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 Salt in Popcorn.

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.

Ordinal Logistic Fit for Salt in Popcorn.jmp

The parameter estimates for Salt and Salt*Salt become the coefficients used to find the critical value. Although it appears as a minimum, it is only a minimum with respect to the probability curves. It is really a maximum in the sense of maximizing the probability of the highest response. The Solution portion of the report is shown under Response Surface in Ordinal Logistic Fit for Salt in Popcorn.jmp, where 1.29 is shown under Critical Value.