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?
The proportional odds model involves, at first, doing some individual logisitic regressions. Logistic regression involves a binary variable so we will introduce a new indicator variable that will given a value of 1 if the rating is equal to or less than one, and 0 if the rating is two or more. We can now use logistic regression to determine proportion of ratings that are 1 or bigger than 1.
Next, just as before, we will introduce a new indicator variable this time that will given a value of 1 if the rating is equal to or less than two, and 0 if the rating is three or more. We can now perform a second logisitic regression that will provide us with a second fitted model used to determine proportion of ratings that are 2 or bigger than 2.
We will continue performing individual logistic regressions in this same manner for the next higher level of rating and so forth... until I get up to 8.
Here is a link to the details of these 8 fitted logistic regression models with the coefficients for each of these highlighted in yellow. ( Details of Fitted Logistic Regression Models )
What do these individual regressions have to do with determining a proportional odds model?
Let's take a look at all of these coefficients from each of these models in summary...
What does this model assume? Are all of the U's equal? In order to answer this question you need to know something about standard deviations. Here is the standard error reported by SAS for the last model shown above...
How does this help?
Here are the SAS results for the Score Test for the Proportional Odds Assumption. Is this significant? What does this tell us?
Where do we go from here?
(Plot of the 8 models here? Dr. Harkness... Curves that are parrallel...)
How might this relate to an Analysis of Covariance?
Advanced Use of Polytomous Logistic Regression
There is something more that you can do with polytomous logistic regression. What if the categorical variable, instead of being a quantitative explanatory variable such as 'fat level' is in this current study, but was strictly categorical? Is polytomous logistic regression still an appropriate approach?