Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative binomial models.

The regression models appropriate for counted data have seen little use in psychology. This article describes problems that occur when ordinary linear regression is used to analyze count data and presents 3 alternative regression models. The simplest, the Poisson regression model, is likely to be misleading unless restrictive assumptions are met because individual counts are usually more variable ("overdispersed") than is implied by the model. This model can be modified in 2 ways to accomodate this problem. In the overdispersed model, a factor can be estimated that corrects the regression model's inferential statistics. In the second alternative, the negative binomial regression model, a random term reflecting unexplained between-subject differences is included in the regression model. The authors compare the advantages of these approaches.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  Jacob Cohen.  Multiple regression as a general data-analytic system. , 1968 .

[3]  R. W. Wedderburn.  Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method , 1974 .

[4]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[5]  Antoine-S Bailly,et al.  Science régionale - Walter Isard, Introduction to régional science. Englewood Cliffs (NJ), Prentice-Hall, 1975 , 1975 .

[6]  John Aitchison,et al.  Statistical Prediction Analysis , 1975 .

[7]  P. Holland,et al.  Discrete Multivariate Analysis. , 1976 .

[8]  David R. Cox,et al.  Some remarks on overdispersion , 1983 .

[9]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[10]  Z. Griliches,et al.  Econometric Models for Count Data with an Application to the Patents-R&D Relationship , 1984 .

[11]  O. D. Duncan,et al.  Linear statistical models and related methods , 1984 .

[12]  S. Weisberg.  Plots, transformations, and regression , 1985 .

[13]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[14]  D. Pierce,et al.  Residuals in Generalized Linear Models , 1986 .

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

[16]  Richard Morton,et al.  A generalized linear model with nested strata of extra-Poisson variation , 1987 .

[17]  J. Lawless.  Negative binomial and mixed Poisson regression , 1987 .

[18]  R. Tsutakawa.  Mixed model for analyzing geographic variability in mortality rates. , 1988, Journal of the American Statistical Association.

[19]  Richard A. Becker,et al.  The New S Language , 1989 .

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

[21]  J. Lawless,et al.  Tests for Detecting Overdispersion in Poisson Regression Models , 1989 .

[22]  J. Chambers,et al.  The New S Language , 1989 .

[23]  John Hinde,et al.  Statistical Modelling in GLIM. , 1989 .

[24]  Norman E. Breslow,et al.  Tests of Hypotheses in Overdispersed Poisson Regression and other Quasi-Likelihood Models , 1990 .

[25]  Scott L. Zeger,et al.  Generalized linear models with random e ects: a Gibbs sampling approach , 1991 .

[26]  Trevor Hastie,et al.  Statistical Models in S , 1991 .

[27]  David Barron,et al.  The Analysis of Count Data: Over-dispersion and Autocorrelation , 1992 .

[28]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[29]  Diane Lambert,et al.  Zero-inflacted Poisson regression, with an application to defects in manufacturing , 1992 .

[30]  John A. Nelder,et al.  Likelihood, Quasi-likelihood and Pseudolikelihood: Some Comparisons , 1992 .

[31]  M. Reiser,et al.  Sociological Methodology 1992. , 1993 .

[32]  K. Land,et al.  AGE, CRIMINAL CAREERS, AND POPULATION HETEROGENEITY: SPECIFICATION AND ESTIMATION OF A NONPARAMETRIC, MIXED POISSON MODEL* , 1993 .

[33]  R. J. O'Hara Hines,et al.  Improved Added Variable and Partial Residual Plots for the Detection of Influential Observations in Generalized Linear Models , 1993 .

[34]  E. Mulvey,et al.  Measuring patient violence in dangerousness research , 1993 .

[35]  Adrian F. M. Smith,et al.  Bayesian Inference for Generalized Linear and Proportional Hazards Models Via Gibbs Sampling , 1993 .

[36]  E. Mulvey,et al.  The accuracy of predictions of violence to others. , 1993, JAMA.

[37]  K. Emmons,et al.  Smoking at home: the impact of smoking cessation on nonsmokers' exposure to environmental tobacco smoke. , 1994, Health psychology : official journal of the Division of Health Psychology, American Psychological Association.

[38]  Joseph Hilbe,et al.  Generalized Linear Models , 1994, International Encyclopedia of Statistical Science.

[39]  Michael Weinstock,et al.  How Well Do Jurors Reason? Competence Dimensions of Individual Variation in a Juror Reasoning Task , 1994 .

[40]  Carolyn Rovee-Collier,et al.  Time Windows in Cognitive Development. , 1995 .