Skip to main content Accessibility help

A Multinomial Framework for Ideal Point Estimation

  • Max Goplerud (a1)


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.


Corresponding author


Hide All

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



Hide All
Agresti, A. 2002. Categorical data analysis . Hoboken, NJ: John Wiley & Sons.
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.
Bailey, M. A., and Maltzman, F.. 2011. The constrained court: Law, politics, and the decisions justices make . Princeton: Princeton University Press.
Bailey, M. A., Strezhnev, A., and Voeten, E.. 2017. Estimating dynamic state preferences from united nations voting data. Journal of Conflict Resolution 61:430456.
Barberá, P. 2015. Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data. Political Analysis 23:7691.
Berinsky, A. J. 1999. The two faces of public opinion. American Journal of Political Science 43:12091230.
Berinsky, A. J. 2002. Silent voices: Social welfare policy opinions and political equality in America. American Journal of Political Science 46:276287.
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.
Bock, R. D. 1972. Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika 37:3051.
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.
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.
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==.
Groseclose, T., and Milyo, J.. 2005. A measure of media bias. The Quarterly Journal of Economics 120:11911237.
Hill, S. J., and Tausanovitch, C.. 2015. A disconnect in representation? comparison of trends in congressional and public polarization. Journal of Politics 77:10581075.
Imai, K., Lo, J., and Olmsted, J.. 2016. Fast estimation of ideal points with massive data. American Political Science Review 110:631656.
Keane, M. P. 1992. A note on identification in the multinomial probit model. Journal of Business & Economic Statistics 10(2):193200.
Lauderdale, B. E., and Clark, T. S.. 2012. The supreme court’s many median justices. American Political Science Review 106(4):847866.
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.
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.
Lo, J. 2013. Voting present: Obama and the Illinois senate 1994–2004. SAGE Open 3:113.
Mare, R. D. 1980. Social background and school continuation decisions. Journal of the American Statistical Association 75(370):295305.
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.
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.
McFadden, D. 1974. Conditional logit analysis of qualitative choice behavior. In Frontiers in econometrics , ed. Zaremmbka, P.. New York: Academic Press.
McFadden, D., and Train, K. E.. 2000. Mixed MNL models for discrete response. Journal of Applied Econometrics 15:447470.
Meng, X.-L., and Rubin, D. B.. 1993. Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika 80(2):267278.
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.
Poole, K. T. 2000. Non-parametric unfolding of binary choice data. Political Analysis 8(3):211232.
Poole, K. T., and Rosenthal, H.. 1997. Congress: A political-economic history of roll call voting . New York: Oxford University Press.
Rivers, D.2003 Identification of multidimensional spatial voting models. Stanford University Typescript.
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.
Scott, J. G., and Sun, L.. 2013. Expectation-maximization for logistic regression. Preprint, arXiv:13060040v1.
Train, K. E. 1998. Recreation demand models with taste differences over people. Land Economics 74:230239.
Treier, S., and Hillygus, D. S.. 2009. The nature of political ideology in the contemporary electorate. Public Opinion Quarterly 73:679703.
Treier, S., and Jackman, S.. 2008. Democracy as a latent variable. American Journal of Political Science 52:201217.
MathJax is a JavaScript display engine for mathematics. For more information see


Related content

Powered by UNSILO
Type Description Title
Supplementary materials

Goplerud supplementary material
Appendix A

 Unknown (702 KB)
702 KB

A Multinomial Framework for Ideal Point Estimation

  • Max Goplerud (a1)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.