Confronting imperfect detection: behavior of binomial mixture models under varying circumstances of visits, sampling sites, detectability, and abundance, in small-sample situations

Binomial mixture models (BMMs) have been increasingly applied to account for imperfect detection and to estimate abundance from count data, but their performance has not been thoroughly evaluated. Here, I conducted simulation experiments to examine parameter estimates in BMMs under various situations. I generated data by assuming that abundance followed a Poisson distribution with an expected value lambda and that the number of detected individuals followed a binomial distribution with an individual detection probability p. In simple simulations without covariates for lambda and p, when the number of sampling sites (n) was between 20 and 160, BMMs could recover lambda and p under the following conditions: 0.1 <=lambda <= 160 and p >= 0.1. However, within these ranges of lambda and p, the estimates were variable under lower values of lambda and p, although the situation improved as n increased. When lambda and p are expected to exceed these ranges and the sample size is small, the results suggest that sampling and/or modeling designs should be reconsidered. I then conducted simulation experiments with covariates. I assumed that lambda increased with a covariate (x) across 20 sampling sites. I varied p, number of visits (v), and their dependency on a covariate. To compare BMMs with analyses that did not accommodate imperfect detection, I fitted ordinary Poisson generalized linear models to mean and maximum counts (GLM(mean) and GLM(max)). The results showed that GLM(max) was superior to GLM(mean) because GLM(mean) underestimated lambda when p was small. GLM(max) underestimated a coefficient of the covariate (slope) when v was negatively correlated with x. BMMs successfully recovered true values of the intercepts, slopes, and lambda in most cases. However, when p and v were small, and when p and lambda were highly negatively correlated due to their inverse dependency on x, estimates from BMMs were more variable.