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 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 |

Number of Threads | 4 |

Model Information | |
---|---|

Data Source | WORK.CHEESE |

Response Variable | y |

Frequency Variable | freq |

Class Parameterization | Reference |

Distribution | Multinomial |

Link Function | Cumulative Logit |

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 |
---|---|

Number of Observations Used | 28 |

Sum of Frequencies Read | 208 |

Sum of Frequencies Used | 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 |
Change | Max Gradient |

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 |