# The HPLOGISTIC Procedure

### Example 9.3 Ordinal Logistic Regression

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;
title 'Multiple Response Cheese Tasting Experiment';
run;
```

Results from the logistic analysis are shown in Output 9.3.1 through Output 9.3.3.

The Response Profile table in Output 9.3.1 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 9.3.1: Proportional Odds Model Regression Analysis

 Multiple Response Cheese Tasting Experiment

The HPLOGISTIC Procedure

Performance Information
Execution Mode Single-Machine

Model Information
Data Source WORK.CHEESE
Response Variable y
Frequency Variable freq
Class Parameterization Reference
Distribution Multinomial
Optimization Technique Newton-Raphson with Ridging

Class Level Information
Class Levels Reference
Value
Values
Additive 4 4 1 2 3 4

 Number of Observations Read 36 28 208 208

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

 You are modeling the probabilities of levels of y having lower Ordered Values in the Response Profile Table.

Output 9.3.2: Proportional Odds Model Regression Analysis

Iteration History
Iteration Evaluations Objective
Function
0 4 2.0668312595 . 0.137412
1 2 1.7319560317 0.33487523 0.062757
2 2 1.7105150048 0.02144103 0.008919
3 2 1.7099716191 0.00054339 0.00035
4 2 1.7099709251 0.00000069 6.981E-7
5 2 1.7099709251 0.00000000 2.98E-12

 Convergence criterion (GCONV=1E-8) satisfied.

Dimensions
Columns in X 11
Number of Effects 2
Max Effect Columns 3
Rank of Cross-product Matrix 11
Parameters in Optimization 11

Fit Statistics
-2 Log Likelihood 711.35
AIC (smaller is better) 733.35
AICC (smaller is better) 734.69
BIC (smaller is better) 770.06

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 148.4539 3 <.0001

The positive value (1.6128) for the parameter estimate for `Additive`=1 in Output 9.3.3 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 9.3.3: 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