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 fivepoint scale from very good (vg) to very bad (vb). An analysis is performed to assess the differences in
the ratings of the three brands. The variable taste
contains the ratings, and the variable 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 data=Icecream 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 to is as follows:

See 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 40.4.1 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 40.4.1: Ordinal Model Information
Model Information  

Data Set  WORK.ICECREAM 
Distribution  Multinomial 
Link Function  Cumulative Logit 
Dependent Variable  taste 
Frequency Weight Variable  count 
Class Level Information  

Class  Levels  Values 
brand  3  ice1 ice2 ice3 
Response Profile  

Ordered Value 
taste  Total Frequency 
1  vg  140 
2  g  162 
3  m  421 
4  b  156 
5  vb  166 
Output 40.4.2 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 40.4.1. That is, Intercept1
is the intercept for the first cumulative logit, , Intercept2
is the intercept for the second cumulative logit, , and so forth.
Output 40.4.2: Parameter Estimates
Analysis Of Maximum Likelihood Parameter Estimates  

Parameter  DF  Estimate  Standard Error  Wald 95% Confidence Limits  Wald 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 40.4.3 indicates that Brand
is highly significant; that is, there are significant differences among the brands. The log odds ratios and odds ratios in
the “ESTIMATE Statement Results” table indicate the relative differences among 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 40.4.3: 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  Mean Estimate  Mean  L'Beta Estimate  Standard Error  Alpha  L'Beta  ChiSquare  Pr > ChiSq  
Confidence Limits  Confidence Limits  
LogOR12  0.7370  0.6805  0.7867  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  0.5950  0.5290  0.6577  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  0.3439  0.2850  0.4081  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 