**Submitted to:** Phytopathology

**Publication Type:** Peer Reviewed Journal

**Publication Acceptance Date:** 6/9/2008

**Publication Date:** 9/30/2008

**Citation:** Gent, D.H., Turechek, W., Mahaffee, W.F. 2008. Spatial and Temporal Stability of the Estimated Parameters of the Binary Power Law. Phytopathology. 98:1107-1117.

**Interpretive Summary:** Hop powdery mildew is an important disease of hops in the Pacific Northwest. Sampling methods for estimating and classifying the incidence of the disease assume that aggregation of the disease is relatively stable among years, cultivars, and geographic regions. In this research, we analyzed long-term data sets and quantified how patterns of the disease can change. Differences in disease aggregation among years were large enough to have practical implications on sampling precision and costs. Collection of data sets over multiple geographic locations, years, and a range of disease incidence may be needed to observe the spectrum of variability in plants diseases observed in a managed agricultural system.

**Technical Abstract:** The binary power law has become a standard approach for describing and quantifying spatial patterns of disease incidence and summarizing the spatial dynamics of disease over the course of an epidemic. However, the portability and temporal stability of parameter estimates of the binary form of the power law are unclear. In this study, incidence of hop powdery mildew on leaves, caused by Podosphaera macularis, collected from 1606 transects in 77 commercial hop yards over nine years was used to assess variability in heterogeneity of disease and the binary power law parameters. Spatial analysis of data sets were conducted at the level of individual rows (row level data set) and multiple rows within a yard (yard level data sets). The beta-binomial distribution parameter ranged from 0 to 0.758 with mean 0.046 (se = 0.002) and median 0.01 at the row level, and 0 to 0.862 with mean 0.046 (se=0.002) and median 0.028 at the yard level. A log-likelihood ratio test indicated that the beta-binomial distribution provided a better fit to the data in 497 of 1606 data sets (30.9%) than the binomial distribution. Similarly, the index of dispersion, D, ranged from 0.4 to 9.17, with mean 1.34 (se =0.015) and median 1.11, and was significantly greater than 1 in 607 data sets (38.9%). At the yard level, a log-likelihood ratio test indicated the beta-binomial distribution fit 425 of 770 data sets (55.2%) better than the binomial distribution. D ranged from 0.5 to 9.08 with mean 1.473 (se = 0.026) and median 0.028. D was significantly greater than 1 in 367 data sets (47.7%). The region where the greatest value of was observed varied among years, being highest in yards in Oregon for 5 of the 8 years. Temporal trends in mean were not apparent. The binary power law provided a good fit to all data sets collected at the row and yard-level, with R2 values ranging from 0.933 to 0.993. At the row level, the intercept parameter ln (Ax) was >0 for years 1999 to 2005 (P < 0.0001) and 2007 (P = 0.0049), but was not significantly greater than 0 in 2006 (P = 0.1245). The slope parameter b was significantly greater than 1 for all row-level data sets collected from 1999 to 2005 (P = 0.05), indicating that heterogeneity changed systematically with disease incidence. In 2006, b was <1 (P = 0.0051) indicating a more regular pattern (under-dispersion) of disease not systematically related to disease incidence. In 2007, b was not significantly different from 1 (P = 0.5399), indicating heterogeneity was not systematically related to disease incidence in this year. Covariance analysis indicated the factor ‘region’ affected ln (Ax) (i.e., height of the regression line) in 2002, 2003, and 2005, and b in 2002 and 2003. ‘Cultivar’ had a significant effect on ln (Ax) in 2002, 2003, and 2007, and b in 2002. At the yard-level, ln (Ax) was significantly greater than 0 for 2000 to 2005 (P < 0.0001). In 2006 and 2007, ln (Ax) was not significantly different from 0 (P = 0.1116 and 0.0553, respectively), indicating a random pattern of disease incidence. The slope parameter b was significantly greater than 1 for 2000 to 2005, but was not significantly different from 1 in 2006 (P = 0.5979) and 2007 (P = 0.8686). Differences in b among years were large enough to have practical implications for sample sizes and precision of fixed and sequential sampling, with nearly twice as many sampling units being required to achieve the specified precision depending on which year parameter estimates from 2004 or 2006 were used to derive the fixed sampling curves. Although the binary power law parameter tended to be relatively stable, spatial and temporal variability of parameters may have practical consequences for sampling precision and costs. Collection of data sets collected over multiple geographic locations, years, and a range of disease incidence may be needed to obser