Lifetime Data Analysis

, Volume 13, Issue 4, pp 533–544 | Cite as

Estimation using penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models

  • Xihong Lin


We consider two estimation schemes based on penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models. The asymptotic bias in regression coefficients and variance components estimated by penalized quasilikelihood (PQL) is studied for small values of the variance components. We show the PQL estimators of both regression coefficients and variance components in Poisson mixed models have a smaller order of bias compared to those for binomial data. Unbiased estimating equations based on quasi-pseudo-likelihood are proposed and are shown to yield consistent estimators under some regularity conditions. The finite sample performance of these two methods is compared through a simulation study.


Asymptotic bias Estimating equations Generalized linear mixed models Laplace expansion Overdispersion Variance components 


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  1. Breslow NE (1984). Extra-Poisson variation in log-linear models. Appl Statist 33: 38–44 CrossRefGoogle Scholar
  2. Breslow NE (1990). Tests of hypotheses in overdispersed Poisson regression and other quasi-likelihood models. J Amer Statist Assoc 85: 565–571 CrossRefGoogle Scholar
  3. Breslow NE and Clayton DG (1993). Approximate inference in generalized linear mixed models. J Amer Statist Assoc 88: 9–25 MATHCrossRefGoogle Scholar
  4. Breslow NE and Lin X (1995). Bias correction in generalized linear mixed models with a single component of dispersion. Biometrika 82: 81–91 MATHCrossRefMathSciNetGoogle Scholar
  5. Davidian M and Carroll RJ (1987). Variance function estimation. J Amer Statist Assoc 82: 1079–1091 MATHCrossRefMathSciNetGoogle Scholar
  6. Green PJ (1987). Penalized likelihood for general semi-parametric regression models. Int Statist Rev 55: 245–259 MATHCrossRefGoogle Scholar
  7. Gumpertz ML and Pantula SG (1992). Nonlinear regression with variance components. J Amer Statist Assoc 87: 201–209 MATHCrossRefMathSciNetGoogle Scholar
  8. Heagerty PJ and Lele SR (1998). A composite likelihood approach to binary spatial data. J Amer Statist Assoc 93: 1099–1111 MATHCrossRefMathSciNetGoogle Scholar
  9. Hinde J (1982) Compound Poisson regression models. In: Gilchrist Berline R (ed) GLIM 82: Proceedings of the International Conference on Generalised Linear Models. Springer-Verlag, pp 109–121Google Scholar
  10. Lawless JF (1987). Negative binomial and mixed Poisson regression. Can J Statist 15: 209–225 MATHCrossRefMathSciNetGoogle Scholar
  11. Lin X and Breslow NE (1996). Bias correction in generalized linear mixed models with multiple components of dispersion. J Amer Statist Assoc 91: 1007–1016 MATHCrossRefMathSciNetGoogle Scholar
  12. Liu Q and Pierce DA (1993). Heterogeneity in Mantel-Haenszel-type models. Biometrika 80: 543–556 MATHCrossRefMathSciNetGoogle Scholar
  13. McCullagh P and Nelder JA (1989). Generalized linear models, 2nd ed. Chapman and Hall, London MATHGoogle Scholar
  14. Miller JJ (1977). Asymptotic properties of maximum likelihood estimates in the mixed model of the analysis of variance. Ann Statist 5: 746–762 MATHCrossRefMathSciNetGoogle Scholar
  15. Morton R (1987). A generalized linear model with nested strata of extra-Poisson variation. Biometrika 74: 247–257MATHCrossRefMathSciNetGoogle Scholar
  16. Nelder JA and Lee Y (1992). Likelihood, quasi-likelihood and pseudolikelihood: some comparisons. J R Statist Soc B 50: 266–268 Google Scholar
  17. Prentice RL and Zhao LP (1991). Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses. Biometrics 47: 825–839 MATHCrossRefMathSciNetGoogle Scholar
  18. Schall R (1991). Estimation in generalized linear models with random effects. Biometrika 40: 917–927 Google Scholar
  19. Solomon PJ and Cox DR (1992). Nonlinear components of variance models. Biometrika 79: 1–11 MATHCrossRefMathSciNetGoogle Scholar
  20. Stiratelli R, Laird N and Ware J (1984). Random effect models for serial observations with binary response. Biometrics 40: 961–971 CrossRefGoogle Scholar
  21. Thall PF and Vail SC (1990). Some covariance models for longitudinal count data with overdispersion. Biometrics 46: 657–671 MATHCrossRefMathSciNetGoogle Scholar
  22. Zeger SL, Liang KY and Albert PS (1988). Models for longitudinal data: a generalized estimating equation approach. Biometrics 44: 1049–1060 MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of BiostatisticsHarvard School of Public HealthBostonUSA

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