In statistical modeling, it is often of interest to evaluate non-negative quantities that capture heterogeneity in the population such as variances, mixing proportions and dispersion parameters. In instances of covariate-dependent heterogeneity, the implied homogeneity hypotheses are nonstandard and existing inferential techniques are not applicable. In this paper, we develop a quasi-score test statistic to evaluate homogeneity against heterogeneity that varies with a covariate profile through a regression model. We establish the limiting null distribution of the proposed test as a functional of mixtures of chi-square processes. The methodology does not require the full distribution of the data to be entirely specified. Instead, a general estimating function for a finite dimensional component of the model that is of interest is assumed but other characteristics of the population are left completely unspecified. We apply the methodology to evaluate the excess zero proportion in zero-inflated models for count data. Our numerical simulations show that the proposed test can greatly improve efficiency over tests of homogeneity that neglect covariate information under the alternative hypothesis. An empirical application to dental caries indices demonstrates the importance and practical utility of the methodology in detecting excess zeros in the data.