Overdispersion Diagnostics for Generalized Linear Models

Abstract Generalized linear models (GLM's) are simple, convenient models for count data, but they assume that the variance is a specified function of the mean. Although overdispersed GLM's allow more flexible mean-variance relationships, they are often not as simple to interpret nor as easy to fit as standard GLM's. This article introduces a convexity plot, or C plot for short, that detects overdispersion and relative variance curves and relative variance tests that help to understand the nature of the overdispersion. Convexity plots sometimes detect overdispersion better than score tests, and relative variance curves and tests sometimes distinguish the source of the overdispersion better than score tests.

[1]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[2]  D. Ruppert,et al.  Optimally bounded score functions for generalized linear models with applications to logistic regression , 1986 .

[3]  John A. Rice,et al.  Displaying the important features of large collections of similar curves , 1992 .

[4]  Daniel Zelterman,et al.  Homogeneity Tests against Central-Mixture Alternatives , 1988 .

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

[6]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.

[7]  Lisa M. Ganio,et al.  Diagnostics for Overdispersion , 1992 .

[8]  Diane Lambert,et al.  Generalizing Logistic Regression by Nonparametric Mixing , 1989 .

[9]  Prem S. Puri,et al.  On Optimal Asymptotic Tests of Composite Statistical Hypotheses , 1967 .

[10]  J. Nelder,et al.  An extended quasi-likelihood function , 1987 .

[11]  B. Efron.  Double Exponential Families and Their Use in Generalized Linear Regression , 1986 .

[12]  Alan E. Gelfand,et al.  A Note on Overdispersed Exponential Families , 1990 .

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

[14]  J. Kalbfleisch,et al.  A Comparison of Cluster-Specific and Population-Averaged Approaches for Analyzing Correlated Binary Data , 1991 .

[15]  K. Roeder,et al.  Residual diagnostics for mixture models , 1992 .

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

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

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

[19]  Gordon K. Smyth,et al.  Generalized linear models with varying dispersion , 1989 .

[20]  J. Cooper.  TOTAL POSITIVITY, VOL. I , 1970 .

[21]  Moshe Shared,et al.  On Mixtures from Exponential Families , 1980 .

[22]  Murray Aitkin,et al.  Variance Component Models with Binary Response: Interviewer Variability , 1985 .

[23]  Allan R. Wilks,et al.  The new S language: a programming environment for data analysis and graphics , 1988 .

[24]  T A Louis,et al.  Random effects models with non-parametric priors. , 1992, Statistics in medicine.

[25]  Daniel F. Heitjan,et al.  Testing and Adjusting for Departures from Nominal Dispersion in Generalized Linear Models , 1993 .

[26]  B. Lindsay.  Exponential family mixture models (with least-squares estimators) , 1986 .