Multidimensional Unfolding

Ordinary multidimensional scaling represents a set of objects in a space so that pairwise dissimilarities match distances among those objects only. Unfolding is a two-mode generalization: it places two different kinds of entities — actors and the stimuli they evaluate — in the same space, so an actor's preference for a stimulus is read as the distance between them. The political-science unfolding model adds the single-peaked assumption that proximity means preference, which is why it can recover both voter and candidate positions from a preference matrix rather than only relative object positions. See the linked generic unfolding-model entry for the broader psychometric treatment.

Optimal classification (OC) is Poole's nonparametric scaling procedure that estimates legislator positions and roll-call cutting planes by maximizing the number of votes correctly classified, using only the rank-order geometry of choices and making no assumption about the error distribution. NOMINATE, by contrast, is a parametric maximum-likelihood model with a specified utility and error structure. OC is more robust to distributional misspecification and is often used to check NOMINATE results; the two typically agree closely on the recovered configuration. See the linked nominate-estimation entry for the parametric counterpart.

Dimensionality is assessed by how much the proportion of correctly classified choices improves as dimensions are added and by inspecting whether additional dimensions are substantively interpretable. A large jump from one to two dimensions followed by diminishing returns suggests a two-dimensional space, as Poole and Rosenthal found for much of U.S. congressional history. Scree-style plots of classification fit, aggregate proportional reduction in error, and the interpretability of cutting-plane patterns together guide the choice, since adding dimensions always improves in-sample fit mechanically.