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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 our website:

use https://stats.idre.ucla.edu/stat/data/ologit.dta, clear

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.

tab apply
        
          apply |      Freq.     Percent        Cum.
----------------+-----------------------------------
       unlikely |        220       55.00       55.00
somewhat likely |        140       35.00       90.00
    very likely |         40       10.00      100.00
----------------+-----------------------------------
          Total |        400      100.00


tab apply, nolab
	      
      apply |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |        220       55.00       55.00
          1 |        140       35.00       90.00
          2 |         40       10.00      100.00
------------+-----------------------------------
      Total |        400      100.00

tab apply pared
                |         pared
          apply |         0          1 |     Total
----------------+----------------------+----------
       unlikely |       200         20 |       220 
somewhat likely |       110         30 |       140 
    very likely |        27         13 |        40 
----------------+----------------------+----------
          Total |       337         63 |       400 



tab apply public

                |        public
          apply |         0          1 |     Total
----------------+----------------------+----------
       unlikely |       189         31 |       220 
somewhat likely |       124         16 |       140 
    very likely |        30         10 |        40 
----------------+----------------------+----------
          Total |       343         57 |       400 

summarize gpa

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         gpa |       400    2.998925    .3979409        1.9          4

table apply, cont(mean gpa sd gpa)

----------------------------------------
          apply |  mean(gpa)     sd(gpa)
----------------+-----------------------
       unlikely |   2.952136     .403594
somewhat likely |   3.030071    .3893446
    very likely |    3.14725    .3560322
----------------------------------------

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

Below we use the ologit command to estimate an ordered logistic regression
model. The i. before pared indicates that pared is a factor
variable (i.e.,
categorical variable), and that it should be included in the model as a series
of indicator variables. The same goes for i.public.

ologit apply i.pared i.public gpa

Iteration 0:   log likelihood = -370.60264  
Iteration 1:   log likelihood =   -358.605  
Iteration 2:   log likelihood = -358.51248  
Iteration 3:   log likelihood = -358.51244  
Iteration 4:   log likelihood = -358.51244  

Ordered logistic regression                       Number of obs   =        400
                                                  LR chi2(3)      =      24.18
                                                  Prob > chi2     =     0.0000
Log likelihood = -358.51244                       Pseudo R2       =     0.0326

------------------------------------------------------------------------------
       apply |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     1.pared |   1.047664   .2657891     3.94   0.000     .5267266    1.568601
    1.public |  -.0586828   .2978588    -0.20   0.844    -.6424754    .5251098
         gpa |   .6157458   .2606311     2.36   0.018     .1049183    1.126573
-------------+----------------------------------------------------------------
       /cut1 |   2.203323   .7795353                      .6754621    3.731184
       /cut2 |   4.298767   .8043147                       2.72234    5.875195
------------------------------------------------------------------------------

In the output above, we first see the iteration log.  At iteration 0,
Stata fits a null model, i.e. the intercept-only model. It then moves on to fit
the full model and stops the iteration process once the difference in log
likelihood between successive iterations become sufficiently small. The final log likelihood (-358.51244)
is displayed again. It can be used
in comparisons of nested models.
Also at the top of the output we see that all 400 observations in our data set
were used in the analysis. The likelihood ratio chi-square of 24.18 with a p-value of 0.0000 tells us that our model as a whole is statistically
significant, as compared to the null model with no predictors.  The pseudo-R-squared
of 0.0326 is also given.

In the table we see the coefficients, their standard errors, z-tests and
their associated p-values, 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 log odds of being in a higher level of apply, given all of the other
variables in the model are held constant.  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 cutpoints shown at the bottom of the
output 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.  The cutpoints are closely related to thresholds, which are
reported by other statistical packages.  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?

We can obtain odds ratios using the or option after the ologit
command.

ologit apply i.pared i.public gpa, or

Iteration 0:   log likelihood = -370.60264
Iteration 1:   log likelihood =   -358.605
Iteration 2:   log likelihood = -358.51248
Iteration 3:   log likelihood = -358.51244

Ordered logistic regression                       Number of obs   =        400
                                                  LR chi2(3)      =      24.18
                                                  Prob > chi2     =     0.0000
Log likelihood = -358.51244                       Pseudo R2       =     0.0326

------------------------------------------------------------------------------
       apply | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       pared |   2.850982     .75776     3.94   0.000      1.69338    4.799927
      public |   .9430059   .2808826    -0.20   0.844     .5259888    1.690644
         gpa |   1.851037   .4824377     2.36   0.018      1.11062    3.085067
-------------+----------------------------------------------------------------
       /cut1 |   2.203323   .7795353                      .6754622    3.731184
       /cut2 |   4.298767   .8043146                       2.72234    5.875195
------------------------------------------------------------------------------

In the output above the results are displayed as proportional odds ratios.
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 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.

You can also use the listcoef command to obtain the odds ratios, as
well as the change in the odds for a standard deviation of the variable.
We have used the help option to get the list at the bottom of the output
explaining each column.  You can use the percent option to see the
percent change in the odds. The listcoeff command was written by Long and
Freese, and you will need to download it by typing search spost (see
How can I use the search command to search for programs and get additional
help? for more information about using search).

listcoef, help

ologit (N=400): Factor Change in Odds 

  Odds of: >m vs <=m

----------------------------------------------------------------------
       apply |      b         z     P>|z|    e^b    e^bStdX      SDofX
-------------+--------------------------------------------------------
       pared |   1.04766    3.942   0.000   2.8510   1.4654     0.3647
      public |  -0.05868   -0.197   0.844   0.9430   0.9797     0.3500
         gpa |   0.61575    2.363   0.018   1.8510   1.2777     0.3979
----------------------------------------------------------------------
       b = raw coefficient
       z = z-score for test of b=0
   P>|z| = p-value for z-test
     e^b = exp(b) = factor change in odds for unit increase in X
 e^bStdX = exp(b*SD of X) = change in odds for SD increase in X
   SDofX = standard deviation of X

listcoef, help percent

ologit (N=400): Percentage Change in Odds 

  Odds of: >m vs <=m

----------------------------------------------------------------------
       apply |      b         z     P>|z|      %      %StdX      SDofX
-------------+--------------------------------------------------------
       pared |   1.04766    3.942   0.000    185.1     46.5     0.3647
      public |  -0.05868   -0.197   0.844     -5.7     -2.0     0.3500
         gpa |   0.61575    2.363   0.018     85.1     27.8     0.3979
----------------------------------------------------------------------
       b = raw coefficient
       z = z-score for test of b=0
   P>|z| = p-value for z-test
       % = percent change in odds for unit increase in X
   %StdX = percent change in odds for SD increase in X
   SDofX = standard deviation of X

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 there are two tests that can be used
to do so.  First, we need to download a user-written command called
omodel (type search omodel).  The first test that we will show
does a likelihood ratio test.  The null hypothesis is that there is no
difference in the coefficients between models, so we “hope” to get a
non-significant result.  Please note that the omodel
command does not recognize factor variables, so the i. is
ommited. The brant command performs a Brant test.
As the note at the bottom of the output indicates, we also “hope” that these
tests are non-significant.  The brant command, like listcoeff,
is part of the spost add-on and can be obtained by typing searchspost. We have used the detail option here,which shows the estimated coefficients for the two equations.  (We have two
equations because we have three categories in our response variable.)
Also, you will note that the likelihood ratio chi-square value of 4.06 obtained
from the omodel command is very close to the 4.34 obtained from the
brant command.

omodel logit apply pared public gpa

Iteration 0:   log likelihood = -370.60264
Iteration 1:   log likelihood =   -358.605
Iteration 2:   log likelihood = -358.51248
Iteration 3:   log likelihood = -358.51244

Ordered logit estimates                           Number of obs   =        400
                                                  LR chi2(3)      =      24.18
                                                  Prob > chi2     =     0.0000
Log likelihood = -358.51244                       Pseudo R2       =     0.0326

------------------------------------------------------------------------------
       apply |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       pared |   1.047664   .2657891     3.94   0.000     .5267266    1.568601
      public |  -.0586828   .2978588    -0.20   0.844    -.6424754    .5251098
         gpa |   .6157458   .2606311     2.36   0.018     .1049183    1.126573
-------------+----------------------------------------------------------------
       _cut1 |   2.203323   .7795353          (Ancillary parameters)
       _cut2 |   4.298767   .8043146 
------------------------------------------------------------------------------

Approximate likelihood-ratio test of proportionality of odds
across response categories:
         chi2(3) =      4.06
       Prob > chi2 =    0.2553

brant, detail

Estimated coefficients from j-1 binary regressions

               y>0         y>1
 pared   1.0596117     .915596
public  -.20055709   .53508208
   gpa   .54824568   .73632132
 _cons  -1.9829709  -4.7544684

Brant Test of Parallel Regression Assumption

    Variable |      chi2   p>chi2    df
-------------+--------------------------
         All |      4.34    0.227     3
-------------+--------------------------
       pared |      0.13    0.716     1
      public |      3.44    0.064     1
         gpa |      0.18    0.672     1
----------------------------------------

A significant test statistic provides evidence that the parallel
regression assumption has been violated.

Both of the above tests indicate that we have not violated the proportional
odds assumption.  If we had, we would want to run our model as a
generalized ordered logistic model using gologit2.  You need to download
gologit2 by typing search gologit2.

We can also obtain predicted probabilities, which are usually easier to
understand than the coefficients or the odds ratios.  We will use the
margins command.
This can be used with either a categorical variable or a continuous variable and
shows the predicted probability for each of the values of the variable
specified.  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.  Hence, if neither of a respondent ‘s parents
have a graduate level education, the predicted probability of applying to
graduate school decreases.  We can see at values each variable is held at
the top of each output.

margins, at(pared=(0/1)) predict(outcome(0)) atmeans

Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==0), predict(outcome(0))

1._at        : pared           =           0
               public          =       .1425 (mean)
               gpa             =    2.998925 (mean)

2._at        : pared           =           1
               public          =       .1425 (mean)
               gpa             =    2.998925 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .5902769   .0268846    21.96   0.000     .5375841    .6429697
          2  |   .3356916   .0549943     6.10   0.000     .2279047    .4434784
------------------------------------------------------------------------------



margins, at(pared=(0/1)) predict(outcome(1)) atmeans

Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==1), predict(outcome(1))

1._at        : pared           =           0
               public          =       .1425 (mean)
               gpa             =    2.998925 (mean)

2._at        : pared           =           1
               public          =       .1425 (mean)
               gpa             =    2.998925 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |    .331053   .0242226    13.67   0.000     .2835775    .3785285
          2  |   .4685299   .0344096    13.62   0.000     .4010883    .5359714
------------------------------------------------------------------------------


margins, at(pared=(0/1)) predict(outcome(2)) atmeans
	
Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==2), predict(outcome(2))

1._at        : pared           =           0
               public          =       .1425 (mean)
               gpa             =    2.998925 (mean)

2._at        : pared           =           1
               public          =       .1425 (mean)
               gpa             =    2.998925 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .0786702   .0132973     5.92   0.000      .052608    .1047323
          2  |   .1957785    .040827     4.80   0.000     .1157591     .275798
------------------------------------------------------------------------------

We can also use the margins command to select values of
a continuous variable and see what the predicted probabilities are at each
point.  Below, we see the predicted probabilities for gpa at 2, 3
and 4.  As you can see, almost for each value of gpa, the highest predicted
probability is for the lowest category of apply and only when gpa is 4, the predicted probability is slightly higher for somewhat likely than unlikely, which makes sense
because most respondents are in that category.  You can also see that the
predicted probability increases for both the middle and highest categories of
apply as gpa increases.

margins, at(gpa=(2/4)) predict(outcome(0)) atmeans

Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==0), predict(outcome(0))

1._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           2

2._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           3

3._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           4

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .6932137    .060112    11.53   0.000     .5753963     .811031
          2  |   .5496956   .0255013    21.56   0.000      .499714    .5996773
          3  |   .3974013   .0665332     5.97   0.000     .2669986    .5278041
------------------------------------------------------------------------------

margins, at(gpa=(2/4)) predict(outcome(1)) atmeans

Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==1), predict(outcome(1))

1._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           2

2._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           3

3._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           4

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .2551558   .0472683     5.40   0.000     .1625116    .3477999
          2  |   .3587569   .0246482    14.56   0.000     .3104474    .4070664
          3  |   .4453892   .0399212    11.16   0.000      .367145    .5236334
------------------------------------------------------------------------------

margins, at(gpa=(2/4)) predict(outcome(2)) atmeans


Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==2), predict(outcome(2))

1._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           2

2._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           3

3._at        : pared           =       .1575 (mean)
               public          =       .1425 (mean)
               gpa             =           4

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .0516305   .0158556     3.26   0.001     .0205541    .0827069
          2  |   .0915475   .0142998     6.40   0.000     .0635204    .1195745
          3  |   .1572095   .0397767     3.95   0.000     .0792486    .2351703
------------------------------------------------------------------------------

Here we loop through the values of apply (0, 1, and 2) and calculate
predicted probabilities when gpa = 3.5, pared = 1, and public
= 1.

forvalues i = 0/2 {
  margins, at(gpa = 3.5 pared = 1 public = 1) predict(outcome(`i'))
}


Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==0), predict(outcome(0))
at           : pared           =           1
               public          =           1
               gpa             =         3.5

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .2807452   .0695883     4.03   0.000     .1443547    .4171357
------------------------------------------------------------------------------

Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==1), predict(outcome(1))
at           : pared           =           1
               public          =           1
               gpa             =         3.5

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .4796188   .0326872    14.67   0.000     .4155531    .5436844
------------------------------------------------------------------------------

Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(apply==2), predict(outcome(2))
at           : pared           =           1
               public          =           1
               gpa             =         3.5

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |    .239636    .063819     3.75   0.000      .114553     .364719
------------------------------------------------------------------------------

Things to consider

See also

References