Probability distributions for marketable pods and white mould on snap bean

Abstract

White mould, caused by Sclerotinia sclerotiorum, is one of the most recalcitrant diseases of snap bean. Probability distributions suitable for describing the total number of marketable pods (hereafter simply referred to as pods) per plant, as well as for the number of pods and stems with white mould per plant, have not been identified. The total number of pods and the number of pods and stems with white mould were measured on a per plant basis in plots of processing snap bean (var. Hystyle) in New York. The total number of pods per plant ranged from 0 to 29, and was best described by the Pólya-Aeppli distribution. The number of pods and the number of stems with white mould per plant were well-described by the negative binomial (NB) distribution. A Sarmanov bivariate distribution with NB marginals was derived and fitted to the joint data on the number of pods and stems with white mould per plant, accounting for correlation between pods and stems with white mould on the same plant. The bivariate distribution was used to formulate an empirical equation for the incidence of plants with white mould as a function of the average number of pods and stems with white mould per plant. The results represented a more complete understanding of the distributional properties of white mould in snap bean.

Hierarchical models for white mould in snap bean

“…With high counts the NB distributional assumptions, upon which popular incidence–count models are based, may not hold, and other distributions may be warranted (Xu & Madden, ). This issue was not a concern with the current study, as the number of pods with white mould per plant did not exceed seven and was well described by the NB distribution (Shah et al ., ). A comparison of the fit of the equations for i pad as a function of m did indicate the tendency for pods with white mould to cluster on the same plant.…”

“…For white mould in processing snap bean, interest lies in predicting low i pod values of about 2–6%, which therefore focuses predictive effort on the lower‐variance regions of the relationship. It may seem paradoxical that the posterior predictive interval for i pod includes zero up to i q(2) ≅ 0.2, but one has to bear in mind that some plants may have stems (but not pods) with white mould, or may not have any pods at all (Shah et al ., ). At i q(2) = 0.1, there was a 92% chance that i pod was less than an assumed processor acceptance threshold of 2%.…”

“…Consistency of the relationships would have to be demonstrated before adoption (Gent et al ., ). Unfortunately, collecting the required data would involve individually evaluating thousands of pods (Shah et al ., ). Instead of relying solely on additional field data, it may be more efficient to design and test new sampling plans using simulation.…”

“…The position of each plant within a transect was noted. A plant was considered affected by white mould if at least one pod or stem had symptoms (Shah et al ., ).…”

“…The number of pods with white mould per plant was modelled by the NB distribution (Shah et al ., ). The NB probability mass function is given bywhere x is an integer variable ≥ 0, and r and p are the parameters of the distribution with r > 0 and 0 < p < 1.…”