Example 39.2: Ordinal Logistic Regression
Consider a study of the effects on taste of various cheese
additives. Researchers tested
four cheese additives and obtained 52 response ratings for each
additive.
Each response was measured on a scale of
nine categories ranging from
strong dislike (1) to excellent taste (9).
The data, given in McCullagh and Nelder (1989, p. 175) in the form of
a twoway frequency table of additive by rating, are saved in the
data set Cheese.
data Cheese;
do Additive = 1 to 4;
do y = 1 to 9;
input freq @@;
output;
end;
end;
label y='Taste Rating';
datalines;
0 0 1 7 8 8 19 8 1
6 9 12 11 7 6 1 0 0
1 1 6 8 23 7 5 1 0
0 0 0 1 3 7 14 16 11
;
The data set Cheese contains the variables y,
Additive, and freq. The variable
y contains the response rating. The variable Additive
specifies the cheese additive (1, 2, 3, or 4). The
variable freq gives the frequency with which each additive
received each rating.
The response variable y is ordinally scaled. A
cumulative logit model is used to investigate the effects of
the cheese additives on taste. The following SAS statements
invoke PROC LOGISTIC to fit this model with y as the
response variable and three indicator variables as explanatory variables,
with the fourth additive as the reference level. With this
parameterization, each Additive parameter compares an additive to
the fourth additive. The
COVB option produces the estimated covariance matrix.
proc logistic data=Cheese;
freq freq;
class Additive (param=ref ref='4');
model y=Additive / covb;
title1 'Multiple Response Cheese Tasting Experiment';
run;
Results of the analysis are shown in Output 39.2.1, and
the estimated covariance matrix is displayed in Output 39.2.2.
Since the strong dislike (y=1) end of the rating scale is
associated with lower Ordered Values in the Response Profile table, the
probability of disliking the additives is modeled.
The score chisquare for testing the proportional
odds assumption is 17.287,
which is not significant with respect to a chisquare distribution with 21
degrees of freedom
(p=0.694). This indicates that the proportional
odds model adequately fits the data.
The positive value
(1.6128) for the parameter estimate for Additive1 indicates
a tendency towards the lowernumbered categories of the first
cheese additive relative to the fourth. In other words, the fourth additive
is better in taste than the first additive. Each of the second and the
third additives is less favorable than the
fourth additive. The relative magnitudes of these slope estimates
imply the preference ordering: fourth, first, third, second.
Output 39.2.1: Proportional Odds Model Regression Analysis
Multiple Response Cheese Tasting Experiment 
Model Information 
Data Set 
WORK.CHEESE 

Response Variable 
y 
Taste Rating 
Number of Response Levels 
9 

Number of Observations 
28 

Frequency Variable 
freq 

Sum of Frequencies 
208 

Link Function 
Logit 

Optimization Technique 
Fisher's scoring 

Response Profile 
Ordered Value 
y 
Total Frequency 
1 
1 
7 
2 
2 
10 
3 
3 
19 
4 
4 
27 
5 
5 
41 
6 
6 
28 
7 
7 
39 
8 
8 
25 
9 
9 
12 
NOTE: 
8 observations having zero frequencies or weights were excluded since they do not contribute to the analysis. 

Model Convergence Status 
Convergence criterion (GCONV=1E8) satisfied. 
Score Test for the Proportional Odds Assumption 
ChiSquare 
DF 
Pr > ChiSq 
17.2866 
21 
0.6936 
Model Fit Statistics 
Criterion 
Intercept Only 
Intercept and Covariates 
AIC 
875.802 
733.348 
SC 
902.502 
770.061 
2 Log L 
859.802 
711.348 
Testing Global Null Hypothesis: BETA=0 
Test 
ChiSquare 
DF 
Pr > ChiSq 
Likelihood Ratio 
148.4539 
3 
<.0001 
Score 
111.2670 
3 
<.0001 
Wald 
115.1504 
3 
<.0001 
Analysis of Maximum Likelihood Estimates 
Parameter 

DF 
Estimate 
Standard Error 
ChiSquare 
Pr > ChiSq 
Intercept 

1 
7.0801 
0.5624 
158.4851 
<.0001 
Intercept2 

1 
6.0249 
0.4755 
160.5500 
<.0001 
Intercept3 

1 
4.9254 
0.4272 
132.9484 
<.0001 
Intercept4 

1 
3.8568 
0.3902 
97.7087 
<.0001 
Intercept5 

1 
2.5205 
0.3431 
53.9704 
<.0001 
Intercept6 

1 
1.5685 
0.3086 
25.8374 
<.0001 
Intercept7 

1 
0.0669 
0.2658 
0.0633 
0.8013 
Intercept8 

1 
1.4930 
0.3310 
20.3439 
<.0001 
Additive 
1 
1 
1.6128 
0.3778 
18.2265 
<.0001 
Additive 
2 
1 
4.9645 
0.4741 
109.6427 
<.0001 
Additive 
3 
1 
3.3227 
0.4251 
61.0931 
<.0001 
Association of Predicted Probabilities and Observed Responses 
Percent Concordant 
67.6 
Somers' D 
0.578 
Percent Discordant 
9.8 
Gamma 
0.746 
Percent Tied 
22.6 
Taua 
0.500 
Pairs 
18635 
c 
0.789 

Output 39.2.2: Estimated Covariance Matrix
Multiple Response Cheese Tasting Experiment 
Estimated Covariance Matrix 
Variable 
Intercept 
Intercept2 
Intercept3 
Intercept4 
Intercept5 
Intercept6 
Intercept7 
Intercept8 
Additive1 
Additive2 
Additive3 
Intercept 
0.316291 
0.219581 
0.176278 
0.147694 
0.114024 
0.091085 
0.057814 
0.041304 
0.09419 
0.18686 
0.13565 
Intercept2 
0.219581 
0.226095 
0.177806 
0.147933 
0.11403 
0.091081 
0.057813 
0.041304 
0.09421 
0.18161 
0.13569 
Intercept3 
0.176278 
0.177806 
0.182473 
0.148844 
0.114092 
0.091074 
0.057807 
0.0413 
0.09427 
0.1687 
0.1352 
Intercept4 
0.147694 
0.147933 
0.148844 
0.152235 
0.114512 
0.091109 
0.05778 
0.041277 
0.09428 
0.14717 
0.13118 
Intercept5 
0.114024 
0.11403 
0.114092 
0.114512 
0.117713 
0.091821 
0.057721 
0.041162 
0.09246 
0.11415 
0.11207 
Intercept6 
0.091085 
0.091081 
0.091074 
0.091109 
0.091821 
0.09522 
0.058312 
0.041324 
0.08521 
0.09113 
0.09122 
Intercept7 
0.057814 
0.057813 
0.057807 
0.05778 
0.057721 
0.058312 
0.07064 
0.04878 
0.06041 
0.05781 
0.05802 
Intercept8 
0.041304 
0.041304 
0.0413 
0.041277 
0.041162 
0.041324 
0.04878 
0.109562 
0.04436 
0.0413 
0.04143 
Additive1 
0.09419 
0.09421 
0.09427 
0.09428 
0.09246 
0.08521 
0.06041 
0.04436 
0.142715 
0.094072 
0.092128 
Additive2 
0.18686 
0.18161 
0.1687 
0.14717 
0.11415 
0.09113 
0.05781 
0.0413 
0.094072 
0.22479 
0.132877 
Additive3 
0.13565 
0.13569 
0.1352 
0.13118 
0.11207 
0.09122 
0.05802 
0.04143 
0.092128 
0.132877 
0.180709 

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.