Spatial and temporal stability of the estimated parameters of the binary power law.

The incidence of hop powdery mildew on leaves, caused by Podosphaera macularis, collected from 1,606 transects in 77 commercial hop yards in Oregon and Washington over 9 years was used to assess variability in heterogeneity of disease and the estimated binary power law parameters. Spatial analyses of data sets were conducted at the level of individual rows (row level) and multiple rows within a yard (yard level). The binary power law provided a good fit to all data sets, with R(2) values ranging from 0.933 to 0.993. At the row level, the intercept parameter ln(A(x)) was >0 for 8 years, but was not significantly greater than 0 in 2006. The parameter b was greater than 1 for all row-level data sets collected from 1999 to 2005, but was <1 in 2006 and not significantly different from 1 in 2007. Covariance analysis indicated the factor 'region' affected ln(A(x)) in 3 years, and b in 2 years. 'Cultivar' had an effect on ln(A(x)) in 3 years and b in year. At the yard level, ln(A(x)) was greater than 0 for 6 years, but in 2006 and 2007, ln(A(x)) was not significantly different from 0. The slope parameter b was greater than 1 in 6 years, but was not significantly different from 1 in 2006 and 2007. Differences in b among years were large enough to have practical implications for sample sizes and precision of fixed and sequential sampling. Although the binary power law parameter tended to be relatively stable, variability of the estimated parameters may have practical consequences for sampling precision and costs.