Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
The LOGISTIC Procedure

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 two-way 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 chi-square for testing the proportional odds assumption is 17.287, which is not significant with respect to a chi-square 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 lower-numbered 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
The LOGISTIC Procedure
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=1E-8) satisfied.
Score Test for the Proportional
Odds Assumption
Chi-Square 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 Chi-Square 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
Chi-Square 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 Tau-a 0.500
Pairs 18635 c 0.789

Output 39.2.2: Estimated Covariance Matrix

Multiple Response Cheese Tasting Experiment
The LOGISTIC Procedure
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

Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
Top
Top

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