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Ecological inference for 2 × 2 tables

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  • Jon Wakefield

Abstract

Summary. A fundamental problem in many disciplines, including political science, sociology and epidemiology, is the examination of the association between two binary variables across a series of 2 × 2 tables, when only the margins are observed, and one of the margins is fixed. Two unobserved fractions are of interest, with only a single response per table, and it is this non‐identifiability that is the inherent difficulty lying at the heart of ecological inference. Many methods have been suggested for ecological inference, often without a probabilistic model; we clarify the form of the sampling distribution and critique previous approaches within a formal statistical framework, thus allowing clarification and examination of the assumptions that are required under all approaches. A particularly difficult problem is choosing between models with and without contextual effects. Various Bayesian hierarchical modelling approaches are proposed to allow the formal inclusion of supplementary data, and/or prior information, without which ecological inference is unreliable. Careful choice of the prior within such models is required, however, since there may be considerable sensitivity to this choice, even when the model assumed is correct and there are no contextual effects. This sensitivity is shown to be a function of the number of areas and the distribution of the proportions in the fixed margin across areas. By explicitly providing a likelihood for each table, the combination of individual level survey data and aggregate level data is straightforward and we illustrate that survey data can be highly informative, particularly if these data are from a survey of the minority population within each area. This strategy is related to designs that are used in survey sampling and in epidemiology. An approximation to the suggested likelihood is discussed, and various computational approaches are described. Some extensions are outlined including the consideration of multiway tables, spatial dependence and area‐specific (contextual) variables. Voter registration–race data from 64 counties in the US state of Louisiana are used to illustrate the methods.

Suggested Citation

  • Jon Wakefield, 2004. "Ecological inference for 2 × 2 tables," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 167(3), pages 385-425, July.
  • Handle: RePEc:bla:jorssa:v:167:y:2004:i:3:p:385-425
    DOI: 10.1111/j.1467-985x.2004.02046_1.x
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    Cited by:

    1. Carolina Plescia & Lorenzo De Sio, 2018. "An evaluation of the performance and suitability of R × C methods for ecological inference with known true values," Quality & Quantity: International Journal of Methodology, Springer, vol. 52(2), pages 669-683, March.
    2. Xiaohui Chang & Rasmus Waagepetersen & Herbert Yu & Xiaomei Ma & Theodore R. Holford & Rong Wang & Yongtao Guan, 2015. "Disease risk estimation by combining case–control data with aggregated information on the population at risk," Biometrics, The International Biometric Society, vol. 71(1), pages 114-121, March.
    3. Hugo Storm & Thomas Heckelei & Ron C. Mittelhammer, 2016. "Bayesian estimation of non-stationary Markov models combining micro and macro data," European Review of Agricultural Economics, Oxford University Press and the European Agricultural and Applied Economics Publications Foundation, vol. 43(2), pages 303-329.
    4. van Dijk, Bram & Paap, Richard, 2008. "Explaining individual response using aggregated data," Journal of Econometrics, Elsevier, vol. 146(1), pages 1-9, September.
    5. Nathan Kallus & Xiaojie Mao & Angela Zhou, 2022. "Assessing Algorithmic Fairness with Unobserved Protected Class Using Data Combination," Management Science, INFORMS, vol. 68(3), pages 1959-1981, March.
    6. Puig, Xavier & Ginebra, Josep, 2014. "A cluster analysis of vote transitions," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 328-344.
    7. Katie Wilson & Jon Wakefield, 2022. "A probabilistic model for analyzing summary birth history data," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 47(11), pages 291-344.
    8. Y. Ma & Ye Zhang, 2014. "Resolution of the Happiness–Income Paradox," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 119(2), pages 705-721, November.
    9. Irene L. Hudson & Linda Moore & Eric J. Beh & David G. Steel, 2010. "Ecological inference techniques: an empirical evaluation using data describing gender and voter turnout at New Zealand elections, 1893–1919," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 173(1), pages 185-213, January.
    10. Beh, Eric J., 2010. "The aggregate association index," Computational Statistics & Data Analysis, Elsevier, vol. 54(6), pages 1570-1580, June.
    11. Zax Jeffrey S., 2012. "Single Regression Estimates of Voting Choices When Turnout is Unknown," Statistics, Politics and Policy, De Gruyter, vol. 4(1), pages 1-22, October.
    12. E. Smoot & S. Haneuse, 2015. "On the analysis of hybrid designs that combine group- and individual-level data," Biometrics, The International Biometric Society, vol. 71(1), pages 227-236, March.
    13. D. James Greiner & Kevin M. Quinn, 2009. "R×C ecological inference: bounds, correlations, flexibility and transparency of assumptions," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 172(1), pages 67-81, January.
    14. Rob Eisinga, 2009. "The beta‐binomial convolution model for 2×2 tables with missing cell counts," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 63(1), pages 24-42, February.
    15. Arie ten Cate, 2014. "Maximum likelihood estimation of the Markov chain model with macro data and the ecological inference model," CPB Discussion Paper 284.rdf, CPB Netherlands Bureau for Economic Policy Analysis.
    16. Roberto Colombi & Antonio Forcina, 2016. "Latent class models for ecological inference on voters transitions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(4), pages 501-517, November.
    17. Sebastien J.‐P. A. Haneuse & And Jonathan C. Wakefield, 2008. "The combination of ecological and case–control data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 73-93, February.
    18. Sebastien J-P. A. Haneuse & Jonathan C. Wakefield, 2007. "Hierarchical Models for Combining Ecological and Case–Control Data," Biometrics, The International Biometric Society, vol. 63(1), pages 128-136, March.
    19. A. Forcina & M. Gnaldi & B. Bracalente, 2012. "A revised Brown and Payne model of voting behaviour applied to the 2009 elections in Italy," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(1), pages 109-119, March.
    20. Antonio Forcina & Davide Pellegrino, 2019. "Estimation of voter transitions and the ecological fallacy," Quality & Quantity: International Journal of Methodology, Springer, vol. 53(4), pages 1859-1874, July.
    21. Hui Huang & Xiaomei Ma & Rasmus Waagepetersen & Theodore R. Holford & Rong Wang & Harvey Risch & Lloyd Mueller & Yongtao Guan, 2014. "A New Estimation Approach for Combining Epidemiological Data From Multiple Sources," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 11-23, March.
    22. Shuai Shao & Göran Kauermann, 2020. "Understanding price elasticity for airline ancillary services," Journal of Revenue and Pricing Management, Palgrave Macmillan, vol. 19(1), pages 74-82, February.

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