The COM-Poisson model for count data: a survey of methods and applications

The Poisson distribution is a popular distribution for modeling count data, yet it is constrained by its equidispersion assumption, making it less than ideal for modeling real data that often exhibit over-dispersion or under-dispersion. The COM-Poisson distribution is a two-parameter generalization of the Poisson distribution that allows for a wide range of over-dispersion and under-dispersion. It not only generalizes the Poisson distribution but also contains the Bernoulli and geometric distributions as special cases. This distribution's flexibility and special properties have prompted a fast growth of methodological and applied research in various fields. This paper surveys the different COM-Poisson models that have been published thus far and their applications in areas including marketing, transportation, and biology, among others. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  Julian J. Faraway,et al.  Extending the Linear Model with R , 2004 .

[2]  Dominique Lord,et al.  Modeling motor vehicle crashes using Poisson-gamma models: examining the effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter. , 2006, Accident; analysis and prevention.

[3]  Joseph B. Kadane,et al.  The Timing of Bid Placement and Extent of Multiple Bidding: An Empirical Investigation Using eBay Online Auctions , 2006, math/0609194.

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

[5]  James A Bonneson,et al.  Calibration of Predictive Models for Estimating Safety of Ramp Design Configurations , 2005 .

[6]  N. Ismail,et al.  Handling Overdispersion with Negative Binomial and Generalized Poisson Regression Models , 2007 .

[7]  Sharad Borle,et al.  The Impact of Survey Participation on Subsequent Customer Behavior: An Empirical Investigation , 2007 .

[8]  Srinivas Reddy Geedipally,et al.  Analyzing Different Parameterizations of the Varying Dispersion Parameter as a Function of Segment Length , 2009 .

[9]  P. McCullagh.  Regression Models for Ordinal Data , 1980 .

[10]  B. G. Heydecker,et al.  Identification of sites for road accident remedial work by Bayesian statistical methods: an example of uncertain inference , 2001 .

[11]  T. Minka,et al.  A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution , 2005 .

[12]  Claudia Czado,et al.  Predictive Model Assessment for Count Data , 2009, Biometrics.

[13]  Simon Washington,et al.  On the nature of over-dispersion in motor vehicle crash prediction models. , 2007, Accident; analysis and prevention.

[14]  Ernst Wit,et al.  Proceedings of the 19th International Workshop on Statistical Modelling , 2004 .

[15]  Ron S. Kenett,et al.  Measures of Association Applied to Operational Risks , 2010 .

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

[17]  J. Goldberg.  Economic impact of motor vehicle crashes. , 2002, Annals of Emergency Medicine.

[18]  Siddharth S. Singh,et al.  A Generalized Framework for Estimating Customer Lifetime Value When Customer Lifetimes are Not Observed , 2007 .

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

[20]  Doohee Nam,et al.  Accident prediction model for railway-highway interfaces. , 2006, Accident; analysis and prevention.

[21]  Galit Shmueli,et al.  Conjugate Analysis of the Conway-Maxwell-Poisson Distribution , 2006 .

[22]  Srinivas Reddy Geedipally,et al.  Extension of the Application of Conway‐Maxwell‐Poisson Models: Analyzing Traffic Crash Data Exhibiting Underdispersion , 2010, Risk analysis : an official publication of the Society for Risk Analysis.

[23]  W. Edwards Deming,et al.  On the Distinction between Enumerative and Analytic Surveys , 1953 .

[24]  Francesca Torti,et al.  Outliers and Robustness for Ordinal Data , 2011 .

[25]  H. Mills.  Marketing as a Science , 1961 .

[26]  F. Famoye.  Restricted generalized poisson regression model , 1993 .

[27]  W. Piegorsch.  Maximum likelihood estimation for the negative binomial dispersion parameter. , 1990, Biometrics.

[28]  Hila Etzion,et al.  Analyzing the Simultaneous Use of Auctions and Posted Prices for Online Selling , 2006, Manuf. Serv. Oper. Manag..

[29]  Anthony C. Atkinson,et al.  Robust Diagnostic Regression Analysis , 2000 .

[30]  Kirthi Kalyanam,et al.  Deconstructing Each Item's Category Contribution , 2007 .

[31]  Karl-Heinz Best.  Probability Distributions of Language Entities* , 2001, J. Quant. Linguistics.

[32]  Luis F. Miranda-Moreno,et al.  Effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter of Poisson-gamma models for modeling motor vehicle crashes: a Bayesian perspective , 2008 .

[33]  J. Faraway.  Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models , 2005 .

[34]  Narayanaswamy Balakrishnan,et al.  Connections of the Poisson weight function to overdispersion and underdispersion , 2008 .

[35]  C. I. Bliss,et al.  FITTING THE NEGATIVE BINOMIAL DISTRIBUTION TO BIOLOGICAL DATA AND NOTE ON THE EFFICIENT FITTING OF THE NEGATIVE BINOMIAL , 1953 .

[36]  Kimberly F. Sellers,et al.  Predicting Censored Count Data with COM-Poisson Regression , 2010 .

[37]  Rainer Winkelmann,et al.  Duration Dependence and Dispersion in Count-Data Models , 1995 .

[38]  Ron S. Kenett,et al.  A Test for Detecting Outlying Cells in the Multinomial Distribution and Two-Way Contingency Tables , 1980 .

[39]  Galit Shmueli,et al.  A Regression Model for Count Data with Observation-Level Dispersion , 2009 .

[40]  Felix Famoye,et al.  Statistical control charts for shifted generalized poisson distribution , 1994 .

[41]  Felix Famoye,et al.  Generalized Poisson-Poisson Mixture Model for Misreported Counts with an Application to Smoking Data , 2021, Journal of Data Science.

[42]  Kimberly F. Sellers,et al.  A Flexible Regression Model for Count Data , 2008, 1011.2077.

[43]  Srinivas Reddy Geedipally,et al.  TRB Paper # 11-2877 Examining the Crash Variances Estimated by the Poisson-Gamma and Conway-Maxwell-Poisson Models , 2010 .

[44]  Marta Pérez-Casany,et al.  Overdispersed and underdispersed Poisson generalizations , 2005 .

[45]  E Hauer,et al.  Overdispersion in modelling accidents on road sections and in empirical bayes estimation. , 2001, Accident; analysis and prevention.

[46]  Kimberly F. Sellers,et al.  Data Dispersion: Now You See It… Now You Don't , 2013 .

[47]  Seth D Guikema,et al.  A Flexible Count Data Regression Model for Risk Analysis , 2008, Risk analysis : an official publication of the Society for Risk Analysis.

[48]  G R Wood,et al.  Generalised linear accident models and goodness of fit testing. , 2002, Accident; analysis and prevention.

[49]  Tom Brijs,et al.  Explaining variation in safety performance of roundabouts. , 2010, Accident; analysis and prevention.

[50]  Srinivas Reddy Geedipally,et al.  Application of the Conway-Maxwell-Poisson generalized linear model for analyzing motor vehicle crashes. , 2008, Accident; analysis and prevention.

[51]  Srinivas Reddy Geedipally,et al.  Characterizing the Performance of the Conway‐Maxwell Poisson Generalized Linear Model , 2012, Risk analysis : an official publication of the Society for Risk Analysis.

[52]  W. R. Buckland.  Statistical Papers , 1971, Nature.

[53]  Vandna Jowaheer,et al.  Estimating Regression Effects in Com Poisson Generalized Linear Model , 2009 .

[54]  Ramayya Krishnan,et al.  A Data Disclosure Policy for Count Data Based on the COM-Poisson Distribution , 2006, Manag. Sci..

[55]  J. Hilbe.  Negative Binomial Regression: Preface , 2007 .

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

[57]  Ron S. Kenett,et al.  Relative Linkage Disequilibrium Applications to Aircraft Accidents and Operational Risks , 2008, Trans. Mach. Learn. Data Min..

[58]  Gabriel Altmann,et al.  Towards a Theory of Word Length Distribution , 1994, J. Quant. Linguistics.

[59]  Ron S. Kenett,et al.  Bootstrap Analysis of Designed Experiments , 2006, Qual. Reliab. Eng. Int..

[60]  Michael J. LuValle,et al.  Generalized Poisson Distributions: Properties and Applications , 1990 .

[61]  Fred L. Mannering,et al.  The statistical analysis of crash-frequency data: A review and assessment of methodological alternatives , 2010 .

[62]  Shaw-Pin Miaou,et al.  Modeling Traffic Crash-Flow Relationships for Intersections: Dispersion Parameter, Functional Form, and Bayes Versus Empirical Bayes Methods , 2003 .

[63]  Joseph C. Nunes,et al.  The Effect of Product Assortment Changes on Customer Retention , 2005 .

[64]  Srinivas Reddy Geedipally,et al.  Examination of Crash Variances Estimated by Poisson–Gamma and Conway–Maxwell–Poisson Models , 2011 .

[65]  Panagiotis Besbeas,et al.  An empirical model for underdispersed count data , 2004 .

[66]  Sharad Borle,et al.  Computing with the COM-Poisson distribution , 2003 .

[67]  Dominique Lord,et al.  Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory. , 2005, Accident; analysis and prevention.

[68]  A. W. Kemp,et al.  Generalized Poisson Distributions: Properties and Applications. , 1992 .

[69]  L. Bortkiewicz,et al.  Das Gesetz der kleinen Zahlen , 1898 .

[70]  Narayanaswamy Balakrishnan,et al.  COM–Poisson cure rate survival models and an application to a cutaneous melanoma data , 2009 .

[71]  Naftali Tishby,et al.  Cortical activity flips among quasi-stationary states. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[72]  Pravin K. Trivedi,et al.  Regression Analysis of Count Data , 1998 .

[73]  Joe N. Perry,et al.  Estimation of the Negative Binomial Parameter κ by Maximum Quasi -Likelihood , 1989 .

[74]  Seth D. Guikema,et al.  Erratum to "A flexible count data regression model for risk analysis" (Risk Analysis (2008) 28, 1, (213-223)) , 2008 .

[75]  Ron S. Kenett.  ‘The COM-Poisson model for count data: a survey of methods and applications’ by K. Sellers, S. Borle and G. Shmueli , 2012 .

[76]  S. Nadarajah.  Useful moment and CDF formulations for the COM–Poisson distribution , 2009 .

[77]  Sharad Borle,et al.  A Model of the Joint Distribution of Purchase Quantity and Timing , 2003 .

[78]  Weiren Wang,et al.  Censored generalized Poisson regression model , 2004, Comput. Stat. Data Anal..

[79]  C. Dean,et al.  Modified pseudo-likelihood estimator of the overdispersion parameter in Poisson mixture models , 1994 .