Robust Inference Based on Quasi-likelihoods for Generalized Linear Models and Longitudinal Data

  • E. Cantoni


In this paper we introduce and develop robust versions of quasi-likelihood functions for model selection via an analysis-of-deviance type of procedure in generalized linear models and longitudinal data analysis. These robust functions are built upon natural classes of robust estimators and can be seen as weighted versions of their classical counterparts. The asymptotic theory of these test statistics is studied and their robustness properties are assessed for both generalized linear models and longitudinal data analysis. The proposed class of test statistics yields reliable inference even under model contamination. The analysis of a real data set completes the article.

Key words

Robust inference Quasi-likelihood functions Estimating equations Generalized linear models Longitudinal data 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. E. Cantoni. Resistant techniques for nonparametric regression,generalized linear and additive models. PhD thesis, University of Geneva, 1999.Google Scholar
  2. E. Cantoni and E. Ronchetti. Robust inference for generalized linear models. J. Am. Statist. Assoc., 96:1022–1030, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  3. P.J. Diggle, K.-Y. Liang, and S.L. Zeger. Analysis of longitudinal data. Clarendon Press, Oxford, 1996.Google Scholar
  4. F.R. Hampel. The influence curve and its role in robust estimation. J. Am. Statist. Assoc., 69: 383–393, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel. Robust statistics: The approach based on influence functions. Wiley, New York, 1986.zbMATHGoogle Scholar
  6. J.J. Hanfelt and K.-Y. Liang. Approximate likelihood ratios for general estimating functions. Biometrika, 82:461–477, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  7. X. He, D.G. Simpson, and S.L. Portnoy. Breakdown robustness of tests. J. Am. Statist. Assoc., 85:446–452, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  8. S. Heritier and E. Ronchetti. Robust bounded-influence tests in general parametric models. J. Am. Statist. Assoc., 89:897–904, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  9. P.J. Huber. Robust estimation of a location parameter. Ann. Math. Statist., 35:73–101, 1964.MathSciNetzbMATHCrossRefGoogle Scholar
  10. N.L. Johnson and S. Kotz. Continuous univariate distributions,volume 2. Houghton-Mifflin, Boston, 1970.zbMATHGoogle Scholar
  11. H.R. Künsch, L.A. Stefanski, and R.J. Carroll. Conditionally unbiased bounded-influence estimation in general regression models, with applications to generalized linear models. J. Am. Statist. Assoc., 84:460–466, 1989.zbMATHGoogle Scholar
  12. K.-Y. Liang and S.L. Zeger. Longitudinal data analysis using generalized linear models. Biometrika, 73:13–22, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  13. P. McCullagh. Quasi-likelihood functions. Ann. Statist., 11:59–67, 1983.MathSciNetCrossRefGoogle Scholar
  14. P. McCullagh and J.A. Nelder. Generalized linear models. Chapman and Hall, London, 2nd edition, 1989.Google Scholar
  15. S. Morgenthaler. Least-absolute-deviations fits for generalized linear models. Biometrika, 79:747–754, 1992.zbMATHCrossRefGoogle Scholar
  16. K. Phelps. Use of the complementary log-log function to describe dose-response relationships in insecticide evaluation field trials. In R. Gilchrist, editor, Proceedings of the International Conference on Generalized Linear Models. Lecture Notes in Statistics, volume 14. Springer, Berlin/New York, 1982.Google Scholar
  17. D. Pregibon. Resistant fits for some commonly used logistic models with medical applications. Biometrics, 38:485–498, 1982.CrossRefGoogle Scholar
  18. J.S. Preisser and B.F. Qaqish. Robust regression for clustered data with applications to binary regression. Biometrics, 55:574–579, 1999.zbMATHCrossRefGoogle Scholar
  19. B.F. Qaqish and J.S. Preisser. Resistant fits for regression with correlated outcomes: An estimating equation approach. J. Statist. Planning and Inf., 75:415–431, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  20. E. Ronchetti and F. Trojani. Robust inference with GMM estimators. J. of Econometrics, 101:37–69, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  21. P.J. Rousseeuw and A.M. Leroy. Robust regression and outlier detection. Wiley, New York, 1987.zbMATHCrossRefGoogle Scholar
  22. A. F. Ruckstuhl and A. H. Welsh. Robust fitting of the binomial model. Manuscript, 1999.Google Scholar
  23. L.A. Stefanski, R.J. Carroll, and D. Ruppert. Optimally bounded score functions for generalized linear models with applications to logistic regression. Biometrika, 73:413–424, 1986.MathSciNetzbMATHGoogle Scholar
  24. R.W.M. Wedderburn. Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61:439–447, 1974.MathSciNetzbMATHGoogle Scholar
  25. S.L. Zeger and K.-Y. Liang. Longitudinal data analysis for discrete and continuous outcomes. Biometrics, 42:121–130, 1986.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • E. Cantoni
    • 1
  1. 1.Econometrics DepartmentUniversity of GenevaSwitzerland

Personalised recommendations