Appendix

In this appendix, I explain how the model is identified and provide a sensitivity analysis. When using an ideal point estimation technique, some anchors are necessary in order to identify the model. In my analysis, two points are fixed a priori: the status quo—i.e., SQ = (−2, −2)—and the original treaty submitted by President Wilson—i.e., TO = (2, 2). No ideal points are fixed a priori. This set of anchors is used to set the direction of the space such that a vote for the original treaty is considered as indicative of support for both multilateralism and imperialism as well as opposition to Irish independence. Both senatorial ideal points and the modified versions of the treaty are estimated relative to those anchors. Instead of fixing a set of bill locations, one can alternatively fix a set of senatorial ideal points to identify the model. However, my experience suggests that the model is identified more effectively by fixing the locations of the two bills than by fixing a set of ideal points. For instance, when I fixed William Borah—a well-known isolationist—at (−2, −2) and Gilbert Hitchcockat (2, 2), other isolationists, such as Robert LaFollette, were estimated to be around (−5, −5), despite the fact that Borah and LaFollette had almost identical voting records during the debate. This indicates that fixing ideal points is not effective in identifying the model.

In addition to fixing the SQ and TO, two more anchors are needed to fully identify the model (see Rivers 2003). However, instead of constraining two additional parameters a priori, I let the multiple Monte Carlo Markov Chain (MCMC) chains run and selected the chains that estimate the positions of the irreconcilables as occupying the southwest corner of the figures, which is consistent with the choice of the anchors discussed above. The advantage of this approach is that it minimizes the number of parameters to be fixed a priori, while still allowing the model to be identified.

An additional reason for fixing the location of the status quo is to avoid having to take into account two layers of uncertainty in computing the supermajority winset of the status quo. If the location of the status quo is estimated, the computation of the supermajority winset of the status quo has to take into account the uncertainty in estimating senatorial positions as well as the uncertainty in estimating the status quo. For this reason, fixing the status quo is computationally convenient while at the same time minimizing the number of ideal points or bill locations that need to be fixed a priori for model identification.

Nevertheless, to check the sensitivity of the estimation to the choice of anchors, I re-estimated the model using a different set of anchors. In this case, the only point I fix is the status quo, at (−2, − 2). All other proposal locations and ideal points are estimated relative to that point. The results are similar to the results that we obtain by fixing the status quo and the original treaty. The results are reported in the online appendix.

The model is estimated with a MCMC method. In order to minimize the role of the priors, I assigned diffuse priors. A uniform distribution with a range of −5 and 5 were assigned as prior distributions for location parameters. WinBUGS (Spiegelhalter et al. 1999) is used to fit the model (see the online appendix for the code). The total number of iterations was 55,000. The first 5000 draws were discarded to remove the effects of initial values. The chains were thinned every 50th draw to reduce autocorrelation. The convergence of the MCMC chains is checked using density plots and traceplots (see the online appendix).