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

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.