Example 29.4: Ordinal Model for Multinomial Data
This example illustrates how you can use the GENMOD procedure to fit a
model to data measured on an ordinal scale. The following statements
create a SAS data set called icecream. The data set contains the
results of a hypothetical taste test of three brands of ice cream.
The three brands are rated for taste on a five point scale from very
good (vg) to very bad (vb). An analysis is performed to assess the
differences in the ratings for the three brands. The variable
taste contains the ratings and brand contains the brands
tested. The variable count contains the number of testers rating
each brand in each category.
The following statements create the icecream data set.
data icecream;
input count brand$ taste$;
datalines;
70 ice1 vg
71 ice1 g
151 ice1 m
30 ice1 b
46 ice1 vb
20 ice2 vg
36 ice2 g
130 ice2 m
74 ice2 b
70 ice2 vb
50 ice3 vg
55 ice3 g
140 ice3 m
52 ice3 b
50 ice3 vb
;
The following statements fit a cumulative logit model to the ordinal
data with the variable taste as the response and the variable
brand as a covariate. The variable count is used as a FREQ
variable.
proc genmod rorder=data;
freq count;
class brand;
model taste = brand / dist=multinomial
link=cumlogit
aggregate=brand
type1
;
estimate 'LogOR12' brand 1 1 / exp;
estimate 'LogOR13' brand 1 0 1 / exp;
estimate 'LogOR23' brand 0 1 1 / exp;
run;
The AGGREGATE=BRAND option in the MODEL statement specifies the
variable brand
as defining multinomial populations for computing deviances and
Pearson chisquares. The RORDER=DATA option specifies that the
taste variable levels be ordered by their order of appearance in the input
data set, that is, from very good (vg) to very bad (vb). By default,
the response is sorted in increasing ASCII order. Always check the
"Response Profiles" table to verify that response levels are
appropriately ordered. The TYPE1 option requests a Type 1 test for
the significance of the covariate brand.
If is the cumulative
probability of the jth or lower taste category, then the odds
ratio comparing x_{1} to x_{2} is as follows:
Refer to McCullagh and Nelder (1989, Chapter 5) for details on the
cumulative logit model. The ESTIMATE statements compute log odds
ratios comparing each of brands. The EXP option in the ESTIMATE
statements exponentiates the log odds ratios to form odds ratio
estimates. Standard errors and confidence intervals are also computed.
Output 29.4.1 displays general information about the model and data,
the levels of the CLASS variable brand, and the total number of
occurrences of the ordered levels of the response variable
taste.
Output 29.4.1: Ordinal Model Information
Model Information 
Data Set 
WORK.ICECREAM 
Distribution 
Multinomial 
Link Function 
Cumulative Logit 
Dependent Variable 
taste 
Frequency Weight Variable 
count 
Observations Used 
15 
Sum Of Frequency Weights 
1045 
Class Level Information 
Class 
Levels 
Values 
brand 
3 
ice1 ice2 ice3 
Response Profile 
Ordered Level 
Ordered Value 
Count 
1 
vg 
140 
2 
g 
162 
3 
m 
421 
4 
b 
156 
5 
vb 
166 

Output 29.4.2 displays estimates of the intercept terms and
covariates and associated statistics. The intercept terms correspond to
the four cumulative logits defined on the
taste categories in the order shown in Output 29.4.1. That is,
Intercept1 is the intercept for the first cumulative logit,
log([(p_{1})/(1p_{1})]),
Intercept2 is the intercept for the second cumulative logit
log([(p_{1}+p_{2})/(1(p_{1}+p_{2}))]), and so forth.
Output 29.4.2: Parameter Estimates
Analysis Of Parameter Estimates 
Parameter 

DF 
Estimate 
Standard Error 
Wald 95% Confidence Limits 
ChiSquare 
Pr > ChiSq 
Intercept1 

1 
1.8578 
0.1219 
2.0967 
1.6189 
232.35 
<.0001 
Intercept2 

1 
0.8646 
0.1056 
1.0716 
0.6576 
67.02 
<.0001 
Intercept3 

1 
0.9231 
0.1060 
0.7154 
1.1308 
75.87 
<.0001 
Intercept4 

1 
1.8078 
0.1191 
1.5743 
2.0413 
230.32 
<.0001 
brand 
ice1 
1 
0.3847 
0.1370 
0.1162 
0.6532 
7.89 
0.0050 
brand 
ice2 
1 
0.6457 
0.1397 
0.9196 
0.3719 
21.36 
<.0001 
brand 
ice3 
0 
0.0000 
0.0000 
0.0000 
0.0000 
. 
. 
Scale 

0 
1.0000 
0.0000 
1.0000 
1.0000 


NOTE: 
The scale parameter was held fixed. 


The Type 1 test displayed in Output 29.4.3 indicates that Brand
is highly significant; that is, there are significant differences
in the brands. The log odds ratios and odds ratios in
the "ESTIMATE Statement Results" table indicate the relative
differences between the brands. For example, the odds ratio
of 2.8 in the "Exp(LogOR12)" row indicates that the odds of brand 1
being in lower taste categories is 2.8 times the odds of brand 2 being
in lower taste categories. Since, in this ordering, the lower categories
represent the more
favorable taste results, this indicates that
brand 1 scored significantly better than brand 2. This is
also apparent from the data in this example.
Output 29.4.3: Type 1 Tests and Odds Ratios
LR Statistics For Type 1 Analysis 
Source 
Deviance 
DF 
ChiSquare 
Pr > ChiSq 
Intercepts 
65.9576 



brand 
9.8654 
2 
56.09 
<.0001 
Contrast Estimate Results 
Label 
Estimate 
Standard Error 
Alpha 
Confidence Limits 
ChiSquare 
Pr > ChiSq 
LogOR12 
1.0305 
0.1401 
0.05 
0.7559 
1.3050 
54.11 
<.0001 
Exp(LogOR12) 
2.8024 
0.3926 
0.05 
2.1295 
3.6878 


LogOR13 
0.3847 
0.1370 
0.05 
0.1162 
0.6532 
7.89 
0.0050 
Exp(LogOR13) 
1.4692 
0.2013 
0.05 
1.1233 
1.9217 


LogOR23 
0.6457 
0.1397 
0.05 
0.9196 
0.3719 
21.36 
<.0001 
Exp(LogOR23) 
0.5243 
0.0733 
0.05 
0.3987 
0.6894 



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