Tests of Hypotheses in Overdispersed Poisson Regression and other Quasi-Likelihood Models

Abstract Test statistics for evaluating the significance of added variables in a regression equation are developed for mixed Poisson models, where the structural parameter φ that determines the mean/variance relationship var(μ; φ) = μ + φ · μ 2 is estimated by the method of moments and the regression coefficients are estimated by quasi-likelihood. The formulas presented for test statistics and related estimating equations are applicable generally to quasi-likelihood models specified by an arbitrary mean value function μ(x; β), together with a variance function V(μ; φ) that contains one or more unknown parameters. Two versions of the Wald and score tests are investigated—one calculated from the usual model-based covariance matrix whose validity depends on correct specification of the variance function, and another using an “empirical” covariance matrix that has a more general asymptotic justification. Monte Carlo simulations demonstrate that the quasi-likelihood/method of moments (QL/M) procedures yield ap...

[1]  M. Woodbury,et al.  A variance components approach to categorical data models with heterogeneous cell populations: analysis of spatial gradients in lung cancer mortality rates in North Carolina counties. , 1981, Biometrics.

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

[3]  Dirk F. Moore,et al.  Modelling the Extraneous Variance in the Presence of Extra‐Binomial Variation , 1987 .

[4]  J. Sacks,et al.  Model robust confidence intervals , 1982 .

[5]  John Hinde,et al.  Compound Poisson Regression Models , 1982 .

[6]  George S. Fishman,et al.  Sampling from the Binomial Distribution on a Computer , 1979 .

[7]  D. A. Williams,et al.  Extra‐Binomial Variation in Logistic Linear Models , 1982 .

[8]  Dirk F. Moore,et al.  Asymptotic properties of moment estimators for overdispersed counts and proportions , 1986 .

[9]  A Tsiatis,et al.  Robust estimation of the variance in moment methods for extra-binomial and extra-Poisson variation. , 1991, Biometrics.

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

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

[12]  N. Inagaki.  Asymptotic relations between the likelihood estimating function and the maximum likelihood estimator , 1973 .

[13]  B H Margolin,et al.  Statistical analysis of the Ames Salmonella/microsome test. , 1981, Proceedings of the National Academy of Sciences of the United States of America.

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

[15]  R. Carroll,et al.  Variance Function Estimation , 1987 .

[16]  Michael P. Keating,et al.  CHAPTER ELEVEN – The Gauss System , 1988 .

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

[18]  R. Royall.  Model robust confidence intervals using maximum likelihood estimators , 1986 .

[19]  D. Ruppert,et al.  Transformation and Weighting in Regression , 1988 .

[20]  David Ruppert,et al.  Robust Estimation in Heteroscedastic Linear Models. , 1982 .

[21]  D. Pregibon.  Score Tests in GLIM with Applications , 1982 .

[22]  M. Væth,et al.  On the use of Wald's test in exponential families , 1985 .

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

[24]  Edward L. Frome,et al.  Regression Analysis of Poisson-Distributed Data , 1973 .

[25]  T R Holford,et al.  The estimation of age, period and cohort effects for vital rates. , 1983, Biometrics.

[26]  L Claxton,et al.  Modeling the Ames test. , 1981, Mutation research.

[27]  Robert V. Foutz,et al.  The Performance of the Likelihood Ratio Test When the Model is Incorrect , 1977 .

[28]  N. Breslow,et al.  Statistical methods in cancer research. Volume II--The design and analysis of cohort studies. , 1987, IARC scientific publications.

[29]  Norman E. Breslow,et al.  Score Tests in Overdispersed GLM’s , 1989 .

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

[31]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

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

[33]  H. Checkoway,et al.  Epidemiologic programs for computers and calculators. Use of Poisson regression models in estimating incidence rates and ratios. , 1985, American journal of epidemiology.