Modeling under or over-dispersed binomial count data by using extended Altham distribution families

Year 2021, , 255 - 274, 04.02.2021

Senay Asma

https://doi.org/10.15672/hujms.671806

Abstract

While aiming particularly at handling under-dispersion, we explore a type of models constructed conservatively using the minimum information of first two moments for the fitting of binomial count data, which could have under, equal or over-dispersion. The extended Altham distribution (EAD) families were presented in this study. The extended Altham families are very close to the binomial distribution under equal dispersion setting, implying that they are alternative models of the binomial distribution. The feature that extended Altham families can reach the full range of dispersion outperforms some commonly used models such as extended beta-binomial and quasi-binomial which have restricted ranges of dispersion. Moreover, the extended Altham family can have double peaks at two boundaries, indicating they are feasible for fitting the double tail inflation phenomenon. This study illustrated the modeling using extended Altham families for both under-dispersed and over-dispersed binomial data resulted from disease cases within the same family.

Keywords

Binomial count data, Kullback-Leibler, exponential family, binomial distribution, dispersion index

References

• [1] P.M.E. Altham, Two Generalizations of the binomial distribution, Appl. Statist. 27 (2), 162-167, 1978.
• [2] P.M.E. Altham and R.K.S. Hankin, Multivariate generalizations of the multiplicative binomial distribution: Introducing the MM package, J. Stat. Softw 46 (12), 1-23, 2012.
• [3] P.M.E. Altham and J.K. Lindsey, Analysis of the human sex ratio by using overdispersion models, Appl. Statist. 47 (1), 149-157, 1998.
• [4] B.J.R. Bailey, A model for function word counts, Appl. Statist. 39 (1), 107-114, 1990.
• [5] S. Chakraborty and K.K. Das, On some properties of a class of weighted quasibinomial distributions, J. Statist. Plann. Inference 136 (1), 159-182, 2006.
• [6] C. Chatfield, Problem Solving: A Statistician’s Guide, Chapman & Hall, London, 1988.
• [7] G.K. Chesterton, Selected Essays of G. K. Chesterton, London: Methuen, 1949.
• [8] P.C. Consul, A simple urn model dependent upon predetermined strategy, Sankhya A Series B 36 (4), 391-399, 1974.
• [9] I. Dobson, B.A. Carreras and D.E. Newman, A probabilistic loading-dependent model of cascading failure and possible implications for blackouts, Proceedings of the 36th Annual Hawaii International Conference on System Sciences (HICSS’03), 2003.
• [10] M. Elwood and A. Coldman, Age of mothers with breast cancer and sex of their children, Br Med J 282 (6265), 734, 1981.
• [11] J.K. Haseman and L.L. Kupper, Analysis of dichotomous response data from certain toxicological experiments, Biometrics 35 (1), 281-293, 1979.
• [12] H. Joe, Multivariate Models and Dependence Concepts, Chapman & Hall, London, 1997.
• [13] L. Macaulay, Literary Essays Contributed to the Edinburgh Review, London: Oxford University Press, 1923.
• [14] A. Mishra, D. Tiwary and S.K. Singh, A class of quasi-binomial distributions, Sankhya A Series B 54 (1), 67-76, 1992.
• [15] R.L. Prentice, Binary regression using an extended beta-binomial distribution, with discussion of correlation induced by covariate measurement errors, J. Amer. Statist. Assoc. 81 (394), 321-327, 1986.
• [16] N. Sastry, A nested frailty model for survival data, with an application to the study of child survival in northest Brazil, J. Amer. Statist. Assoc. 92 (438), 426-434, 1997.
• [17] R. Viveros-Aguilera, K. Balasubramanian and N. Balakrishnan, Binomial and negative binomial analogues under correlated Bernoulli trials, Amer. Statist. 48 (3), 243-247, 1994.
• [18] R.R. Wilcox, A review of the beta-binomial model and its extensions, J. Educ. Stat. 6 (1), 3-32, 1981.
• [19] D. Zelterman, Discrete Distributions: Applications in the Health Sciences, John Wiley & Sons, West Sussex, 2004.