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Prediction in multilevel generalized linear models

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  • Anders Skrondal
  • Sophia Rabe‐Hesketh

Abstract

Summary. We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix. For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random‐intercept models under a range of conditions.

Suggested Citation

  • Anders Skrondal & Sophia Rabe‐Hesketh, 2009. "Prediction in multilevel generalized linear models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 172(3), pages 659-687, June.
  • Handle: RePEc:bla:jorssa:v:172:y:2009:i:3:p:659-687
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    File URL: https://doi.org/10.1111/j.1467-985X.2009.00587.x
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    2. Michele Battisti & Andrea Mario Lavezzi & Lucio Masserini & Monica Pratesi, 2018. "Resisting the extortion racket: an empirical analysis," European Journal of Law and Economics, Springer, vol. 46(1), pages 1-37, August.
    3. Wunder, Christoph & Riphahn, Regina, 2013. "Welfare transitions before and after reforms of the German welfare system," Annual Conference 2013 (Duesseldorf): Competition Policy and Regulation in a Global Economic Order 79715, Verein für Socialpolitik / German Economic Association.
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    5. Abdulrahman, Abdulallah S & Johnston, Robert J, 2016. "Systematic Non-Response in Stated Preference Choice Experiments: Implications for the Valuation of Climate Risk Reductions," 2016 Annual Meeting, July 31-August 2, Boston, Massachusetts 235465, Agricultural and Applied Economics Association.
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    8. repec:spr:psycho:v:82:y:2017:i:3:d:10.1007_s11336-017-9555-z is not listed on IDEAS
    9. Arthur Charpentier & Mathieu Pigeon, 2016. "Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective)," Risks, MDPI, Open Access Journal, vol. 4(2), pages 1-18, May.
    10. Casey Codd & Robert Cudeck, 2014. "Nonlinear Random-Effects Mixture Models for Repeated Measures," Psychometrika, Springer;The Psychometric Society, vol. 79(1), pages 60-83, January.
    11. Daidone, Silvio & Street, Andrew, 2013. "How much should be paid for specialised treatment?," Social Science & Medicine, Elsevier, vol. 84(C), pages 110-118.
    12. Heckman, James J. & Karapakula, Ganesh, 2019. "The Perry Preschoolers at Late Midlife: A Study in Design-Specific Inference," IZA Discussion Papers 12362, Institute of Labor Economics (IZA).
    13. Regina T. Riphahn & Christoph Wunder, 2016. "State dependence in welfare receipt: transitions before and after a reform," Empirical Economics, Springer, vol. 50(4), pages 1303-1329, June.
    14. Caliendo, Marco & Künn, Steffen & Uhlendorff, Arne, 2016. "Earnings exemptions for unemployed workers: The relationship between marginal employment, unemployment duration and job quality," Labour Economics, Elsevier, vol. 42(C), pages 177-193.
    15. Glenn Ellison & Ashley Swanson, 2016. "Do Schools Matter for High Math Achievement? Evidence from the American Mathematics Competitions," American Economic Review, American Economic Association, vol. 106(6), pages 1244-1277, June.
    16. Schurer, S. & Yong, J., 2012. "Personality, well-being and the marginal utility of income: What can we learn from random coefficient models?," Health, Econometrics and Data Group (HEDG) Working Papers 12/01, HEDG, c/o Department of Economics, University of York.
    17. Gerard J Van Den Berg & Barbara Hofmann & Arne Uhlendorff, 2016. "The Role of Sickness in the Evaluation of Job Search Assistance and Sanctions," Working Papers 2016-17, Center for Research in Economics and Statistics.
    18. Germana Corrado & Luisa Corrado & Giuseppe De Michele & Francesco Salustri, 2017. "Are Perceptions of Corruption Matching Reality? Theory and Evidence from Microdata," CEIS Research Paper 420, Tor Vergata University, CEIS, revised 14 Dec 2017.
    19. repec:kap:jrisku:v:54:y:2017:i:3:d:10.1007_s11166-017-9261-3 is not listed on IDEAS
    20. Daniele Pacifico, 2014. "On the role of unobserved preference Heterogeneity in discrete choice Models of labour supply," Working Papers 6, Department of the Treasury, Ministry of the Economy and of Finance.
    21. Michele Battisti & Andrea Mario Lavezzi & Lucio Masserini & Monica Pratesi, 2018. "Resisting the extortion racket: an empirical analysis," European Journal of Law and Economics, Springer, vol. 46(1), pages 1-37, August.
    22. Van Oirbeek, R. & Lesaffre, E., 2012. "Assessing the predictive ability of a multilevel binary regression model," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1966-1980.
    23. Tamura, Karin Ayumi & Giampaoli, Viviana, 2013. "New prediction method for the mixed logistic model applied in a marketing problem," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 202-216.
    24. Elbers, Chris & van der Weide, Roy, 2014. "Estimation of normal mixtures in a nested error model with an application to small area estimation of poverty and inequality," Policy Research Working Paper Series 6962, The World Bank.
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