Count data

Mathematics

Poisson distribution

Parametric statistics

Statistics

Goodness of fit

Overdispersion

Engineering

Beta-binomial distribution

Econometrics

Negative binomial distribution

Parametric model

Multinomial distribution

Sensitivity (control systems)

Electronic engineering

Binomial distribution

Mixture model

Methods in Ecology and Evolution

Binomial N‐mixture models are commonly applied to analyse population survey data. By estimating detection probabilities, N‐mixture models aim at extracting information about abundances in terms of absolute and not just relative numbers. This separation of detection probability and abundance relies on parametric assumptions about the distribution of individuals among sites and of detections of individuals among repeat visits to sites. Current methods for checking assumptions are limited, and their computational complexity has hindered evaluations of their performance. We use simulations and a case study to assess the sensitivity of binomial N‐mixture models to overdispersion in abundance and in detection, develop computationally efficient graphical goodness of fit checks to detect it, and evaluate the ability of the checks to identify overdispersion. The simulations show that if the parametric assumptions are not exact the bias in estimated abundances can be severe: underestimation if there is overdispersion in abundance relative to the fitted model and overestimation if there is overdispersion in detection. Our goodness‐of‐fit checks performed well in detecting lack of fit when the abundance distribution was overdispersed, but struggled to detect lack of fit when detections were overdispersed. We show that the inability to detect lack of fit due to overdispersed detection is caused by a fundamental similarity between N‐mixture models with beta‐binomial detections and N‐mixture models with negative binomial abundances. The strong biases that can occur in the binomial N‐mixture model when the distribution of individuals among sites, or the detection model, is mis‐specified implies that checking goodness of fit is essential for sound inference about abundance. To check the assumptions we provide computationally efficient goodness of fit checks that are available in an R‐package nmixgof . However, even when a binomial N‐mixture model appears to fit the data well, estimates are not robust in the presence of overdispersion. We show that problems can occur even when estimated detection probabilities are high, and that previously reported problems with negative binomial models cannot always be diagnosed by checking the sensitivity of abundance estimates to numerical cutoff values used in likelihood computations.

Sensitivity of binomial N‐mixture models to overdispersion: The importance of assessing model fit