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

Abstract

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.

Time‐for‐space substitution in N‐mixture modeling and population monitoring

“…We evaluated goodness of fit of the global model (i.e., the model with all the covariates and in which other candidate models are nested) using a Pearson chi‐square test (MacKenzie and Bailey ), using a parametric bootstrap procedure (5,000 re‐sampling). Moreover, we also evaluated model fit by computing a quasi‐coefficient of variation (QCV) following Duarte et al () and inspecting residuals following Knape et al (). We ranked all candidate models with Akaike's Information Criterion corrected for small samples (AIC c ).…”

The importance of simulation assumptions when evaluating detectability in population models

“…We found dynamic N-mixture models underestimated N by ≥18%, and absolute bias inN decreased when we assumed all individuals were adults because yearlings attended leks less frequently than adults and therefore were less likely to be counted. Dynamic N-mixture models still underestimated N by ≥9% in scenarios without yearlings due to individuals that did not attend leks, and additional data and models to estimate annual lek attendance (p s ) and yearling abundance could be useful when estimates of absolute population size are needed , Knape et al 2018. In contrast, unmodeled heterogeneity from age and attendance had little effect onk from N-mixture models and so benefits from auxiliary data were more limited.…”

“…Furthermore, goodness-of-fit tests may perform worse when detectability is low and sample sizes are small (Duarte et al 2018), although we did not evaluate this directly in our study. Additional simulation studies and more targeted discrepancy measures, graphical methods, and cross-validation could provide further insights into model adequacy (K ery and Royle 2016, Conn et al 2018, Knape et al 2018).…”

Multiyear monitoring of bird communities in chlorpyrifos‐treated orchards in Spain and the United Kingdom: Spatial and temporal trends in species composition, abundance, and site fidelity

“…The best model among this set of models was chosen by the lowest Akaike information criterion value (Burnham and Anderson 2002). The goodness of fit of the best N-mixture model/per species was assessed by residual plots, QQ plots, and an estimate of overdispersion (ĉ), as suggested by Knape et al (2018). Model predictions were plotted with 95% confidence intervals, which might underestimate the uncertainty around the estimates but provides a reasonable approximation (Bates et al 2015).…”

“…The best model among this set of models was chosen by the lowest Akaike information criterion value (Burnham and Anderson ). The goodness of fit of the best N ‐mixture model/per species was assessed by residual plots, QQ plots, and an estimate of overdispersion (ĉ), as suggested by Knape et al ().…”