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

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 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 difference between small and medium is 10
ounces, between medium and large 8, and between large and extra large 12.

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 simulated some data for this example
and it can be obtained from here: ologit.sav
This hypothetical data set has a three-level variable called apply
(coded 0, 1, 2), that we will use as our outcome 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 public and 0 private, and
gpa, which is the student’s grade point average.

Let’s start with the descriptive statistics of these variables.

get file "D:dataologit.sav".
freq var = apply pared public.descriptives var = gpa.

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 are empty
or extremely small.  If any are, we may have difficulty running our model.
There are two ways in SPSS that we can do this.  The first way is to make
simple crosstabs.  The second way is to use the cellinfo option on
the /print subcommand.  You should use the cellinfo option
only with categorical predictor variables; the table will be long and difficult
to interpret if you include continuous predictors.

crosstabs
/tables = apply by pared
/tables = apply by public.
plum apply with pared public 
/link = logit
/print = cellinfo.

None of the cells is too small or empty (has no cases), so we will run our
model. In the syntax below, we have included the link = logit
subcommand, even though it is the default, just to remind ourselves that we are
using the logit link function. Also note that if you do not include the
print subcommand, only the Case Processing Summary table is provided in the
output.

plum apply with pared public gpa
/link = logit
/print = parameter summary.

In the Case Processing Summary table, we see the number and percentage
of cases in each level of our response variable.
These numbers look fine, but we would be concerned if one level
had very few cases in it. We also 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, SPSS does a listwise
deletion of cases with missing values.  Next we see the Model Fitting
Information table, which gives the -2 log likelihood for the intercept-only and
final models. The -2 log likelihood can be used in comparisons of nested
models, but we won’t show an example of that here.

In the Parameter Estimates table we see the coefficients, their standard errors, the
Wald test and associated p-values (Sig.), and the 95% confidence interval of the coefficients.
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 ordered 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. The thresholds are shown at the top of the parameter estimates output, and they
indicate where the latent variable is cut to make the three groups that we
observe in our data. Note that this latent variable is continuous.
In general, these are not used in the interpretation of the results. Some
statistical packages call the thresholds “cutpoints” (thresholds and cutpoints
are the same thing); other packages, such as SAS report intercepts, which are
the negative of the thresholds. In this example, the intercepts would be
-2.203 and -4.299. For further information, please see the Stata FAQ:
How can I convert Stata’s parameterization of ordered probit and logistic models to one in which a constant is estimated?

As of version 15 of SPSS, you cannot directly obtain the proportional odds
ratios from SPSS. You can either use the SPSS Output Management System (OMS)
to capture the parameter estimates and exponentiate them, or you can calculate
them by hand. Please see Ordinal Regression by
Marija J. Norusis for examples of how to do this. The commands for using
OMS and calculating the proportional odds ratios is shown below. For more
information on how to use OMS, please see our SPSS FAQ:
How can I output my results to a data file in SPSS? Please note that
the single quotes in the square brackets are important, and you will get an
error message if they are omitted or unbalanced.

oms select tables
 /destination format = sav outfile = "D:ologit_results.sav"
 /if commands = ['plum'] subtypes = ['Parameter Estimates'].

plum apply with pared public gpa
/link = logit
/print = parameter.

omsend.

get file "D:ologit_results.sav".
rename variables Var2 = Predictor_Variables.
* the next command deletes the thresholds from the data set.
select if Var1 = "Location".
exe.
* the command below removes unnessary variables from the data set.
* transformations cannot be pending for the command below to work, so
* the exe.
* above is necessary.
delete variables Command_ Subtype_ Label_ Var1.
compute expb = exp(Estimate).
compute Lower_95_CI = exp(LowerBound).
compute Upper_95_CI = exp(UpperBound).
execute.

In the column expb we see the results presented as proportional odds
ratios (the coefficient exponentiated). We have also calculated the lower
and upper 95% confidence interval.
We would interpret these 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 low and middle categories of apply
versus the high category 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.

One of the assumptions underlying ordered logistic (and ordered probit)
regression is that the relationship between each pair of outcome groups is the
same. In other words, ordered 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 to describe the
relationship between each pair of outcome groups. We need to test the
proportional odds assumption, and we can use the tparallel option on the
print subcommand. The null hypothesis of this chi-square test is
that there is no difference in the coefficients between models, so we hope to
get a non-significant result.

plum apply with pared public gpa
/link = logit
/print = tparallel.

The above test indicates that we have not violated the proportional odds
assumption. If the proportional odds assumption was violated, we may want to go with
multinomial logistic regression.

We we use these formulae to calculate the predicted probabilities for each level
of the outcome, apply. Predicted probabilities are usually easier to
understand than the coefficients or the odds ratios.

$$P(Y = 2) = left(frac{1}{1 + e^{-(a_{2}+b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3})}}right)$$
$$P(Y = 1) = left(frac{1}{1 + e^{-(a_{1}+b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3})}}right) – P(Y = 2)$$
$$P(Y = 0) = 1 – P(Y = 1) – P(Y = 2)$$

We will calculate the predicted probabilities using SPSS’ Matrix language.
We will use pared as an example with a categorical predictor. Here we will
see how the probabilities of membership to each category of apply change
as we vary pared and hold the other variable at their means.
As you can see, the predicted probability of being in the lowest category of apply
is 0.59 if neither parent has a graduate
level education and 0.34 otherwise. For the middle category of apply, the
predicted probabilities are 0.33 and 0.47, and for the highest category of
apply, 0.078 and 0.196 (annotations were added to the output for clarity). Hence, if neither of a respondent’s parents
have a graduate level education, the predicted probability of applying to
graduate school decreases. Note that the intercepts are the negatives of the
thresholds. For a more detailed explanation of how to interpret the predicted probabilities and its relation to the odds ratio, please refer to FAQ: How do I interpret the coefficients in an ordinal logistic regression?

Matrix.
* intercept1 intercept2 pared public gpa.
* these coefficients are taken from the output.
compute b = {-2.203 ; -4.299 ; 1.048 ; -.059 ; .616}.
* overall design matrix including means of public and gpa.
compute x = {{0, 1, 0; 0, 1, 1}, make(2, 1, .1425), make(2, 1, 2.998925)}.
compute p3 = 1/(1 + exp(-x * b)).
* overall design matrix including means of public and gpa.
compute x = {{1, 0, 0; 1, 0, 1}, make(2, 1, .1425), make(2, 1, 2.998925)}.
compute p2 = (1/(1 + exp(-x * b))) - p3.
compute p1 = make(NROW(p2), 1, 1) - p2 - p3.
compute p = {p1, p2, p3}.
print p / FORMAT = F5.4 / title = "Predicted Probabilities for Outcomes 0 1 2 for pared 0 1 at means".
End Matrix.

Run MATRIX procedure: 
 
Predicted Probabilities for Outcomes 0 1 2 for pared 0 1 at means 
		(apply=0) 	(apply=1)	(apply=2)
(pared=0)	.5900 		.3313 		.0787 
(pared=1)	.3354 		.4687 		.1959 
 
------ END MATRIX -----

Below, we see the predicted probabilities for gpa at 2, 3
and 4. You can see that the
predicted probability increases for both the middle and highest categories of
apply as gpa increases (annotations were added to the output for clarity). For a more detailed explanation of how to interpret the predicted probabilities and its relation to the odds ratio, please refer to FAQ: How do I interpret the coefficients in an ordinal logistic regression?

Matrix.
* intercept1 intercept2 pared public gpa.
* these coefficients are taken from the output.
compute b = {-2.203 ; -4.299 ; 1.048 ; -.059 ; .616}.
* overall design matrix including means of pared and public.
compute x = {make(3, 1, 0), make(3, 1, 1), make(3, 1, .1575), make(3, 1, .1425), {2; 3; 4}}.
compute p3 = 1/(1 + exp(-x * b)).
* overall design matrix including means of pared and public.
compute x = {make(3, 1, 1), make(3, 1, 0), make(3, 1, .1575), make(3, 1, .1425), {2; 3; 4}}.
compute p2 = (1/(1 + exp(-x * b))) - p3.
compute p1 = make(NROW(p2), 1, 1) - p2 - p3.
compute p = {p1, p2, p3}.
print p / FORMAT = F5.4 / title = "Predicted Probabilities for Outcomes 0 1 2 for gpa 2 3 4 at means".
End Matrix.

Run MATRIX procedure: 
 
Predicted Probabilities for Outcomes 0 1 2 for gpa 2 3 4 at means 
		(apply=0) 	(apply=1)	(apply=2)
(gpa=2) 	.6930 		.2553 		.0516 
(gpa=3) 	.5494 		.3590 		.0916 
(gpa=4) 	.3971 		.4456 		.1573 

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

Things to consider

References