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

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.