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| The LOGISTIC Procedure |
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|
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