### SAS Data Analysis Examples Ordinal 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 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 here.

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 = "D:\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


The FREQ Procedure Cumulative Cumulative APPLY Frequency Percent Frequency Percent ---------------------------------------------------------- 0 220 55.00 220 55.00 1 140 35.00 360 90.00 2 40 10.00 400 100.00 Cumulative Cumulative PARED Frequency Percent Frequency Percent ---------------------------------------------------------- 0 337 84.25 337 84.25 1 63 15.75 400 100.00 Cumulative Cumulative PUBLIC Frequency Percent Frequency Percent ----------------------------------------------------------- 0 343 85.75 343 85.75 1 57 14.25 400 100.00
proc means data = "D:\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.

• 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 ordinal 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.
• Ordinal probit regression:  This is very, very similar to running an ordinal logistic regression.  The main difference is in the interpretation of the coefficients.

#### 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 = "D:\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 = "D:\ologit" desc;
class pared(ref='0') public(ref='0') / param=reference;
model apply = pared public gpa;
run;
The LOGISTIC Procedure

Model Information

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

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 = "C:\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***
Estimate

Standard
Standard                               Error of
Label                       APPLY  Estimate     Error  z Value  Pr > |z|      Mean      Mean

Pr prob apply=2 at pared=0  2       -2.5108    0.3104    -8.09    <.0001   0.07511   0.02156

Estimate

Standard
Standard                               Error of
Label                       APPLY  Estimate     Error  z Value  Pr > |z|      Mean      Mean

Pr prob apply=2 at pared=1  2       -1.4629    0.3545    -4.13    <.0001    0.1880   0.05412

Estimate

Standard
Standard                               Error of
Label                            APPLY  Estimate     Error  z Value  Pr > |z|      Mean      Mean

Pr prob apply=1 or 2 at pared=0  1       -0.4153    0.2733    -1.52    0.1286    0.3976   0.06546

Estimate

Standard
Standard                               Error of
Label                            APPLY  Estimate     Error  z Value  Pr > |z|      Mean      Mean

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 proc logistic statement and supply a new estimate statement to get the probabilities of being in apply category 0.

proc logistic data = "C:\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***
Estimate

Standard
Standard                               Error of
Label                       APPLY  Estimate     Error  z Value  Pr > |z|      Mean      Mean

Pr prob apply=0 at pared=0  0        0.4153    0.2733     1.52    0.1286    0.6024   0.06546

Estimate

Standard
Standard                               Error of
Label                       APPLY  Estimate     Error  z Value  Pr > |z|      Mean      Mean

Pr prob apply=0 at pared=1  0       -0.6325    0.3451    -1.83    0.0668    0.3469   0.07818



#### Things to consider

• Perfect prediction:  Perfect prediction means that only 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 ordinal logistic and ordinal 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 (a small cell), 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.

#### References

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