Political Analysis Advance Access originally published online on August 13, 2007
Political Analysis 2008 16(1):41-69; doi:10.1093/pan/mpm017
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Bayesian and Likelihood Inference for 2 x 2 Ecological Tables: An Incomplete-Data Approach
Department of Politics, Princeton University, Princeton, NJ 08544
Department of Sociology, University of Colorado at Boulder, Boulder, CO 80309 e-mail: ying.lu{at}colorado.edu
Department of Politics, Princeton University, Princeton, NJ 08544 e-mail: abstraus{at}princeton.edu
e-mail: kimai{at}princeton.edu (corresponding author)
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 x 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).
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 x 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.