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For one thing, there are equally many parameters (binary choice probabilities) as there are empirical cells (binary choice proportions), yet we can tell from the figures that the models can be extremely restrictive, especially in highdimensional spaces, and hence, must be testable. As explained in Davis-Stober and Brown (2011) one cannot simply count parameters to evaluate the complexity of these types of models.
The second reason, returning to data like those of DM1 and DM 13 in Figure 3 is that the best fitting parameters, i.the maximum-likelihood estimate, satisfying an order-constrained model may lie on a face, an edge, or even a vertex of the shaded modal choice cube. This becomes even more complicated in higher dimensional spaces, where the modal choice model has surfaces of many different dimensions.