Updated: 17 August 2015

Alpha-NOMINATE is a new form of NOMINATE that is fully Bayesian and is meant to replace W-NOMINATE which is now about 32 years old (the multidimensional version, written by Nolan McCarty and Keith Poole is 24 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} Senate. There have been 262 total votes in the Senate as of the August recess of which 220 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 2000 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.

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The mean was 0.9916 strongly indicating that the Senators’ utility functions were Gaussian.

The next plot shows the estimated ideal points for the 100 Senators along with their 95% Credible Intervals. On the left, Senator Sanders (I-VT) is located at -2.74 just off the left edge of the plot. His credible interval runs from about -3.31 to -1.89. Off the right end and not visible are Senator Cruz (R-TX) at 3.84 with a credible interval that runs all the way from 0.958 to 5.398 and Senator Paul (R-KY) at 5.175 with a credible interval that ranges from 4.766 to 5.750.

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The median of the Republican Party is 0.7562 with a standard deviation of 0.0331 and the median of the Democratic Party is -0.9318 with a standard deviation of 0.0583. The probabilities for the median Senator are 0.575 for Murkowski (R-AK), 0.158 for Graham (R-SC), and 0.128 for Heller (R-NV). Assuming that the Republicans vote as a bloc, the probabilities for the the Filibuster Pivot are 0.251 for King (I-ME), 0.218 for Tester (D-MT), 0.197 for Bennet (D-CO), and 0.154 for Warner (D-VA). Again, assuming that the Republicans vote as a bloc, the probabilities for the Veto Override Pivot are 0.174 for Nelson (D-FL), 0.130 for Coons (D-DE), 0.125 for Feinstein (D-CA), and 0.115 for Shaheen (D-NH). President Obama will likely have the votes to prevent a veto override of his nuclear deal with Iran.

[…] According to political scientists on the Voteview site, these five have already been among the most likely members of their party to vote with Republicans. Still, most of them usually tend to reliably side with Democrats on key partisan matters and just defect on a few high-profile issues to burnish their reputation for independence. (Even Manchin, the most conservative Democrat in the chamber, was just added to the party’s leadership Wednesday.) […]

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