Consider a study of the effects of various cheese additives on taste. 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
by using the following program. 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.
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 response variable y
is ordinally scaled. A cumulative logit model is used to investigate the effects of the cheese additives on taste. The following
statements invoke PROC HPLOGISTIC 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.
proc hplogistic data=Cheese; freq freq; class Additive(ref='4') / param=ref ; model y=Additive; title 'Multiple Response Cheese Tasting Experiment'; run;
Results from the logistic analysis are shown in Output 10.3.1 through Output 10.3.3.
The "Response Profile" table in Output 10.3.1 shows that the strong dislike (y
=1) end of the rating scale is associated with lower Ordered Values in the "Response Profile" table; hence the probability
of disliking the additives is modeled.
Output 10.3.1: Proportional Odds Model Regression Analysis
Output 10.3.2: Proportional Odds Model Regression Analysis
The positive value (1.6128) for the parameter estimate for Additive
=1 in Output 10.3.3 indicates a tendency toward the lower-numbered categories of the first cheese additive relative to the fourth. In other words,
the fourth additive tastes better than the first additive. Similarly, the second and third additives are both less favorable
than the fourth additive. The relative magnitudes of these slope estimates imply the preference ordering: fourth, first, third,
second.
Output 10.3.3: Proportional Odds Model Regression Analysis
Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Parameter | Taste Rating |
Estimate | Standard Error |
DF | t Value | Pr > |t| |
Intercept | 1 | -7.0802 | 0.5640 | Infty | -12.55 | <.0001 |
Intercept | 2 | -6.0250 | 0.4764 | Infty | -12.65 | <.0001 |
Intercept | 3 | -4.9254 | 0.4257 | Infty | -11.57 | <.0001 |
Intercept | 4 | -3.8568 | 0.3880 | Infty | -9.94 | <.0001 |
Intercept | 5 | -2.5206 | 0.3453 | Infty | -7.30 | <.0001 |
Intercept | 6 | -1.5685 | 0.3122 | Infty | -5.02 | <.0001 |
Intercept | 7 | -0.06688 | 0.2738 | Infty | -0.24 | 0.8071 |
Intercept | 8 | 1.4930 | 0.3357 | Infty | 4.45 | <.0001 |
Additive 1 | 1.6128 | 0.3805 | Infty | 4.24 | <.0001 | |
Additive 2 | 4.9646 | 0.4767 | Infty | 10.41 | <.0001 | |
Additive 3 | 3.3227 | 0.4218 | Infty | 7.88 | <.0001 |