The limits of the binary power law describing spatial variability for incidence data



Taylor's power law (TPL), an empirical law relating the observed variance to mean density (or abundance), has found wide applicability for characterizing heterogeneity in many disciplines. However, when the density variable has an upper bound, the TPL does not hold and the binary power law (BPL) needs to be used instead. The BPL has been shown to describe the heterogeneity of numerous plant disease epidemic systems. In this study, a generic stochastic simulator was used to study the extent to which the BPL can satisfactorily describe incidence data. Results showed that the symmetrical BPL does hold whenever there is a positive correlation among neighbours on the probability of a plant becoming infected, or where disease development is not influenced by the neighbours. These results held for a wide range of neighbourhood sizes, strengths of neighbourhood influence, and size of the sampling quadrats. However, the symmetrical BPL did not hold when there is a negative influence among neighbours. The more general asymmetrical BPL (ABPL) fitted the data with positive or negative neighbourhood influence, but because a negative neighbourhood effect is generally unlikely for plant epidemics, the symmetrical BPL is preferred over the ABPL because of its parsimony. The magnitude of the estimated BPL parameters increased with increasing neighbourhood influence and sampling-quadrat size. However, except when the power parameter equals 1, inferring specific underlying mechanisms generating the data or comparing BPL estimates from different studies is difficult, because of the large effect of sampling on the BPL parameter estimates.