How well does the regression model work for this data? What if someone did not know about polytomous logistic regression and relied solely on a regression approach to predict optimum fat levels in ice cream? How close does this come?

If we knew nothing about polytomous logisitic regression, how good of a job would ordinary multiple regression do on developing a predictive model for this data? Here is the data, rearranged from the table showing the replications.

When we have replications we need to do a weighted least squares regression, one that is weighted to account for these replications.

Here is the output from Minitab for this calculating a quadratic regression equation of 'Rating' on 'Fat' :

What level of 'Fat' (U) maximizes the average rating? The fitted regression equation is given by

\(Y = 4.2768977 + 36.2380923U - 125.2658465U^2 \)

We can find the maximizing value by differentiating this equation and setting the result equal to 0:

\(\frac{dY}{dU}=36.2380923-250.0531693U=0 \rightarrow \hat{U}=36.2380923 / 250.0531693 =0.144648\)

So, using this least weighted squares regression approach we find that a fat level of 0.144648 yields the highest average rating of participants.