1 |    124 |     16 |    140
---------+--------+--------+
       2 |     30 |     10 |     40
---------+--------+--------+
Total         343       57      400

None of the cells is too small or empty (has no cases), so we will run our
model.

proc logistic data = ologit desc;
class pared(ref='0') public(ref='0') / param=reference;
model apply = pared public gpa;
run;

The LOGISTIC Procedure

                         Model Information

Data Set                      ologit            Written by SAS
Response Variable             APPLY
Number of Response Levels     3
Model                         cumulative logit
Optimization Technique        Fisher's scoring

Number of Observations Read         400
Number of Observations Used         400

          Response Profile

 Ordered                      Total
   Value        APPLY     Frequency

       1            2            40
       2            1           140
       3            0           220

Probabilities modeled are cumulated over the lower Ordered Values.

                    Model Convergence Status

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

Score Test for the Proportional Odds Assumption

Chi-Square       DF     Pr > ChiSq

    4.8446        3         0.1835

         Model Fit Statistics

                             Intercept
              Intercept            and
Criterion          Only     Covariates

AIC             745.205        727.025
SC              753.188        746.982
-2 Log L        741.205        717.025

The LOGISTIC Procedure

        Testing Global Null Hypothesis: BETA=0

Test                 Chi-Square       DF     Pr > ChiSq

Likelihood Ratio        24.1804        3         <.0001
Score                   23.4804        3         <.0001
Wald                    24.3337        3         <.0001

              Analysis of Maximum Likelihood Estimates

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

Intercept 2     1     -4.2983      0.8092       28.2189        <.0001
Intercept 1     1     -2.2029      0.7844        7.8869        0.0050
PARED           1      1.0478      0.2684       15.2350        <.0001
PUBLIC          1     -0.0585      0.2886        0.0411        0.8393
GPA             1      0.6156      0.2626        5.4963        0.0191

           Odds Ratio Estimates

             Point          95% Wald
Effect    Estimate      Confidence Limits

PARED        2.851       1.685       4.826
PUBLIC       0.943       0.536       1.661
GPA          1.851       1.106       3.096

Association of Predicted Probabilities and Observed Responses

Percent Concordant     60.0    Somers' D    0.210
Percent Discordant     39.0    Gamma        0.213
Percent Tied            1.1    Tau-a        0.119
Pairs                 45200    c            0.605

In the output above, 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.  By default, SAS does a listwise
deletion of cases with missing values.  The Response Profile shows the
value that SAS used when conducting the analysis (given in the Ordered Value
column), the value of the original variable, and the number of cases in each
level of the outcome variable.  (If you want SAS to use the values that you
have assigned the outcome variable, then you would want to use the order =
data option on the proc logistic statement.)  The note below this table reminds us that
the “Probabilities modeled are cumulated over the lower Ordered Values.”
It is helpful to remember this when interpreting the output.  Next we see
that the model converged (you should not try to interpret any output if the
model has not converged), and we also see that the test of the proportional odds
assumption is non-significant.  One of the assumptions underlying ordinal
logistic (and ordinal probit) regression is that the relationship between each
pair of outcome groups is the same.  In other words, ordinal logistic
regression assumes that the coefficients that describe the relationship between,
say, the lowest versus all higher categories of the response variable are the
same as those that describe the relationship between the next lowest category
and all higher categories, etc.  This is called the proportional odds
assumption or the parallel regression assumption.  Because the relationship
between all pairs of groups is the same, there is only one set of coefficients
(only one model).  If this was not the case, we would need different models
(such as a generalized ordered logit model) to describe the relationship between
each pair of outcome groups.  The table showing the Model Fit Statistics provides the AIC, SC and -2 log
likelihood.  These can be used in the comparison of nested models.  In
the next table we see various tests of the overall model; they all indicated
that the model is statistically significant.

In the table Analysis of Maximum Likelihood Estimates, we see the degrees of
freedom, coefficients, their standard errors, the Wald chi-square test and
associated p-values.
Both pared and gpa are statistically significant; public is
not.  So for pared, we would say that for a one unit
increase in pared (i.e., going from 0 to 1), we expect a 1.05 increase in
the log odds of being in a higher level of apply, given all of the other variables in the model are
held constant.  For gpa, we would say that for a one unit increase
in gpa, we would expect a 0.62 increase in the log odds of being in a
higher level of apply, given that all of the other variables in the model
are held constant.  In the next table we see the results presented as
proportional odds ratios (the coefficient exponentiated) and the 95% confidence
intervals for the proportional odds ratios.  We would interpret the
proportional odds ratios pretty much as we would odds ratios from a binary
logistic regression.  For pared, we would say that for a one unit increase
in pared, i.e., going from 0 to 1, the odds of high apply versus the combined
middle and low categories are 2.85 greater, given that all of the other
variables in the model are held constant.  Likewise, the odds of the
combined middle and high categories versus low apply is 2.85 times greater,
given that all of the other variables in the model are held constant.  For a one unit
increase in gpa, the odds of the high category of apply
versus the low and middle categories of apply are 1.85 times greater, given that the
other variables in the model are held constant.  Because of the
proportional odds assumption (see below for more explanation), the same
increase, 1.85 times, is found between low apply and the combined
categories of middle and high apply.

We can also obtain predicted probabilities, which are usually easier to
understand than the coefficients or the odds ratios.  We will use the
estimate statement.  To use the estimate statement, we supply
values of our predictor variables to be multiplied by the regression
coefficients, which are for our current model the intercept for apply =
2, the intercept for apply = 1, the coefficient for public = 1 ,
the coefficient for pared = 1, and the coefficient for gpa.  Here we will
see how the probabilities of membership to the categories of apply change
as we vary pared and hold public at 1 and gpa at its mean
of 2.9989.

proc logistic data = ologit desc;;
class pared(ref='0') public(ref='0')/ param = reference;
model apply = pared public gpa;
estimate "Pr prob apply=2 at pared=0" intercept 1 public 1 gpa 2.9989 / ilink category='2';
estimate "Pr prob apply=2 at pared=1" intercept 1 pared 1 public 1 gpa 2.9989 / ilink category='2';
estimate "Pr prob apply=1 or 2 at pared=0" intercept 1 public 1 gpa 2.9989 / ilink category='1';
estimate "Pr prob apply=1 or 2 at pared=1" intercept 1 pared 1 public 1 gpa 2.9989 / ilink category='1';
run;

***SOME OUTPUT OMITTED AND LAYOUT MODIFIED***							

Label	                        APPLY	Estimate   Standard Error      z Value	   Pr > |z|	  Mean	   Standard Error
							                                                         of Mean
Pr prob apply=2 at pared=0	  2	-2.5108	           0.3104	 -8.09	     <.0001	0.07511	         0.02156
Pr prob apply=2 at pared=1	  2	-1.4629	           0.3545	 -4.13	     <.0001	0.188	         0.05412
Pr prob apply=1 or 2 at pared=0	  1	-0.4153	           0.2733	 -1.52	     0.1286	0.3976	         0.06546
Pr prob apply=1 or 2 at pared=1	  1	 0.6325	           0.3451	  1.83	     0.0668	0.6531	         0.07818

The predicted probabilities are listed in the “Mean” column.  All
predicted probabilities discussed below were calculated at public = 1 and
gpa = 2.9989.  As you can see, the predicted probability of
being in the highest category of apply (apply = 2) is 0.07511 if neither parent has a graduate
level education and 0.1880 otherwise.  For membership to either the
highest or middle category of apply (apply = 1 or 2), the
predicted probabilities are 0.3976 and 0.6531, for parents without graduate
level education and with graduate level education, respectively.  Predicted
probabilities of being in the middle category alone can be calculated by
subtracting the predicted probabilities of (apply = 1 or 2) from the
probability of (apply = 2).  Thus, the probability of belonging to
the middle apply category when parents do not have graduate level
education is 0.3976 – 0.07511 = 0.32249. Predicted probabilities of being in the
lowest apply category can be obtained in 2 ways.  First, we can
subtract the probability of being in either the highest or middle apply
category from 1.  For example, the probability of being in the lowest apply
group (apply = 0) when parents do not have graduate education is 1 – 0.3976 =
0.6024.  Alternatively, we can change the reference apply category
to 2 by removing the desc option from the proclogistic
statement and supply a new estimate statement to get the probabilities of
being in apply category 0.

proc logistic data = ologit;
class pared(ref='0') public(ref='0')/ param = reference;
model apply = pared public gpa;
estimate "Pr prob apply=0 at pared=0" intercept 1 public 1 gpa 2.9989 / ilink category='0';
estimate "Pr prob apply=0 at pared=1" intercept 1 pared 1 public 1 gpa 2.9989 / ilink category='0';
run;

***SOME OUTPUT OMITTED AND LAYOUT MODIFIED***							

Label	                        APPLY	Estimate   Standard Error      z Value	   Pr > |z|	  Mean	   Standard Error
							                                                         of Mean
Pr prob apply=0 at pared=0        0      0.4153            0.2733         1.52     0.1286         0.6024         0.06546
Pr prob apply=0 at pared=1        0     -0.6325            0.3451        -1.83     0.0668         0.3469         0.07818

Things to consider

See also

References