Comments Regarding the Binary Power Law for Heterogeneity of Disease Incidence

Authors: Turechek, William; MADDEN, L - The Ohio State University; Gent, David - Dave; XU, X - East Malling Research

Submitted to: Phytopathology
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 7/27/2011
Publication Date: 12/1/2011
Citation: Turechek, W., Madden, L.V., Gent, D.H., Xu, X.M. 2011. Comments Regarding the Binary Power Law for Heterogeneity of Disease Incidence. Phytopathology. 101:1396-1407.

Interpretive Summary: The binary power law (BPL) has been successfully used to characterize small-scale aggregation of disease incidence for many plant pathosystems. With the BPL, the log of the observed variance is a linear function of the log of the theoretical variance for a binomial distribution and the estimated scale and slope parameters provide information on the characteristics of aggregation. In two articles published in Phytopathology in 2009, Gosme and Lucas used their stochastic simulation model, Cascade, to reveal attributes of this relationship that have never been observed with real-world data. We evaluated their findings by utilizing a general spatially-explicit stochastic simulator developed by Xu and Ridout in 1998 and through an assessment of published BPL results. Our results confirmed part of their findings, but showed that they occur only under very specific set of circumstances. The implications are important to epidemiologists that describe the spatial properties of plant disease epidemics.

Technical Abstract: The binary power law (BPL) has been successfully used to characterize heterogeneity (over dispersion or small-scale aggregation) of disease incidence for many plant pathosystems. With the BPL, the log of the observed variance is a linear function of the log of the theoretical variance for a binomial distribution over the range of incidence values and the estimated scale (a) and slope (b) parameters provide information on the characteristics of aggregation. When b = 1, the interpretation is that the degree of aggregation remains constant over the range of incidence values observed, otherwise aggregation is variable. In two articles published in this journal in 2009, Gosme and Lucas used their stochastic simulation model, Cascade, to show a multi-phasic (split-line) relationship of the variances, with straight-line (linear) relationships on a log-log scale within each phase. In particular, they showed a strong break-point in the lines at very low incidence, with a b considerably above 1 in the first line segment (corresponding to a range of incidence values usually not observed in the field), and b being equal or very close to 1 in the next segment (corresponding to the range of incidence values usually observed). We evaluated their findings by utilizing a general spatially-explicit stochastic simulator developed by Xu and Ridout in 1998, with a wide range of median dispersal distances for the contact distribution and number of plants in the sampling units (quadrants), and through an assessment of published BPL results. The simulation results showed that the split-line phenomenon can occur, with a break point at incidence values of around 0.01, but the split is most obvious for short median dispersal distances and large quadrant sizes. However, values of b in the second phase were almost always greater than 1, and only approached 1 with extremely short median dispersal distances and small quadrant sizes. An appraisal of published results showed no evidence of multiple phases (although the minimum incidence may generally be too high to observe the break), and estimates of b were almost always larger than 1. Thus, it appears that the results from the Cascade simulation model represent a special epidemiological case, corresponding primarily to a roughly nearest-neighbor population-dynamic process. Implications of a multi-phasic BPL property may be important and are discussed.