There are a bunch of methods floating around for estimating ideal points of legislators and judges. We’ve done some work on the logistic regression (“3-parameter Rasch”) model, and it might be helpful to see some references to other approaches.
I don’t have any unified theory of these models, and I don’t really have any good reason to prefer any of these models to any others. Just a couple of general comments: (1) Any model that makes probabilistic predictions can be judged on its own terms by comparing to actual data. (2) When a model is multidimensional, the number of dimensions is a modeling choice. (In our paper, we use 1-dimensional models but in any given application we would consider that as just a starting point. More dimensions will explain more of the data, which is a good thing.) I do not consider the number of dimensions to be, in any real sense, a “parameter” to be estimated.
Now, on to the models.
Most of us are familiar with the Poole and Rosenthal model for ideal points in roll-call voting. The website has tons of data and some cool dynamic graphics.
For a nice overview of distance-based models, see Simon Jackman’s webpage on ideal-point models. This page has a derivation of the model from first principles along with code for fitting it yourself.
Aleks Jakulin has come up with his own procedure for hierarchical classification of legislators using roll-call votes and has lots of detail and cool pictures on his website. He also discusses the connection of these measures to voting power.
Jan de Leeuw has a paper on ideal point estimation as an example of principal component analysis. The paper is mostly about computation but it has an interesting discussion of some general ideas about how to model this sort of data.
Any other good references on this stuff? Let us know.