### Stata Data Analysis Examples Ordered Logistic Regression

#### 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 http://www.ats.ucla.edu/stat/data/ologit.dta, clear

This hypothetical data set has a thee 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.

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.

• OLS regression:  This analysis is problematic because the assumptions of OLS are violated when it is used with a non-interval outcome variable.
• ANOVA:  If you use only one continuous predictor, you could "flip" the model around so that, say, gpa was the outcome variable and apply was the predictor variable.  Then you could run a one-way ANOVA.  This isn't a bad thing to do if you only have one predictor variable (from the logistic model), and it is continuous.
• Multinomial logistic regression:  This is similar to doing ordered logistic regression, except that it is assumed that there is no order to the categories of the outcome variable (i.e., the categories are nominal).  The downside of this approach is that the information contained in the ordering is lost.
• Ordered probit regression:  This is very, very similar to running an ordered logistic regression.  The main difference is in the interpretation of the coefficients.

#### 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 findit spost (see How can I use the findit command to search for programs and get additional help? for more information about using findit).

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 findit 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 findit spost. 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 ologit 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 findit 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, for each value of gpa, the highest predicted probability is for the lowest category of apply, 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

• Perfect prediction:  Perfect prediction means that one value of a predictor variable is associated with only one value of the response variable.  If this happens, Stata will usually issue a note at the top of the output and will drop the cases so that the model can run.
• Sample size:  Both ordered logistic and ordered probit, using maximum likelihood estimates, require sufficient sample size.  How big is big is a topic of some debate, but they almost always require more cases than OLS regression.
• 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, the model may become unstable or it might not run at all.
• Pseudo-R-squared:  There is no exact analog of the R-squared found in OLS.  There are many versions of pseudo-R-squares.  Please see Long and Freese 2005 for more details and explanations of various pseudo-R-squares.
• Diagnostics:  Doing diagnostics for non-linear models is difficult, and ordered logit/probit models are even more difficult than binary models.