Lifetime Data Analysis

, Volume 13, Issue 4, pp 497–512 | Cite as

Generalized linear mixed models: a review and some extensions

  • C. B. DeanEmail author
  • Jason D. Nielsen


Breslow and Clayton (J Am Stat Assoc 88:9–25,1993) was, and still is, a highly influential paper mobilizing the use of generalized linear mixed models in epidemiology and a wide variety of fields. An important aspect is the feasibility in implementation through the ready availability of related software in SAS (SAS Institute, PROC GLIMMIX, SAS Institute Inc., URL, 2007), S-plus (Insightful Corporation, S-PLUS 8, Insightful Corporation, Seattle, WA, URL, 2007), and R (R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, URL, 2006) for example, facilitating its broad usage. This paper reviews background to generalized linear mixed models and the inferential techniques which have been developed for them. To provide the reader with a flavor of the utility and wide applicability of this fundamental methodology we consider a few extensions including additive models, models for zero-heavy data, and models accommodating latent clusters.


Generalized linear mixed model Random effects Longitudinal data analysis Penalized quasi-likelihood 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramowitz, M, Stegun, IA (eds) (1984) Handbook of mathematical functions with formulas, graphs, and mathematical tables. A Wiley-Interscience Publication, John Wiley, New YorkzbMATHGoogle Scholar
  2. Ainsworth L, Dean CB (2007) Detection of local and global outliers in mapping studies. Environmetrics. doi: 10.1002/env.851
  3. Barndorff-Nielsen O, Cox DR (1979) Edgeworth and saddle-point approximations with statistical applications (with discussion). J R Stat Soc Ser B: Methodol 41: 279–299zbMATHMathSciNetGoogle Scholar
  4. Bates D, Sarkar D (2007) lme4: Linear mixed-effects models using S4 classes. R package version 0.99875–1Google Scholar
  5. Breslow N (1989) Score tests in overdispersed GLM’s. In: Decarli A, Francis BJ, Gilchrist R, Seeber GUH, (eds) Statistical modelling. Springer-Verlag Inc., pp 64–74Google Scholar
  6. Breslow NE, Clayton DG (1993) Approximate inference in generalized linear mixed models. J Am Stat Assoc 88: 9–25zbMATHCrossRefGoogle Scholar
  7. Breslow NE, Lin X (1995) Bias correction in generalised linear mixed models with a single component of dispersion. Biometrika 82: 81–91zbMATHCrossRefMathSciNetGoogle Scholar
  8. Casella G, George EI (1992) Explaining the Gibbs sampler. Am Stat 46: 167–174CrossRefMathSciNetGoogle Scholar
  9. Chen J, Zhang D, Davidian M (2002) A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution. Biostatistics (Oxford) 3: 347–360zbMATHCrossRefGoogle Scholar
  10. Cheng KF, Wu JW (1994) Testing goodness of fit for a parametric family of link functions. J Am Stat Assoc 89: 657–664zbMATHCrossRefMathSciNetGoogle Scholar
  11. Cook RD, Weisberg S (1989) Regression diagnostics with dynamic graphics (C/R: P293–311). Technometrics 31: 277–291CrossRefMathSciNetGoogle Scholar
  12. Davidian M, Carroll RJ (1987) Variance function estimation. J Am Stat Assoc 82: 1079–1091zbMATHCrossRefMathSciNetGoogle Scholar
  13. Dean CB, Balshaw R (1997) Efficiency lost by analyzing counts rather than event times in Poisson and overdispersed Poisson regression models. J Am Stat Assoc 92: 1387–1398zbMATHCrossRefMathSciNetGoogle Scholar
  14. Dean C, Lawless JF, Willmot GE (1989) A mixed Poisson-inverse–Gaussian regression model. Can J Stat 17: 171–181zbMATHCrossRefMathSciNetGoogle Scholar
  15. Dubin JA, Han L, Fried TR (2007) Triggered sampling could help improve longitudinal studies of persons with elevated mortality risk. J Clin Epidemiol 60: 288–93CrossRefGoogle Scholar
  16. Green PJ, Silverman BW (1994) Nonparametric regression and generalized linear models: a roughness penalty approach. Chapman & Hall Ltd.Google Scholar
  17. Harville DA (1977) Maximum likelihood approaches to variance component estimation and to related problems. J Am Stat Assoc 72: 320–338zbMATHCrossRefMathSciNetGoogle Scholar
  18. Hastie T, Tibshirani R (1999) Generalized additive models. Chapman & Hall Ltd.Google Scholar
  19. Henderson R, Shimakura S (2003) A serially correlated gamma frailty model for longitudinal count data. Biometrika 90: 355–366zbMATHCrossRefMathSciNetGoogle Scholar
  20. Insightful Corporation (2007) S-PLUS 8. Insightful Corporation, Seattle, WA. URL, accessed on 25 October 2007
  21. Jiang J (1998) Consistent estimators in generalized linear mixed models. J Am Stat Assoc 93: 720–729zbMATHCrossRefGoogle Scholar
  22. Jørgensen B (1982) Statistical properties of the generalized inverse Gaussian distribution. Lecture Notes in Statistics, vol 9. Springer-Verlag, New YorkGoogle Scholar
  23. Kleinman K, Lazarus R, Platt R (2004) A generalized linear mixed models approach for detecting incident clusters of disease in small areas, with an application to biological terrorism. Am J Epidemiol 159: 217–24CrossRefGoogle Scholar
  24. Laird NM (1991) Topics in likelihood-based methods for longitudinal data analysis. Statistica Sinica 1: 33–50zbMATHGoogle Scholar
  25. Laird NM, Louis TA (1982) Approximate posterior distributions for incomplete data problems. J R Stat Soc Ser B: Methodol 44: 190–200zbMATHMathSciNetGoogle Scholar
  26. Lambert D (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34: 1–14zbMATHCrossRefGoogle Scholar
  27. Lawless JF (1987) Negative binomial and mixed P. Can J Stat 15: 209–225zbMATHCrossRefMathSciNetGoogle Scholar
  28. Lawless JF, Zhan M (1998) Analysis of interval-grouped recurrent-event data using piecewise constant rate functions. Can J Stat 26: 549–565zbMATHCrossRefGoogle Scholar
  29. Liang K-Y, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73: 13–22zbMATHCrossRefMathSciNetGoogle Scholar
  30. Lin X, Breslow NE (1996) Bias correction in generalized linear mixed models with multiple components of dispersion. J Am Stat Assoc 91: 1007–1016zbMATHCrossRefMathSciNetGoogle Scholar
  31. Lin X, Zhang D (1999) Inference in generalized additive mixed models by using smoothing splines. J R Stat Soc Ser B: Stat Methodol 61: 381–400zbMATHCrossRefMathSciNetGoogle Scholar
  32. Lin X, Harlow SD, Raz J, Harlow SD (1997) Linear mixed models with heterogeneous within-cluster variances. Biometrics 53: 910–923zbMATHCrossRefMathSciNetGoogle Scholar
  33. Lindstrom MJ, Bates DM (1988) Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data. J Am Stat Assoc 83: 1014–1022, Corr: 94V89, p 1572zbMATHCrossRefMathSciNetGoogle Scholar
  34. Martin TG, Wintle BA, Rhodes JR, Kuhnert PM, Field SA, Low-Choy SJ, Tyre AJ, Possingham HP (2005) Zero tolerance ecology: improving ecological inference by modelling the source of zero observations. Ecol Lett 8: 1235–1246CrossRefGoogle Scholar
  35. McCullagh P, Nelder JA (1989) Generalized linear models. Chapman & Hall Ltd.Google Scholar
  36. McCulloch CE (1997) Maximum likelihood algorithms for generalized linear mixed models. J Am Stat Assoc 92: 162–170zbMATHCrossRefMathSciNetGoogle Scholar
  37. Nelder JA, Pregibon D (1987) An extended quasi-likelihood function. Biometrika 74: 221–232zbMATHCrossRefMathSciNetGoogle Scholar
  38. Nielsen JD, Dean CB (2007) Clustered mixed nonhomogeneous Poisson process spline models for the analysis of recurrent event panel data. Biometrics. doi: 10.1111/j.1541-0420.2007.00940.x Google Scholar
  39. Nodtvedt A, Dohoo I, Sanchez J, Conboy G, DesCjteaux L, Keefe G, Leslie K, Campbell J (2002) The use of negative binomial modelling in a longitudinal study of gastrointestinal parasite burdens in Canadian dairy cows. Can J Vet Res 66: 249–257Google Scholar
  40. Pierce DA, Schafer DW (1986) Residuals in generalized linear models. J Am Stat Assoc 81: 977–986zbMATHCrossRefMathSciNetGoogle Scholar
  41. Pinheiro JC, Bates DM (2000) Mixed-effects models in S and S-PLUS. Springer-Verlag Inc.Google Scholar
  42. R Development Core Team (2006) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL, accessed on 25 October 2007
  43. Raudenbush SW, Yang M-L, Yosef M (2000) Maximum likelihood for generalized linear models with nested random effects via high-order, multivariate Laplace approximation. J Comput Graph Stat 9: 141–157CrossRefMathSciNetGoogle Scholar
  44. Rich-Edwards JW, Kleinman KP, Strong EF, Oken E, Gillman MW (2005) Preterm delivery in Boston before and after September 11th, 2001. Epidemiology 16: 323–327CrossRefGoogle Scholar
  45. Rosen O, Jiang W, Tanner MA (2000) Mixtures of marginal models. Biometrika 87: 391–404zbMATHCrossRefMathSciNetGoogle Scholar
  46. Sartori N, Severini TA (2004) Conditional likelihood inference in generalized linear mixed models. Statistica Sinica 14: 349–360zbMATHMathSciNetGoogle Scholar
  47. SAS Institute (2007) PROC GLIMMIX. SAS Institute Inc. URL, accessed on 25 October 2007
  48. Shun Z, McCullagh P (1995) Laplace approximation of high dimensional integrals. J R Stat Soc Ser B: Methodol 57: 749–760zbMATHMathSciNetGoogle Scholar
  49. Sichel HS (1974) On a distribution representing sentence-length in written prose. J R Stat Soc Ser A 137: 25–34CrossRefGoogle Scholar
  50. Simons JS, Neal DJ, Gaher RM (2006) Risk for marijuana-related problems among college students: an application of zero-inflated negative binomial regression. Am J Drug Alcohol Abuse 32: 41–53CrossRefGoogle Scholar
  51. Song PX-K, Fan Y, Kalbfleisch JD (2005) Maximization by parts in likelihood inference. J Am Stat Assoc 100: 1145–1158zbMATHCrossRefMathSciNetGoogle Scholar
  52. Stangle DE, Smith DR, Beaudin SA, Strawderman MS, Levitsky DA, Strupp BJ (2007) Succimer chelation improves learning, attention, and arousal regulation in lead-exposed rats but produces lasting cognitive impairment in the absence of lead exposure. Environ Health Perspect 115: 201–209CrossRefGoogle Scholar
  53. Tchetgen EJ, Coull BA (2006) A diagnostic test for the mixing distribution in a generalised linear mixed model. Biometrika 93: 1003–1010CrossRefMathSciNetGoogle Scholar
  54. Tierney L, Kass RE, Kadane JB (1989) Approximate marginal densities of nonlinear functions. Biometrika 76: 425–433, Corr: V78, p233–234zbMATHCrossRefMathSciNetGoogle Scholar
  55. Tjur T (1982) A connection between Rasch’s item analysis model and a multiplicative Poisson model. Scand J Stat 9: 23–30MathSciNetzbMATHGoogle Scholar
  56. Venables WN, Ripley BD (2002) Modern applied statistics with S, 4th edn. Springer, New York. URL, accessed on 25 October 2007
  57. Vonesh EF, Wang H, Nie L, Majumdar D (2002) Conditional second-order generalized estimating equations for generalized linear and nonlinear mixed-effects models. J Am Stat Assoc 97: 271–283zbMATHCrossRefMathSciNetGoogle Scholar
  58. Waagepetersen R (2006) A simulation-based goodness-of-fit test for random effects in generalized linear mixed models. Scand J Stat 33: 721–731zbMATHCrossRefMathSciNetGoogle Scholar
  59. Wedderburn RWM (1974) Quasi-likelihood functions, generalized linear models, and the Gauss–Newton method. Biometrika 61: 439–447zbMATHMathSciNetGoogle Scholar
  60. White H (1982) Maximum likelihood estimation of misspecified models. Econometrica 50: 1–26zbMATHCrossRefMathSciNetGoogle Scholar
  61. Wood S (2006) mgcv: GAMs with GCV smoothness estimation and GAMMs by REML/PQL. R package version 1.3–24Google Scholar
  62. Zeger SL, Karim MR (1991) Generalized linear models with random effects: a Gibbs sampling approach. J Am Stat Assoc 86: 79–86CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Statistics and Actuarial ScienceSimon Fraser UniversityBurnabyCanada
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

Personalised recommendations