Plant disease incidence: Inverse sampling, sequential sampling, and confidence intervals when observed mean incidence is zero

Sequential and inverse sampling equations were developed for estimating the mean proportion of diseased plants (or plant units), disease incidence (p), where the data were obtained by cluster sampling. With cluster sampling, the disease status of all n plants in each of N sampling units is determined. Derived sampling equations were applicable for up to three ways of specifying reliability or precision of estimated p, and the following conditions of spatial heterogeneity: i) random pattern, with data described by the binomial distribution; ii) aggregated pattern, with data described by the beta- binomial distribution, and constant degree of aggregation (assessed with the e index of the beta-binomial); and iii) aggregated pattern, with data described by the binary form of the power law, in which the observed (empirical) variance of incidence is a power function of the theoretical variance for a binomial distribution. For the third situation, aggregation can vary with p, such that o is a function of the power law parameters. A selection of the sequential estimation equations were evaluated by the simulated sampling from: 1) data sets of the incidence of grape vines infected by Eutypa lata, and 2) simulated data with beta-binomial distributions. Results on achieved (observed) coefficient of variation of estimated p(C), difference between observed and true p, and average sample number, were similar to that found for the analogous equations developed for noncluster simple random sampling of count data with no upper bound (e.g. insect counts). Equations also were developed to calculate confidence intervals for p as a function of n, N, and o, when all sampled observations are disease free. These equations used the approximation of the negative binomial to the beta-binomial distribution that applies when p is small.