Empirical binomial sampling plans: model calibration and testing using Williams’ method III for generalized linear models with overdispersion

Binomial sampling plans that use presence/absence data for estimating pest population density are commonly used in cropprotection when counting individual pest units is not cost effective. These plans are often based on the empirical relationship between the proportion of presences, p, and a count-based estimate of the mean population density, (m)\tilde]\tilde \mu , given by ln{ - ln(1 - p) } = a₀ + a₁ ln((m)\tilde] )\ln \left\{ { - \ln (1 - p)} \right\} = \alpha _0 + \alpha _1 ln(\tilde \mu ), which is typically fitted as a simple linear regression. However, correctly incorporating all of (i) binomial sampling errors, (ii) biological errors (i.e., overdispersion), and (iii) errors in variables is not possible using linear regression. Here, model calibration and testing is carried out using William’s method III for fitting a binomial generalized linear model with overdispersion (GLMw) in order to handle (i) and (ii), and simulation is used to study the effect of using the sample estimate of μ as the predictor variable. Calculation of the operating characteristics function of the decision rule for an action threshold of p ₀ is compared for linear and GLM_w models, with the former shown to substantially underestimate the probability of correct decisions and overestimate the probability of incorrect decisions. A binomial sampling plan for populations of the leaf beetle Chrysophtharta bimaculata, a defoliator of Eucalyptusnitens plantations, is used to demonstrate the methods.