Estimating Regression Effects in Com Poisson Generalized Linear Model
Com Poisson distribution is capable of modeling the count responses irrespective of their mean variance relation and the parameters of this distribution when fitted to a simple cross sectional data can be efficiently estimated using maximum likelihood (ML) method. In the regression setup, however, ML estimation of the parameters of the Com Poisson based generalized linear model is computationally intensive. In this paper, we propose to use quasilikelihood (QL) approach to estimate the effect of the covariates on the Com Poisson counts and investigate the performance of this method with respect to the ML method. QL estimates are consistent and almost as efficient as ML estimates. The simulation studies show that the efficiency loss in the estimation of all the parameters using QL approach as compared to ML approach is quite negligible, whereas QL approach is lesser involving than ML approach.
[1] G. Bates and J. Neymann, " Contribution to the theory of accident
processes", Publications in Statistics Vol 1, 215-253, 1952, University
of California.
[2] N. Breslow and X. Lin , " Bias correction in generalized linear mixed
models", Biometrics Vol 82, 81-91, 1995.
[3] R. Conway and W. Maxwell, " A queuing model with state dependent
service rates", Journal of Industrial Engineering, Vol 12, 132-136,
1962.
[4] C. Dean and R. Balshaw, " Efficiency lost by analyzing counts rather
than even times in Poisson and overdispersed Poisson regression
models", Journal of American Statistical Association, Vol 92, 1387-
1398, 1997.
[5] D. Firth and L. Harris, " Quasi-likelihood for multiplicative random
effects", Biometrics, Vol 78, 545-555, 1991.
[6] S. Guikema, " Formulating informative, data(based priors for failure
probability estimation in reliability analysis", Reliability Engineering
and System Safety, Vol 92, 490-502, 2007.
[7] S. Guikema and J. Gofelt, " A flexible count data regression model for
risk analysis", Risk analysis, Vol 28, 213-223, 2008.
[8] J. Kadane, G. Shmueli, G. Minka, T. Borle and P. Boatwright, "
Conjugate analysis of the Conway Maxwell Poisson distribution",
Bayesian analysis, Vol 1, 363-374, 2006.
[9] J. Klein, " Semi parametric estimation of random effects using Cox
model based on the EM algorithm", Biometrics , Vol 48, 795-806, 1992.
[10] D. Lord , S. Washington and J. Ivan, " Poisson, Poisson(Gamma and
Zero inflated regression models of motor vehicle crashes: Balancing
statistical fit and theory", Accident analysis and Prevention, Vol 37, 35-
46, 2005.
[11] T. Minka, G. Shmueli, J. Kadane, J. Borle and P. Boatwright,"
Computing with the Com Poisson distribution", Technical report CMU
statistics department, http:// www.stat.cmu.edu/tr/tr776/tr776.html, 2003.
[12] D. Moore, " Asymptotic properties of moment estimators for
overdispersed counts and proportions", Biometrics Vol 73, 583-588,
1986.
[13] G. Shmueli, T. Minka, J. Borle and P. Boatwright, " A useful
distribution for fitting discrete data", Applied Statistics, Journal of Royal
Statistical Society, 2005.
[14] B. Sutradhar and K. Das, " A higher order approximation to likelihood
inference in the Poisson mixed model", Statist, Prob, Lett, Vol 52, 59-
67, 2001.
[15] R. Wedderburn, " Quasi-likelihood functions, generalized linear models
and the Gauss Newton method, Biometrics, Vol 61, 439-447, 1974.
[1] G. Bates and J. Neymann, " Contribution to the theory of accident
processes", Publications in Statistics Vol 1, 215-253, 1952, University
of California.
[2] N. Breslow and X. Lin , " Bias correction in generalized linear mixed
models", Biometrics Vol 82, 81-91, 1995.
[3] R. Conway and W. Maxwell, " A queuing model with state dependent
service rates", Journal of Industrial Engineering, Vol 12, 132-136,
1962.
[4] C. Dean and R. Balshaw, " Efficiency lost by analyzing counts rather
than even times in Poisson and overdispersed Poisson regression
models", Journal of American Statistical Association, Vol 92, 1387-
1398, 1997.
[5] D. Firth and L. Harris, " Quasi-likelihood for multiplicative random
effects", Biometrics, Vol 78, 545-555, 1991.
[6] S. Guikema, " Formulating informative, data(based priors for failure
probability estimation in reliability analysis", Reliability Engineering
and System Safety, Vol 92, 490-502, 2007.
[7] S. Guikema and J. Gofelt, " A flexible count data regression model for
risk analysis", Risk analysis, Vol 28, 213-223, 2008.
[8] J. Kadane, G. Shmueli, G. Minka, T. Borle and P. Boatwright, "
Conjugate analysis of the Conway Maxwell Poisson distribution",
Bayesian analysis, Vol 1, 363-374, 2006.
[9] J. Klein, " Semi parametric estimation of random effects using Cox
model based on the EM algorithm", Biometrics , Vol 48, 795-806, 1992.
[10] D. Lord , S. Washington and J. Ivan, " Poisson, Poisson(Gamma and
Zero inflated regression models of motor vehicle crashes: Balancing
statistical fit and theory", Accident analysis and Prevention, Vol 37, 35-
46, 2005.
[11] T. Minka, G. Shmueli, J. Kadane, J. Borle and P. Boatwright,"
Computing with the Com Poisson distribution", Technical report CMU
statistics department, http:// www.stat.cmu.edu/tr/tr776/tr776.html, 2003.
[12] D. Moore, " Asymptotic properties of moment estimators for
overdispersed counts and proportions", Biometrics Vol 73, 583-588,
1986.
[13] G. Shmueli, T. Minka, J. Borle and P. Boatwright, " A useful
distribution for fitting discrete data", Applied Statistics, Journal of Royal
Statistical Society, 2005.
[14] B. Sutradhar and K. Das, " A higher order approximation to likelihood
inference in the Poisson mixed model", Statist, Prob, Lett, Vol 52, 59-
67, 2001.
[15] R. Wedderburn, " Quasi-likelihood functions, generalized linear models
and the Gauss Newton method, Biometrics, Vol 61, 439-447, 1974.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:59826", author = "Vandna Jowaheer and Naushad A. Mamode Khan", title = "Estimating Regression Effects in Com Poisson Generalized Linear Model", abstract = "Com Poisson distribution is capable of modeling the count responses irrespective of their mean variance relation and the parameters of this distribution when fitted to a simple cross sectional data can be efficiently estimated using maximum likelihood (ML) method. In the regression setup, however, ML estimation of the parameters of the Com Poisson based generalized linear model is computationally intensive. In this paper, we propose to use quasilikelihood (QL) approach to estimate the effect of the covariates on the Com Poisson counts and investigate the performance of this method with respect to the ML method. QL estimates are consistent and almost as efficient as ML estimates. The simulation studies show that the efficiency loss in the estimation of all the parameters using QL approach as compared to ML approach is quite negligible, whereas QL approach is lesser involving than ML approach.
", keywords = "Com Poisson, Cross-sectional, Maximum
Likelihood, Quasi likelihood", volume = "3", number = "5", pages = "377-5", }
