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I would like to understand the difference between the binomial, multinomial and ordinal models when it comes to interpreting the "distance" between classes, in terms of response probabilities.

To illustrate my question, I have prepared the following graph (just for the sake of example)

enter image description here

Imagine I have use the logit link, and plotted the cumulative curves corresponding to 3 classes with a series of binomial regressions (first case), then I applied multinomial and ordinal regressions (cases 2 and 3, say for 4 classes, one of which is a reference class - not represented because the cumulative probability is 1).

In every case, I will be able to estimate a mean response (probabilities) for each class, $\mu_i$. But the parameters will be different - understandably, as the formulation of the odds in the link function is different.

Now what I dont understand, is how to interpret the distance between the classes (in green in my graph), in other words all the probabilities that are smaller than $\mu_2$ but greater than $\mu_1$, for example. What are the differences in interpretation according to the model I am choosing?

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For the binomial case we have just two classes. What makes the difference between classes is then the intercept. For ordinal regression this is again simple if the proportional odds assumption holds for your data.

For a multenter image description hereinomial problem it is different since the apart from the intercept different functions separate classes. Have a look at the image I have attached.

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  • $\begingroup$ So in the ordinal case, can I say that the distance between each class, or the difference between two thresholds (at $i+1$ and $i$), corresponds to the range of probabilities of $class = class_i$? $\endgroup$
    – Neodyme
    Dec 18 '13 at 7:15
  • $\begingroup$ and if I understand your graph correctly this wouldnt be true for the multinomial and binary logistic cases? $\endgroup$
    – Neodyme
    Dec 18 '13 at 7:17

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