I've been reading about Item Response Theory during the past few weeks and I'd like to use it to examine how my scales are functioning. The response categories are ordinal. If I understood well, the Graded Response Model is particularly appealing because it is more flexible than other alternatives such as the Partial Credit Model or the Rating Scale Model, providing different "difficulty" parameters per category within each item, and different "discrimination" parameters for each item. Moreover, I'm confident that the response categories are ordered (expressing different degrees of agreement with a statement) so cumulative logit rather than adjacent category models seem adequate. (I might be wrong in various parts of this reasoning though!).
1) Is there a way to fit Graded Response Models in Stata 13, maybe using the gsem command or gsem builder? If so, can anyone shed light on how to do it and how to process the outputs to draw Category Response Curves, Item Information Functions and Standard Error of Measurement function for the scale?
2) If it is not possible to do it with gsem, is it possible to do it with gllamm? In this article it is explained how to fit PCM and RSM but not Graded Response Models. This post suggests it is possible to do it with the thresh() option, in order to relax the proportional odds assumption/constraint, but I have no clue about what to put inside the thresh() option. It seems I have to define equations for the thresholds, and again I'm clueless here.
3) My final question is a conceptual one: If I understood well, the Graded Response Model gives a "discrimination" parameter for each item and C-1 "difficulty" parameters for the C response categories. Is there a model which provides different "discrimination" parameters for each response category as well? Would this be a nominal model? Why is it reasonable to constrain discrimination parameters to be equal for different response categories of an item? (maybe this is not really a "constraint"?)