Twenty-Five Years of the Binary Power Law for Characterizing Heterogeneity of Disease Incidence

Abstract

Spatial pattern, an important epidemiological property of plant diseases, can be quantified at different scales using a range of methods. The spatial heterogeneity (or overdispersion) of disease incidence among sampling units is an especially important measure of small-scale pattern. As an alternative to Taylor's power law for the heterogeneity of counts with no upper bound, the binary power law (BPL) was proposed in 1992 as a model to represent the heterogeneity of disease incidence (number of plant units diseased out of n observed in each sampling unit, or the proportion diseased in each sampling unit). With the BPL, the log of the observed variance is a linear function of the log of the variance for a binomial (i.e., random) distribution. Over the last quarter century, the BPL has contributed to both theory and multiple applications in the study of heterogeneity of disease incidence. In this article, we discuss properties of the BPL and use it to develop a general conceptualization of the dynamics of spatial heterogeneity in epidemics; review the use of the BPL in empirical and theoretical studies; present a synthesis of parameter estimates from over 200 published BPL analyses from a wide range of diseases and crops; discuss model fitting methods, and applications in sampling, data analysis, and prediction; and make recommendations on reporting results to improve interpretation. In a review of the literature, the BPL provided a very good fit to heterogeneity data in most publications. Eighty percent of estimated slope (b) values from field studies were between 1.06 and 1.51, with b positively correlated with the BPL intercept parameter. Stochastic simulations show that the BPL is generally consistent with spatiotemporal epidemiological processes and holds whenever there is a positive correlation of disease status of individuals composing sampling units.

Spatiotemporal dynamics and modelling support the case for area‐wide management of citrus greasy spot in a Brazilian smallholder farming region

“…The binary power law, an adapted form of the Taylor power law for proportional data where variances do not increase monotonically with means (Madden et al , ), was used to assess the spatial heterogeneity of disease incidence. If the disease incidence was found to be aggregated (indicated by a slope, b < 1), this would favour the hypothesis that auto‐infection was an important driver of CGS incidence in groves.…”

“…Whilst the index of dispersion considers individual data sets, the binary power law can be used to assess multiple data sets (Madden et al , ). The binary power law uses observed variance and expected binomial variance to estimate the spatial heterogeneity of CGS:…”

“…The binary power law, an adapted form of the Taylor power law for proportional data where variances do not increase monotonically with means (Madden et al, 2018), was used to assess the spatial heterogeneity of disease incidence. If the disease incidence was found to F I G U R E 1 Locations in Recôncavo of Bahia, Brazil (our Hierarchical Level 4) that were sampled for disease.…”

Spatiotemporal dynamics and modelling support the case for area-wide management of citrus greasy spot in a Brazilian smallholder farming region

“…The Binary Power Law, an adapted form of the Taylor Power Law for proportional 189 data where variances do not increase monotonically with means (Madden et al, 2018), was 190 used to assess the spatial heterogeneity of disease incidence. If the disease incidence was 191 found to be aggregated (indicated by a slope, b, < 1), this would favour the hypothesis that 192 auto-infection was an important driver of CGS incidence in groves.…”

“…Whilst the index of dispersion considers individual datasets, the Binary Power Law 196 (BPL) can be used to assess multiple data sets (Madden et al, 2018). The BPL uses observed 197 variance and expected binomial variance to estimate the spatial heterogeneity of CGS.…”