The Poisson Inverse Gaussian Regression Model in the Analysis of Clustered Counts Data

We explore the possibility of modeling clustered count data using the Poisson Inverse Gaussian distribution. We develop a regression model, which relates the number of mastitis cases in a sample of dairy farms in Ontario, Canada, to various farm level covariates, to illustrate the methodology. Residual plots are constructed to explore the quality of the fit. We compare the results with a negative binomial regression model using maximum likelihood estimation, and to the generalized linear mixed regression model fitted in SAS.

[1]  James S. Harris,et al.  Tables of integrals , 1998 .

[2]  Pravin K. Trivedi,et al.  Regression Analysis of Count Data: Preface , 1998 .

[3]  C. Dean.  Testing for Overdispersion in Poisson and Binomial Regression Models , 1992 .

[4]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[5]  Barry H. Margolin,et al.  Testing Goodness of Fit for the Poisson Assumption When Observations are Not Identically Distributed , 1985 .

[6]  Thomas F. Coleman,et al.  On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds , 1994, Math. Program..

[7]  G. Molenberghs,et al.  Multivariate Clustered Data Analysis in Developmental Toxicity Studies , 2001 .

[8]  N. Breslow.  Extra‐Poisson Variation in Log‐Linear Models , 1984 .

[9]  R. Wolfinger,et al.  Generalized linear mixed models a pseudo-likelihood approach , 1993 .

[10]  P. Albert,et al.  Models for longitudinal data: a generalized estimating equation approach. , 1988, Biometrics.

[11]  C. Mcgilchrist.  Estimation in Generalized Mixed Models , 1994 .

[12]  T. Coleman,et al.  On the Convergence of Reflective Newton Methods for Large-scale Nonlinear Minimization Subject to Bounds , 1992 .

[13]  Emmanuel Lesaffre,et al.  On the effect of the number of quadrature points in a logistic random effects model: an example , 2001 .

[14]  M. Segal,et al.  An Assessment of Approximate Maximum Likelihood Estimators in Generalized Linear Mixed Models , 1997 .

[15]  R. Schall.  Estimation in generalized linear models with random effects , 1991 .

[16]  M. Kendall.  Theoretical Statistics , 1956, Nature.

[17]  J. Crank.  Tables of Integrals , 1962 .

[18]  P. A. P. Moran,et al.  On asymptotically optimal tests of composite hypotheses , 1970 .

[19]  P. Thall,et al.  Some covariance models for longitudinal count data with overdispersion. , 1990, Biometrics.

[20]  John A. Nelder,et al.  Generalized linear models. 2nd ed. , 1993 .

[21]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[22]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..

[23]  H. Goldstein.  Nonlinear multilevel models, with an application to discrete response data , 1991 .

[24]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[25]  T. Ten Have,et al.  Comparison of two approaches to analyzing correlated binary data in developmental toxicity studies. , 1995, Teratology.

[26]  T. Stukel,et al.  Comparison of methods for the analysis of longitudinal interval count data. , 1993, Statistics in medicine.

[27]  H. Barkema,et al.  Management practices associated with the incidence rate of clinical mastitis. , 1999, Journal of dairy science.

[28]  R. Burnett,et al.  Nonlinear regression models for correlated count data , 1992 .

[29]  K. Christiansen,et al.  Estimating the costs associated with endemic diseases of dairy cattle , 1999, Journal of Dairy Research.

[30]  Eugene Demidenko,et al.  Asymptotic Properties of Nonlinear Mixed-Effects Models , 1997 .

[31]  J M Neuhaus,et al.  Assessing change with longitudinal and clustered binary data. , 2001, Annual review of public health.

[32]  Gordon E. Willmot,et al.  The Poisson-Inverse Gaussian distribution as an alternative to the negative binomial , 1987 .