Summarizing the regional incidence of seed-borne fungi with the β–binomial distribution

Abstract

A theoretical probability distribution conveys more information than the mean in summarizing data. We investigated the ability of two discrete probability distributions, binomial and β–binomial, to describe the incidence (proportion) of seed-borne fungi among seed lots. The fit of the distributions to 185 data sets was assessed by either χ2 analysis or a dithered Kolmogorov-Smirnov goodness-of-fit test. The data sets represented a range of fungi, crops, and geographic regions. The binomial distribution was an adequate fit to only 36% of the data sets, whereas the β–binomial distribution adequately fit 85% of the data sets (P > 0.05). The β–binomial was a better fit than the binomial in 72% of data sets (P < 0.01) based on the likelihood-ratio test, indicating that there was greater variability in seed infection than expected for a binomial (i.e., random) distribution. For a subset of 25 data sets on wheat-seed infection by Fusarium graminearum Schwabe, a binary power law analysis indicated that heterogeneity of seed infection (summarized by the θ parameter of the β–binomial) was a function of mean incidence. Therefore, in most instances, the β–binomial captures the observed heterogeneity in the incidence of seed-borne fungi.