Estimating bird and bat fatalities caused by wind-turbine facilities is challenging when carcass counts are rare and produce counts that are either exactly zero or very near zero. The rarity of found carcasses is exacerbated when live members of a particular species are rare and when carcasses degrade quickly, are removed by scavengers, or are not detected by observers. With few observed carcass counts, common statistical methods like logistic, Poisson, or negative binomial regression are unreliable (statistically biased) and often fail to provide answers (i.e., fail to converge). Here, we propose a binomial N-mixture model that estimates fatality rates as well as the total number of carcass counts when these rates are expanded. Our model extends the ‘evidence of absence' model (Huso et al., 2015; Dalthorp, Huso, and Dail, 2017) by relating carcass deposition rates to study covariates and by incorporating terms that naturally scale counts from facilities of different sizes. Our model, which we call Evidence of Absence Regression (EoAR), can estimate the total number of birds or bats killed at a single wind energy facility or a fleet of wind energy facilities based on covariate values. Furthermore, with accurate prior distributions, the model's results are extremely robust to sparse data and unobserved combinations of covariate values. In this paper, we describe the model, show its low bias and high precision via computer simulation, and apply it to bat carcass counts observed at 21 wind energy facilities in Iowa.