The GENMOD Procedure

Example 42.4 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 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$;
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
   estimate 'LogOR12' brand 1 -1 / exp;
   estimate 'LogOR13' brand 1  0  -1 / exp;
   estimate 'LogOR23' brand 0  1  -1 / exp;

The AGGREGATE=BRAND option in the MODEL statement specifies the variable brand as defining multinomial populations for computing deviances and Pearson chi-squares. 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 $\gamma _ j(\mb {x}) = \mr {Pr}(\mr {taste} \le j)$ is the cumulative probability of the jth or lower taste category, then the odds ratio comparing $\mb {x}_1$ to $\mb {x}_2$ is as follows:

\[  \frac{\gamma _ j(\mb {x}_1)/(1-\gamma _ j(\mb {x}_1))}{\gamma _ j(\mb {x}_2)/(1-\gamma _ j(\mb {x}_2))} = \exp [(\mb {x}_1-\mb {x}_2)^\prime \bbeta ]  \]

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 42.4.1 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 42.4.1: Ordinal Model Information

The GENMOD Procedure

Model Information
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
taste Total
1 vg 140
2 g 162
3 m 421
4 b 156
5 vb 166

Output 42.4.2 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 42.4.1. That is, Intercept1 is the intercept for the first cumulative logit, $\log (\frac{p_1}{1-p_1})$, Intercept2 is the intercept for the second cumulative logit, $\log (\frac{p_1+p_2}{1-(p_1+p_2)})$, and so forth.

Output 42.4.2: Parameter Estimates

Analysis Of Maximum Likelihood Parameter Estimates
Parameter   DF Estimate Standard Error Wald 95% Confidence Limits Wald Chi-Square 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 42.4.3 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 42.4.3: Type 1 Tests and Odds Ratios

LR Statistics For Type 1 Analysis
Source Deviance DF Chi-Square 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 Chi-Square 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