Abstract
A general framework for the analysis of count data (with covariates) is proposed using formulations for the transition rates of a state-dependent birth process. The form for the transition rates incorporates covariates proportionally, with the residual distribution determined from a smooth non-parametric state-dependent form. Computation of the resulting probabilities is discussed, leading to model estimation using a penalized likelihood function. Two data sets are used as illustrative examples, one representing underdispersed Poisson-like data and the other overdispersed binomial-like data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball F. 1995. A note on variation in birth processes. Math. Scientist 20: 50–55.
Ball F. and Donnelly P. 1987. Interparticle correlation in death processes with application to variability in compartmental models. Adv. Appl. Prob. 18:755–766.
Bartlett M.S. 1978. AnIntroduction to Stochastic Processes. Cambridge University Press, Cambridge.
Brown T.C. and Donnelly P. 1993. On conditional intensities and on interparticle correlations in nonlinear death processes. Adv. Appl. Prob. 25: 255–260.
Consul P.C. 1989. Generalized Poisson Distributions. Marcel Dekker Inc., New York.
Cox D.R. and Miller H.D. 1965. The Theory of Stochastic Processes. Methuen, London.
Cox D.R. and Reid N. 1987. Parameter orthogonality and approximate conditional inference (with discussion). J. R. Statist. Soc. B 49: 1–39.
Daniels H.E. 1982. The saddlepoint approximation for a general birth process. J. Appl. Prob. 19: 20–28.
Dean C.B. 1998. Over-dispersion. In: Armitage P. and Colton T. (Eds.), Encyclopedia of Biostatistics, Wiley, London, pp. 3226–3232.
Donnelly P., Kurtz T., and Marjoram P. 1993. Correlation and variability in birth processes. J. Appl. Prob. 30: 275–284.
Efron B. 1978. Regression and ANOVA with zero-one data: Measures of residual variation. J. Am. Statist. Ass. 73: 113–121.
Efron B. 1986. Double exponential families and their use in generalized linear regression. J. Am. Statist. Ass. 81: 709–721.
Eubank R.L. 1988. Spline Smoothing and Nonparametric Regression. Marcel Dekker Inc., New York.
Faddy M.J. 1997a. Extended Poisson process modelling and analysis of count data. Biometrical Journal 39: 431–440.
Faddy M.J. 1997b. On extending the negative binomial distribution and the number of weekly winners of the UK national lottery. Math. Scientist 22: 77–82.
Faddy M.J. 1998a. Markov process modelling and analysis of discrete data. Appl. Stochastic Models and Data Analysis 13: 217–223.
Faddy M.J. 1998b. Stochastic models for analysis of species abundance data. In: Fletcher D.J., Kavalieris L. and Manly B.F.J. (Eds.), Statistics in Ecology and Environmental Monitoring 2. University of Otago Press, Dunedin, pp. 33–40.
Faddy M.J. and Bosch R.J. 2001. Likelihood-based modelling and analysis of data under-dispersed relative to the Poisson distribution. Biometrics 57: 620–624.
Faddy M.J. and Fenlon J.S. 1999. Stochastic modelling of the invasion process of nematodes in fly larvae. J. R. Statist. Soc. C 48: 31–37.
Good I.J. and Gaskins R.A. 1971. Nonparametric roughness penalties for probability densities. Biometrika 58: 255–277.
Green P.J. 1987. Penalized likelihood for general semi-parametric regression models. Int. Statist. Rev. 55: 245–259.
Green P.J. and Silverman B.W. 1994. Nonparametric Regression and Generalized Linear Models. Chapman and Hall, London.
Hastie T.J. and Tibshirani R.J. 1990. Generalized Additive Models. Chapman and Hall, London.
Lindsey J.K. 1999. Models for Repeated Measurements, 2nd edition. Clarendon Press, Oxford.
McCaughran D.A. and Arnold D.W. 1976. Statistical models for numbers of implantation sites and embryonic deaths in mice. Toxicol. and Appl. Pharmacol. 38: 325–333.
McCullagh P. and Nelder J.A. 1989. Generalized Linear Models, 2nd edition. Chapman and Hall, London.
Moler C.B. and van Loan C.F. 1978. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20: 801–836.
Nelson D.L. 1975. Some remarks on generalizations of the negative binomial and Poisson distributions. Technometrics 17: 135–136.
O’Sullivan F., Yandell B.S., and Raynor J. Jr. 1986. Automatic smoothing of regression functions in generalized linear models. J. Am. Statist. Ass. 81: 96–103.
Podlich H.M., Faddy M.J., and Smyth G.K. 1999. Likelihood computations for extended Poisson process models. InterStat, Sept., no. 1, 15 p.
Reinsch C.H. 1967. Smoothing by spline functions. Numer. Math. 10: 177–183.
Sidje R.B. 1998. EXPOKIT: Software package for computing matrix exponentials. ACM Transactions on Mathematical Software 24: 130–156.
Silverman B.W. 1985. Some aspects of the spline smoothing approach to non-parametric regression curve fitting. J. R. Statist. Soc. B 47: 1–52.
Smyth G.K. and Podlich H.M. 2002. An improved saddlepoint approximation based on the negative binomial distribution for the general birth process. Computational Statistics 17: 17–28.
Stewart W.J. 1994. Introduction to the Numerical Solution of Markov Chains. Princeton University Press, New Jersey.
Tapia R.A. and Thompson J.R. 1978. Nonparametric Probability Density Estimation. Johns Hopkins University Press, Baltimore.
Toscas P.J. and Faddy M.J. 2003. A likelihood-based analysis of longitudinal count data using a generalised Poisson model. Statistical Modelling 3: 99–108.
Tukey J.W. 1949. One degree of freedom for nonadditivity. Biometrics 5: 232–242.
Verbyla A.P., Cullis B.R., Kenward M.G., and Welham S.J. 1999. The analysis of designed experiments and longitudinal data by using smoothing splines (with discussion). J. R. Statist. Soc. C 48: 269–311.
Wecker W.E. and Ansley C.F. 1983. The signal extraction approach to nonlinear regression and spline smoothing. J. Am. Statist. Ass. 78: 81–89.
Winkelman R. 1997. Econometric Analysis of Count Data. Springer, Berlin.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Podlich, H.M., Faddy, M.J. & Smyth, G.K. Semi-parametric extended Poisson process models for count data. Statistics and Computing 14, 311–321 (2004). https://doi.org/10.1023/B:STCO.0000039480.66002.5a
Issue Date:
DOI: https://doi.org/10.1023/B:STCO.0000039480.66002.5a