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Second-order generalized estimating equations for correlated count data

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Abstract

Generalized estimating equations have been widely used in the analysis of correlated count data. Solving these equations yields consistent parameter estimates while the variance of the estimates is obtained from a sandwich estimator, thereby ensuring that, even with misspecification of the so-called working correlation matrix, one can draw valid inferences on the marginal mean parameters. That they allow misspecification of the working correlation structure, though, implies a limitation of these equations should scientific interest also be in the covariance or correlation structure. We propose herein an extension of these estimating equations such that, by incorporating the bivariate Poisson distribution, the variance-covariance matrix of the response vector can be properly modelled, which would permit inference thereon. A sandwich estimator is used for the standard errors, ensuring sound inference on the parameters estimated. Two applications are presented.

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References

  • Agresti A (2002) Categorical data analysis, 2nd edn. Wiley, New York

    Book  MATH  Google Scholar 

  • Breslow N (1984) Extra-poisson variation in log-linear models. Applied Statistics 33:38–44

    Article  Google Scholar 

  • Breslow N, Clayton D (1993) Approximate inference in generalized linear mixed models. J Am Stat Assoc 88:9–25

    MATH  Google Scholar 

  • Crowder M (1995) On the use of a working correlation matrix in using generalised linear models for repeated measures. Biometrika 82:407–410

    Article  MATH  Google Scholar 

  • Diggle P, Heagerty P, Liang K, Zeger S (2002) Analysis of longitudinal data, 2nd edn. Oxford Science Publications, Clarendon Press, Oxford

    MATH  Google Scholar 

  • Fahrmeir L, Tutz G (1994) Multivariate statistical modelling based on generalized linear models. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Fahrmeir L, Tutz G (2001) Multivariate statistical modeling based on generalized linear models. Springer series in statistics, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Fan J, Huang T, Li R (2007) Analysis of longitudinal data with semiparametric estimation of covariance function. J Am Stat Assoc 102(478):632–641. doi:10.1198/016214507000000095

    Article  MathSciNet  MATH  Google Scholar 

  • Fitzmaurice G, Laird N, Ware J (2004) Applied longitudinal analysis. Wiley, New York

    MATH  Google Scholar 

  • Hardin J, Hilbe J (2003) Generalized estimating equations. Chapman and Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Iddi S, Molenberghs G (2013) A marginalized model for zero-inflated, overdispersed and correlated count data. Electron J Appl Stat Anal 6:149–165

    MathSciNet  Google Scholar 

  • Karlis D (2003) An em algorithm for multivariate poisson distribution and related models. J Appl Stat 30:63–77

    Article  MathSciNet  MATH  Google Scholar 

  • Kassahun W, Neyens T, Molenberghs G, Faes C, Verbeke G (2012) Modeling overdispersed longitudinal binary data using a combined beta and normal random-effects model. Arch Public Health 70(1):1

    Article  Google Scholar 

  • Kim H, Shults J (2010) %qls SAS macro: a SAS macro for analysis of correlated data using quasi-least squares. J Stat Softw 35:1–22

    Article  Google Scholar 

  • Kocherlakota S, Kocherlakota K (1992) Bivariate discrete distributions. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Kocherlakota S, Kocherlakota K (2001) Regression in the bivariate poisson distribution. Commun Stat Theory Methods 30:815–825

    Article  MathSciNet  MATH  Google Scholar 

  • Lakshminarayana J, Pandit S, Rao K (1999) On a bivariate poisson distribution. Commun Stat Theory Methods 28:267–276

    Article  MATH  Google Scholar 

  • Lambert D (1992) Zero-inflated poisson regression, with an application to defects in manufacturing. Technometrics 34:1–13

    Article  MATH  Google Scholar 

  • Lawless J (1987) Negative binomial and mixed poisson regression. Can J Stat 15:209–225

    Article  MathSciNet  MATH  Google Scholar 

  • Lee Y, Nelder J (2004) Conditional and marginal models: another view. Stat Sci 19:219–228

    Article  MathSciNet  MATH  Google Scholar 

  • Leng C, Zhang W, Pan J (2010) Semiparametric mean-covariance regression analysis for longitudinal data. J Am Stat Assoc 105(489):181–193. doi:10.1198/jasa.2009.tm08485

    Article  MathSciNet  MATH  Google Scholar 

  • Leppik I, Dreifuss F, Bowman-Cloyd T et al (1985) A double-blind crossover evaluation of progabide in partial seizures. Neurology 35:285

    Article  Google Scholar 

  • Lesaffre E, Asefa M, Verbeke G (1999) Assessing the goodness-of-fit of the Laird and Ware model-an example: the Jimma infant survival differential longitudinal study. Stat Med 18:835–854

    Article  Google Scholar 

  • Liang K, Zeger S (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22

    Article  MathSciNet  MATH  Google Scholar 

  • Liang K, Zeger S, Qaqish B (1992) Multivariate regression analyses for categorical data. J R Stat Soc B 54:3–40

    MathSciNet  MATH  Google Scholar 

  • Lindsey J, Lambert P (1998) On the appropriateness of marginal models for repeated measurements in clinical trials. Stat Med 17:447–469

    Article  Google Scholar 

  • Lipsitz S, Laird N, Harrington D (1991) Generalized estimating equations for correlated binary data: using the odds ratio as a measure of association. Biometrika 78:153–160

    Article  MathSciNet  Google Scholar 

  • McCullagh P, Nelder J (1989) Generalized linear models. Chapman & Hall, London

    Book  MATH  Google Scholar 

  • Molenberghs G, Verbeke G (2005) Models for discrete longitudinal data. Springer, New York

    MATH  Google Scholar 

  • Molenberghs G, Verbeke G, Demétrio C (2007) An extended random effects approach to modeling repeated overdispersed count data. Lifetime Data Anal 13:513–531

    Article  MathSciNet  MATH  Google Scholar 

  • Molenberghs G, Verbeke G, Demétrio C, Vieira A (2010) A family of generalized linear models for repeated measures with normal and conjugate random effects. Stat Sci 25:325–347

    Article  MathSciNet  MATH  Google Scholar 

  • Nelder J, Wedderburn R (1972) Generalized linear models. J R Stat Soc B 135:370–384

    Google Scholar 

  • Prentice R (1988) Correlated binary regression with covariates specific to each binary observation. Biometrics 44:1033–1048

    Article  MathSciNet  MATH  Google Scholar 

  • Prentice R, Zhao L (1991) Estimating equations for parameters in means and covariances of multivariate discrete and continuous response. Biometrics 47:825–839

    Article  MathSciNet  MATH  Google Scholar 

  • Ridout M, Hinde J, Demetrio C (2001) A score test for a zero-inflated poisson regression model against zero-inflated negative binomial alternatives. Biometrics 57:219–233

    Article  MathSciNet  MATH  Google Scholar 

  • Solis-Trapala I, Farewell V (2005) Regression analysis of overdispersed correlated count data with subject specific covariates. Stat Med 24:2557–2575

    Article  MathSciNet  Google Scholar 

  • Sun W, Shults J, Leonard M (2009) A note on the use of unbiased estimating equations to estimate correlation in analysis of longitudinal trials. Biom J 51:5–18

    Article  MathSciNet  Google Scholar 

  • Thall P, Vail S (1990) Some covariance models for longitudinal count data with overdispersion. Biometrics 46:657–671

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Y, Carey V (2004) Unbiased estimating equations from working correlation models for irregularly timed repeated measures. J Am Stat Assoc 99:845–852

    Article  MathSciNet  MATH  Google Scholar 

  • Winkelmann R (2008) Econometric analysis of count data. Springer, Berlin

    MATH  Google Scholar 

  • Wolfinger R, O’Connell M (1993) Generalized linear mixed models: a pseudo-likelihood approach. J Stat Comput Simul 48:233–243

    Article  MATH  Google Scholar 

  • Ye H, Pan J (2006) Modelling of covariance structures in generalized estimating equations for longitudinal data. Biometrika 93:927–941

    Article  MathSciNet  Google Scholar 

  • Zhao L, Prentice R (1990) Correlated binary regression using a quadratic exponential model. Biometrika 77:642–648

    Article  MathSciNet  Google Scholar 

  • Ziegler A (2011) Generalized estimating equations. Springer, New York

    Book  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. I-Biostat, Universiteit Hasselt, Martelarenlaan 42, 3500, Hasselt, Belgium

    George Kalema, Geert Molenberghs & Wondwosen Kassahun

  2. School of Statistics and Applied Economics, Makerere University, P.O. Box 7062, Kampala, Uganda

    George Kalema

  3. I-Biostat, KU Leuven, Kapucijnenvoer 35, 3000, Louvain, Belgium

    Geert Molenberghs

  4. Department of Epidemiology and Biostatistics, Jimma University, Jimma, Ethiopia

    Wondwosen Kassahun

Authors
  1. George Kalema
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  2. Geert Molenberghs
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  3. Wondwosen Kassahun
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Correspondence to George Kalema.

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Kalema, G., Molenberghs, G. & Kassahun, W. Second-order generalized estimating equations for correlated count data. Comput Stat 31, 749–770 (2016). https://doi.org/10.1007/s00180-015-0599-1

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