Estimating bird and bat fatalities caused by wind-turbine facilities is challenging when fatalities are rare and the number of observed carcasses is either exactly zero or very near zero. The rarity of found carcasses is exacerbated when particular species are rare, when carcasses degrade quickly, when they are removed by scavengers, or when they are not detected by observers. With few observed fatalities, common statistical methods like logistic, Poisson, or negative binomial regression are biased and prone to fail due to complete or quasi-complete separation. Here, we propose a binomial N-mixture model to estimate fatality rates and totals that incorporates study covariates and separate information on probability of detection. Our model extends the 'evidence of absence' model (Huso et al., 2015) by relating carcass deposition rates to study covariates and by incorporating the number of turbines. Our model, which we call Evidence of Absence Regression (EoAR), can retrospectively and prospectively estimate the total number of birds or bats killed at a single wind-power facility or a fleet of wind-power facilities given covariates in the relation. Furthermore, with accurate prior distributions the model's results are extremely robust to complete or quasi-complete separation. In this paper, we describe the model, show its low bias and high precision via computer simulation, and apply it to bat fatalities observed on 21 wind power facilities in Iowa.