NOMINATE

W-NOMINATE (Weighted NOMINATE) estimates static ideal points for a single legislature (e.g., one congress), so its scores are not directly comparable across separate time periods. DW-NOMINATE (Dynamic, Weighted) models each legislator's position as a smooth polynomial over time and pools information across many congresses, putting legislators from different eras on a common scale. For cross-time polarization analysis, DW-NOMINATE is the appropriate tool; W-NOMINATE suits a single body.

NOMINATE assumes a Gaussian (bell-shaped) deterministic utility that flattens for outcomes far from the legislator, whereas the item-response ideal point model uses a quadratic utility that keeps increasing the penalty with distance. The normal form captures the empirical pattern that very distant alternatives are roughly equally unattractive, and it gave NOMINATE strong fit to congressional data. In practice the two functional forms yield highly correlated estimates.

Because NOMINATE is fit by maximum likelihood rather than MCMC, it lacks a posterior distribution. Carroll and colleagues use the parametric bootstrap: many vote matrices are simulated from the fitted model's predicted probabilities, NOMINATE is re-estimated on each, and the variability of the recovered coordinates yields standard errors and confidence regions. This reveals that legislators with few or lopsided votes have much less precisely estimated positions.