Zero-inflated regression models have emerged as a popular tool within the parametric framework to characterize count data with excess zeros. Despite their increasing popularity, much of the literature on real applications of these models has centered around the latent class formulation where the mean response of the so-called at-risk or susceptible population and the susceptibility probability are both related to covariates. While this formulation in some instances provides an interesting representation of the data, it often fails to produce easily interprétable covariate effects on the overall mean response. In this article, we propose two approaches that circumvent this limitation. The first approach consists of estimating the effect of covariates on the overall mean from the assumed latent class models, while the second approach formulates a model that directly relates the overall mean to covariates. Our results are illustrated by extensive numerical simulations and an application to an oral health study on low income African-American children, where the overall mean model is used to evaluate the effect of sugar consumption on caries indices.