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The Explicit Dependence of Quadrat Variance on the Ratio of Clump Size to Quadrat Size

May 2005 , Volume 95 , Number  5
Pages  452 - 462

Francis J. Ferrandino

Associate Scientist, Department of Plant Pathology and Ecology, The Connecticut Agricultural Experiment Station, P.O. Box 1106, New Haven 06504

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Accepted for publication 14 December 2004.

In the past decade, it has become common practice to pool mapped binary epidemic data into quadrats. The resultant “quadrat counts” can then be analyzed by fitting them to a probability distribution (i.e., betabinomial). Often a binary form of Taylor's power law is used to relate the quadrat variance to the quadrat mean. The fact that there is an intrinsic dependence of such analyses on quadrat size and shape is well known. However, a clear-cut exposition of the direct connection between the spatial properties of the two-dimensional pattern of infected plants in terms of the geometry of the quadrat and the results of quadrat-based analyses is lacking. This problem was examined both empirically and analytically. The empirical approach is based on a set of stochastically generated “mock epidemics” using a Neyman-Scott cluster process. The resultant spatial point-patterns of infected plants have a fixed number of disease foci characterized by a known length scale (monodisperse) and saturated to a known disease level. When quadrat samples of these epidemics are fit to a beta-binomial distribution, the resulting measures of aggregation are totally independent of disease incidence and most strongly dependent on the ratio of the length scale of the quadrat to the length scale of spatial aggregation and to a lesser degree on disease saturation within individual foci. For the analytical approach, the mathematical form for the variation in the sum of random variates is coupled to the geometry of a quadrat through an assumed exponential autocorrelation function. The net result is an explicit equation expressing the intraquadrat correlation, quadrat variance, and the index of dispersion in terms of the ratio of the quadrat length scale to the correlative length scale.

© 2005 The American Phytopathological Society