Modeling and Multivariate Methods - SAS

224 Performing Logistic Regression on Nominal and Ordinal Responses Chapter 7
Example of a Quadratic Ordinal Logistic Model
Example of a Quadratic Ordinal Logistic Model
The Ordinal Response Model can fit a quadratic surface to optimize the probabilities of the higher or lower
responses. The arithmetic in terms of the structural parameters is the same as that for continuous responses.
Up to five factors can be used, but this example has only one factor, for which there is a probability plot.
Consider the case of a microwave popcorn manufacturer who wants to find out how much salt consumers
like in their popcorn. To do this, the manufacturer looks for the maximum probability of a favorable
response as a function of how much salt is added to the popcorn package. An experiment controls salt
amount at 0, 1, 2, and 3 teaspoons, and the respondents rate the taste on a scale of 1=low to 5=high. The
optimum amount of salt is the amount that maximizes the probability of more favorable responses. The ten
observations for each of the salt levels are shown in Table 7.2.
Table 7.2 Salt in Popcorn
Salt
Amount
Salt Rating Response
no salt 1 3 2 4 2 2 1 4 3 4
1 tsp. 4 5 3 4 5 4 5 5 4 5
2 tsp. 4 3 5 1 4 2 5 4 3 2
3 tsp. 3 1 2 3 1 2 1 2 1 2
Use Fit Model with the Salt in Popcorn.jmp sample data to fit the ordinal taste test to the surface effect of
salt. Use Taste Test as Y. Highlight Salt in the Select Columns box, and then select Macros >
Response Surface.
The report shows how the quadratic model fits the response probabilities. The curves, instead of being
shifted logistic curves, become a folded pile of curves where each curve achieves its optimum at the same
point. The critical value is at Mean(X)–0.5 *b1/b2 where b1 is the linear coefficient and b2 is the quadratic
coefficient. This formula is for centered X. From the Parameter Estimates table, you can compute the
optimum as 1.5 - 0.5* (0.5637/1.3499) = 1.29 teaspoons of salt.