Abundance estimation is central to avian ecology. For replicated counts, Royle (2004) developed a model to estimate abundance adjusted for detectability. Hitherto, it was unknown whether the same covariate was allowed to affect both abundance and detectability. This situation was disconcerting, because relationships between abundance and such covariates describing, for example, habitat, lie at the heart of ecology. I test this by simulation and provide additional guidelines on the model as well as code to fit it in a Bayesian mode of analysis. I simulated 1,000 data sets mimicking the Swiss breeding-bird survey "Monitoring Haufige Brutvogel" (three surveys in each of 268 quadrats). Elevation affected abundance negatively and detectability positively, resulting in a hump-shaped relationship between counts and elevation. I used WinBUGS to fit the model and estimate parameters, including quad rat-specific abundance and total abundance, across all 268 quadrats. For every parameter, the model recovered estimates that showed no indication of bias. The mean error in the estimated total population size across all quadrats was only 2%, whereas the summed maximum counts, a conventional abundance estimate, underestimated total population size by 43%. In contrast to maximum counts, the binomial mixture model revealed the true negative relationship between abundance and elevation. This model is a promising new alternative to capture-recapture or distance sampling methods to estimate bird abundance free of distorting effects of detectability. It has perhaps the fewest requirements, needing neither individual identification nor distance information to "convert" simple counts ("relative abundance") into estimates of true abundance. It ought to be seriously considered in future bird-survey schemes.
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