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A Multinomial Framework for Ideal Point Estimation

Published online by Cambridge University Press:  09 October 2018

Max Goplerud*
Harvard University, Department of Government, 1737 Cambridge Street, Cambridge, MA 02138, USA. Email:


This paper creates a multinomial framework for ideal point estimation (mIRT) using recent developments in Bayesian statistics. The core model relies on a flexible multinomial specification that includes most common models in political science as “special cases.” I show that popular extensions (e.g., dynamic smoothing, inclusion of covariates, and network models) can be easily incorporated whilst maintaining the ability to estimate a model using a Gibbs Sampler or exact EM algorithm. By showing that these models can be written and estimated using a shared framework, the paper aims to reduce the proliferation of bespoke ideal point models as well as extend the ability of applied researchers to estimate models quickly using the EM algorithm. I apply this framework to a thorny question in scaling survey responses—the treatment of nonresponse. Focusing on the American National Election Study (ANES), I suggest that a simple but principled solution is to treat questions as multinomial where nonresponse is a distinct (modeled) category. The exploratory results suggest that certain questions tend to attract many more invalid answers and that many of these questions (particularly when signaling out particular social groups for evaluation) are masking noncentrist (typically conservative) beliefs.

Copyright © The Author(s) 2018. Published by Cambridge University Press on behalf of the Society for Political Methodology. 

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Author’s note: I thank Kosuke Imai, Gary King, Michael Peress, Marc Ratkovic, Dustin Tingley, Christopher Warshaw, Xiang Zhou and discussants at MPSA 2017 for helpful comments on earlier versions of this paper. All errors remaining are my own. Code to implement the models in the paper, and the mIRT more generally, can be found at

Contributing Editor: Jonathan N. Katz


Agresti, A. 2002. Categorical data analysis . Hoboken, NJ: John Wiley & Sons.CrossRefGoogle Scholar
Ansolabehere, S., Rodden, J., and Snyder, J. M.. 2008. The strength of issues: Using multiple mesures to gauge preference stability, ideological constraint, and issue voting. American Political Science Review 102(2):215232.CrossRefGoogle Scholar
Bailey, M. A., and Maltzman, F.. 2011. The constrained court: Law, politics, and the decisions justices make . Princeton: Princeton University Press.Google Scholar
Bailey, M. A., Strezhnev, A., and Voeten, E.. 2017. Estimating dynamic state preferences from united nations voting data. Journal of Conflict Resolution 61:430456.CrossRefGoogle Scholar
Barberá, P. 2015. Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data. Political Analysis 23:7691.CrossRefGoogle Scholar
Berinsky, A. J. 1999. The two faces of public opinion. American Journal of Political Science 43:12091230.CrossRefGoogle Scholar
Berinsky, A. J. 2002. Silent voices: Social welfare policy opinions and political equality in America. American Journal of Political Science 46:276287.CrossRefGoogle Scholar
Biane, P., Pitman, J., and Yor, M.. 2001. Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bulletin of the American Mathematical Society 38:435466.CrossRefGoogle Scholar
Bock, R. D. 1972. Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika 37:3051.Google Scholar
Carroll, R., Lewis, J. B., Lo, J., Poole, K. T., and Rosenthal, H.. 2009. Measuring bias and uncertainty in DW-nominate ideal point estimates via the parametric bootstrap. Political Analysis 17(3):261275.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M., and Rubin, D. B.. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 39:138.CrossRefGoogle Scholar
Goplerud, M.2018. Replication Data for: A Multinomial Framework for Ideal Point Estimation, doi:10.7910/DVN/LD0ITE, Harvard Dataverse, V1, UNF:6:BZpPtqgMaRyXpWQa5b/NqA==.CrossRefGoogle Scholar
Groseclose, T., and Milyo, J.. 2005. A measure of media bias. The Quarterly Journal of Economics 120:11911237.CrossRefGoogle Scholar
Hill, S. J., and Tausanovitch, C.. 2015. A disconnect in representation? comparison of trends in congressional and public polarization. Journal of Politics 77:10581075.CrossRefGoogle Scholar
Imai, K., Lo, J., and Olmsted, J.. 2016. Fast estimation of ideal points with massive data. American Political Science Review 110:631656.CrossRefGoogle Scholar
Keane, M. P. 1992. A note on identification in the multinomial probit model. Journal of Business & Economic Statistics 10(2):193200.Google Scholar
Lauderdale, B. E., and Clark, T. S.. 2012. The supreme court’s many median justices. American Political Science Review 106(4):847866.CrossRefGoogle Scholar
Lewis, J. B., and Poole, K. T.. 2004. Measuring bias and uncertainty in ideal point estimates via the parametric bootstrap. Political Analysis 12(2):105127.CrossRefGoogle Scholar
Linderman, S. W., Johnson, M. J., and Adams, R. P.. 2015. Dependent multinomial models made easy: Stick-breaking augmentation. In Advances in Neural Information Processing Systems 28: 29th Annual Conference on Neural Information Processing Systems 2015 , ed. Cortes, C., Lee, D. D., Garnett, R., Lawrence, N. D., and Sugiyama, M.. Red Hook, NY: Curran Associates, pp. 34563464.Google Scholar
Lo, J. 2013. Voting present: Obama and the Illinois senate 1994–2004. SAGE Open 3:113.CrossRefGoogle Scholar
Mare, R. D. 1980. Social background and school continuation decisions. Journal of the American Statistical Association 75(370):295305.CrossRefGoogle Scholar
Martin, A. D., and Quinn, K. M.. 2002. Dynamic ideal point estimation via Markov chain Monte Carlo for the U.S. supreme court, 1953–1999. Political Analysis 10(2):134153.CrossRefGoogle Scholar
Martin, A. D., Quinn, K. M., and Park, J. H.. 2011. MCMCpack: Markov chain Monte Carlo in R. Journal of Statistical Software 42(9):121.CrossRefGoogle Scholar
McFadden, D. 1974. Conditional logit analysis of qualitative choice behavior. In Frontiers in econometrics , ed. Zaremmbka, P.. New York: Academic Press.Google Scholar
McFadden, D., and Train, K. E.. 2000. Mixed MNL models for discrete response. Journal of Applied Econometrics 15:447470.3.0.CO;2-1>CrossRefGoogle Scholar
Meng, X.-L., and Rubin, D. B.. 1993. Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika 80(2):267278.CrossRefGoogle Scholar
Polson, N. G., Scott, J. G., and Windle, J.. 2013. Bayesian inference for logistic models using pólya–gamma latent variables. Journal of the American Statistical Association 108(504):13391349.CrossRefGoogle Scholar
Poole, K. T. 2000. Non-parametric unfolding of binary choice data. Political Analysis 8(3):211232.CrossRefGoogle Scholar
Poole, K. T., and Rosenthal, H.. 1997. Congress: A political-economic history of roll call voting . New York: Oxford University Press.Google Scholar
Rivers, D.2003 Identification of multidimensional spatial voting models. Stanford University Typescript.Google Scholar
Rosas, G., Shomer, Y., and Haptonstahl, S. R.. 2015. No news is news: Nonignorable nonresponse in roll-call data analysis. American Journal of Political Science 59(2):511528.CrossRefGoogle Scholar
Scott, J. G., and Sun, L.. 2013. Expectation-maximization for logistic regression. Preprint, arXiv:13060040v1.Google Scholar
Train, K. E. 1998. Recreation demand models with taste differences over people. Land Economics 74:230239.CrossRefGoogle Scholar
Treier, S., and Hillygus, D. S.. 2009. The nature of political ideology in the contemporary electorate. Public Opinion Quarterly 73:679703.CrossRefGoogle Scholar
Treier, S., and Jackman, S.. 2008. Democracy as a latent variable. American Journal of Political Science 52:201217.CrossRefGoogle Scholar
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