Update: 3 December 2015. While we await the completion of voting this week we revisit the latest results from alpha-NOMINATE. A multidimensional version of alpha-NOMINATE will be completed within a few months and we will be showing both one and two dimensional results when it is completed.

As we discussed in earlier posts, alpha-NOMINATE is a new form of NOMINATE that is fully Bayesian and is meant to replace W-NOMINATE which is now about 33 years old (the multidimensional version, written by Nolan McCarty and Keith Poole is almost 25 years old). NOMINATE was designed by Keith Poole and Howard Rosenthal during 1982-1983. It used a random utility model with a Gaussian deterministic utility function (see pages 14 – 15 of the linked 1983 paper) and logistic error (random draws from the log of the inverse exponential). The Gaussian deterministic utility function is able to capture non-voting due to indifference **and** alienation.

Alpha-NOMINATE is a mixture model in which legislators’ utility functions are allowed to be a mixture of the two most commonly assumed utility functions: the quadratic function and the Gaussian function assumed by NOMINATE. The “Alpha” is a parameter estimated by Alpha-NOMINATE that varies from 0 (Quadratic Utility) to 1 (Gaussian Utility). Hence, in one dimension with Alpha = 0, Alpha-NOMINATE is identical to the popular IRT model. Thus Alpha-NOMINATE can actually test whether or not legislators’ utility functions are Quadratic or Gaussian.

Below we apply Alpha-NOMINATE to the 114^{th} House. There have been 643 total votes in the House as of the Thanksgiving recess of which 569 are scalable (at least 2.5% in the minority; that is, votes that are 97-3 to 50-50). We used the R version of Alpha-NOMINATE to perform the analysis. We used 4000 samples from a slice sampler in one dimension with a burn-in of 1000. The first graph shows the Trace and Density plots for alpha.

*Click image to enlarge*

The mean of alpha is 0.99987 with a standard deviation of 0.000013 strongly indicating that the Representatives’ utility functions were Gaussian.

Below is a smoothed histogram of the 3000 configurations after burn-in. The divide between Democrats and Republicans is a very deep one.

*Click image to enlarge*

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