In this paper we introduce the Hinde-Demétrio (HD) regression models for analyzing overdispersed count data and, mainly, investigate the e¤ect of dispersion parameter. The HD distributions are discrete additive exponential dispersion models (depending on canonical and dispersion parameters) with a third real index parameter p and have been characterized by its unit variance function + p . For p equals to 2; 3; , the corresponding distributions are concentrated on nonnegative integers, overdispersed and zero-in ‡ated with respect to a Poisson distribution having the same mean. The negative binomial (p = 2), strict arcsine (p = 3) and Poisson (p ! 1) distributions are particular count HD families. From generalized linear modelling framework, the e¤ect of dispersion parameter in the HD regression models, among other things, is pointed out through the double mean parametrization: unit and standard means. In the particular additive model, this e¤ect must be negligible within an adequate HD model for …xed integer p. The estimation of the integer p is also examined separately. The results are illustrated and discussed on a horticultural data set.