Zero-inflated sum of Conway-Maxwell-Poissons (ZISCMP) regression

Abstract

While excess zeros are often thought to cause data over-dispersion (i.e. when the variance exceeds the mean), this implication is not absolute. One should instead consider a flexible class of distributions that can address data dispersion along with excess zeros. This work develops a zero-inflated sum-of-Conway-Maxwell-Poissons (ZISCMP) regression as a flexible analysis tool to model count data that express significant data dispersion and contain excess zeros. This class of models contains several special case zero-inflated regressions, including zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), zero-inflated binomial (ZIB), and the zero-inflated Conway-Maxwell-Poisson (ZICMP). Through simulated and real data examples, we demonstrate class flexibility and usefulness. We further utilize it to analyze shark species data from Australia's Great Barrier Reef to assess the environmental impact of human action on the number of various species of sharks.

Keywords:

• Count data modelling
• discrete data
• over-dispersion
• under-dispersion
• zero-inflated negative binomial
• zero-inflated Poisson

2010 Mathematics Subject Classifications:

• 62J12
• 62F10
• 62-07

Acknowledgements

The authors thank the anonymous reviewers for their insightful comments that helped improve the manuscript. The authors also thank Sophie Lockwood (Georgetown University) for her assistance.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Kimberly F. Sellers http://orcid.org/0000-0001-6516-0548

Derek S. Young http://orcid.org/0000-0002-3048-3803

Notes

1 We are assuming $\mathbf{b}=\mathbf{1}$, which is a ‘zero-inflated Bernoulli’ distribution, i.e. a Bernoulli distribution with an adjusted success probability to account for the excess zeros.

2 We had to carefully select parameter values for the ZIGP simulations to ensure that the constraints on the parameters stated in [9] were met, otherwise, this could create difficulties for the random variable generation function included in the ZIGP package [27]. Note that this package has been archived by CRAN and, thus, is not maintained for errors.

3 We wrote the random generation structure in terms of p and ν as this is more illustrative than the regression parameters of, respectively, ${\delta }_0$ and ${\gamma }_0$.

Additional information

Funding

Support for Kimberly Sellers was provided in part by the American Statistical Association (ASA)/ National Science Foundation (NSF)/ Census Research Program, U. S. Census Bureau Contract #YA1323-14-SE-0122. This paper is released to inform interested parties of research and to encourage discussion. The views expressed are those of the authors and not necessarily those of the U.S. Census Bureau.