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Bayesian and Likelihood Inference for 2 × 2 Ecological Tables: An Incomplete-Data Approach

  • Kosuke Imai (a1), Ying Lu (a2) and Aaron Strauss (a3)


Ecological inference is a statistical problem where aggregate-level data are used to make inferences about individual-level behavior. In this article, we conduct a theoretical and empirical study of Bayesian and likelihood inference for 2 × 2 ecological tables by applying the general statistical framework of incomplete data. We first show that the ecological inference problem can be decomposed into three factors: distributional effects, which address the possible misspecification of parametric modeling assumptions about the unknown distribution of missing data; contextual effects, which represent the possible correlation between missing data and observed variables; and aggregation effects, which are directly related to the loss of information caused by data aggregation. We then examine how these three factors affect inference and offer new statistical methods to address each of them. To deal with distributional effects, we propose a nonparametric Bayesian model based on a Dirichlet process prior, which relaxes common parametric assumptions. We also identify the statistical adjustments necessary to account for contextual effects. Finally, although little can be done to cope with aggregation effects, we offer a method to quantify the magnitude of such effects in order to formally assess its severity. We use simulated and real data sets to empirically investigate the consequences of these three factors and to evaluate the performance of our proposed methods. C code, along with an easy-to-use R interface, is publicly available for implementing our proposed methods (Imai, Lu, and Strauss, forthcoming).


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Authors' note: This article is in the part based on two working papers by Imai and Lu, “Parametric and Nonparamateric Bayesian Models for Ecological Inference in 2 × 2 Tables” and “Quantifying Missing Information in Ecological Inference.” Various versions of these papers were presented at the 2004 Joint Statistical Meetings, the Second Cape Cod Monte Carlo Workshop, the 2004 Annual Political Methodology Summer Meeting, and the 2005 Annual Meeting of the American Political Science Association. We thank anonymous referees, Larry Bartels, Wendy Tam Cho, Jianqing Fan, Gary King, Xiao-Li Meng, Kevin Quinn, Phil Shively, David van Dyk, Jon Wakefield, and seminar participants at New York University (the Northeast Political Methodology conference), at Princeton University (Economics Department and Office of Population Research), and at the University of Virginia (Statistics Department) for helpful comments.



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Achen, C. H., and Shively, W. P. 1995. Cross-level inference. Chicago, IL: University of Chicago Press.
Antoniak, C. E. 1974. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. The Annals of Statistics 2: 1152–74.
Benoit, Kenneth and King, Gary. 2003. EzI: A(n easy) program for ecological inference. Cambridge, Mass.: Harvard University. Available from: (accessed August 8, 2007).
Brown, P. J., and Payne, C. D. 1986. Aggregate data, ecological regression, and voting transitions. Journal of the American Statistical Association 81: 452–60.
Burden, B. C., and Kimball, D. C. 1998. A new approach to the study of ticket splitting. American Political Science Review 92: 533–44.
Bush, C. A., and MacEachern, S. N. 1996. A semiparametric Bayesian model for randomized block designs. Biometrika 83: 275–85.
Cho, W. K. T. 1998. Iff the assumption fits …: A comment on the King ecological inference solution. Political Analysis 7: 143–63.
Cho, W. K. T., and Gaines, B. J. 2004. The limits of ecological inference: The case of split-ticket voting. American Journal of Political Science 48: 152–71.
Copas, J., and Eguchi, S. 2005. Local model uncertainty and incomplete-data bias. Journal of the Royal Statistical Society, Series B (Methodological) 67: 459513.
Cross, P. J., and Manski, C. F. 2002. Regressions, short and long. Econometrica 70: 357–68.
Dempster, A. P., Laird, N. M., and Rubin, D. B. 1977. Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, Methodological 39: 137.
Dey, D., Müller, P., and Sinha, D., eds. 1998. Practical nonparametric and semiparametric Bayesian statistics. New York: Springer-Verlag Inc.
Duncan, O. D., and Davis, B. 1953. An alternative to ecological correlation. American Sociological Review 18: 665–6.
Escobar, M. D., and West, M. 1995. Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association 90: 577–88.
Ferguson, T. S. 1973. A Bayesian analysis of some nonparametric problems. The Annals of Statistics 1: 209–30.
Freedman, D. A., Klein, S. P., Sacks, J., Smyth, C. A., and Everett, C. G. 1991. Ecological regression and voting rights (with discussion). Evaluation Review 15: 673816.
Freedman, D.A., Ostland, M., Roberts, M. R., and Klein, S. P. 1998. Review of “A Solution to the Ecological Inference Problem.” Journal of the American Statistical Association 93: 1518–22.
Gelman, A., Park, D. K., Ansolabehere, S., Price, P. N., and Minnite, L. C. 2001. Models, assumptions and model checking in ecological regressions. Journal of the Royal Statistical Society, Series A 164: 101–18.
Gill, J., and Casella, G. 2006. Markov chain Monte Carlo methods for models with nonparametric priors. Technical report, University of California, Davis.
Goodman, L. 1953. Ecological regressions and behavior of individuals. American Sociological Review 18: 663–6.
Grofman, B. 1991. Statistics without substance: A critique of Freedman et al. and Clark and Morrison. Evaluation Review 15: 746–69.
Heitjan, D. F., and Rubin, D. B. 1991. Ignorability and coarse data. The Annals of Statistics 19: 2244–53.
Herron, M. C., and Shotts, K. W. 2004. Logical inconsistency in EI-based second stage regressions. American Journal of Political Science 48: 172–83.
Imai, K., and King, G. 2004. Did illegal overseas absentee ballots decide the 2000 U.S. presidential election? Perspectives on Politics 2: 537–49.
Imai, K., Lu, Y., and Strauss, A. eco: R package for ecological inference in 2 × 2 tables. Journal of Statistical Software (forthcoming).
Judge, G. G., Miller, D. J., and Cho, W. K. T. 2004. An information theoretic approach to ecological estimation and inference. In Ecological inference: New methodological strategies, ed. King, G., Rosen, O., and Tanner, M., 162–87. Cambridge: Cambridge University Press.
King, G. 1997. A solution to the ecological inference problem: Reconstructing individual behavior from aggregate data. Princeton, NJ: Princeton University Press.
King, G. 1999. Comment on “review of ‘a solution to the ecological inference problem’.” Journal of the American Statistical Association 94: 352–5.
King, G., Rosen, O., and Tanner, M. A. 1999. Binomial-beta hierarchical models for ecological inference. Sociological Methods & Research 28: 6190.
King, G., Rosen, O., and Tanner, M. A., eds. 2004. Ecological inference: New methodological strategies. Cambridge: Cambridge University Press.
Kong, A., Meng, X.-L., and Nicolae, D. L. 2005. Quantifying relative incomplete information for hypothesis testing in statistical and genetic studies. Unpublished manuscript, Department of Statistics, Harvard University.
Larson, R., Hostetler, R. P., and Edwards, B. H. 2002. Calculus: Early transcendental functions. 3rd ed. Boston, MA: Houghton Mifflin Company.
Meng, X.-L., and Rubin, D. B. 1991. Using EM to obtain asymptotic variance-covariance matrices: The SEM algorithm. Journal of the American Statistical Association 86: 899909.
Meng, X.-L., and Rubin, D. B. 1993. Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika 80: 267–78.
Mukhopadhyay, S., and Gelfand, A. E. 1997. Dirichlet process mixed generalized linear models. Journal of the American Statistical Association 92: 633–9.
Neyman, J., and Scott, E. L. 1948. Consistent estimation from partially consistent observations. Econometrica 16: 132.
Orchard, T., and Woodbury, M. A. 1972. A missing information principle: Theory and applications. Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability 1: 697715.
Robinson, W. S. 1950. Ecological correlations and the behavior of individuals. American Sociological Review 15: 351–7.
Rosen, O., Jiang, W., King, G., and Tanner, M. A. 2001. Bayesian and frequentist inference for ecological inference: The R × C case. Statistica Neerlandica 55: 134–56.
van Dyk, D. A., Meng, X.-L., and Rubin, D. B. 1995. Maximum likelihood estimation via the ECM algorithm: Computing the asymptotic variance. Statistica Sinica 5: 5575.
Wakefield, J. 2004a. Ecological inference for 2 × 2 tables (with discussion). Journal of the Royal Statistical Society, Series A 167: 385445.
Wakefield, J. 2004b. Prior and likelihood choices in the analysis of ecological data. In Ecological inference: New methodological strategies, ed. King, Gary, Rosen, Ori, and Tanner, Martin, 1350. Cambridge: Cambridge University Press.
West, M., Müller, P., and Escobar, M. D. 1994. Hierarchical priors and mixture models, with application in regression and density estimation. In Aspects of uncertainty: A tribute to D. V. Lindley, ed. Smith, A. F. M. and Freedman, P. R., 363–86. London: John Wiley & Sons.
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Political Analysis
  • ISSN: 1047-1987
  • EISSN: 1476-4989
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