Comment-Using spatial and temporal patterns of

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of Armillaria root disease to formulate management recommendations for Ontario's black spruce (Picea mariana) seed orchards1. G. Hughes and L.V. Madden.
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DISCUSSION

Comment—Using spatial and temporal patterns of Armillaria root disease to formulate management recommendations for Ontario’s black spruce (Picea mariana) seed orchards1 G. Hughes and L.V. Madden

A recent article in the Canadian Journal of Forest Research (Bruhn et al. 1996) discussed spatial and temporal patterns of mortality due to Armillaria root disease. We comment on this article, highlighting, in particular, aspects of the analysis that relate to other research on spatial patterns of plant disease incidence and the statistical analysis of aggregated disease incidence data. Incidence data are based on the classification of individuals into one of two classes. In the case of the study by Bruhn et al. (1996) trees were classified as either killed by Armillaria root disease or otherwise. Thus, we use the terms disease incidence, or incidence, for general statements, and the terms incidence of tree mortality, or mortality, when referring specifically to the Armillaria root disease data of Bruhn et al. (1996).

of tree mortality within planted clusters. To this end, observations of the number of trees killed by Armillaria in each fourtree cluster (the adopted sampling unit) in each orchard were collected. A resampling procedure was then used to provide expected frequencies under the null hypothesis of a random pattern of mortality. The subsequent χ2 goodness-of-fit tests showed that the observed frequencies deviated from the expected. In fact, a resampling procedure is not required to generate expected frequencies for grouped binary data under the hypothesis of randomness. If every tree in an orchard has an equal and constant probability (π) of mortality due to Armillaria, the number of trees killed (X) out of n in a sampling unit then has the binomial distribution n Prob(X = x) =   πx(1 − π)n−x x  in which x takes the values 0, 1, 2, ..., n. The observed proportion of trees killed by Armillaria (referred to, specifically, as the incidence of tree mortality and, generally, as disease incidence) provides an estimate of π, and the resulting probabilities can be multiplied by N, the number of sampling units, to give expected binomial frequencies. For the calculation of expected frequencies, it is a requirement that all sampling units have the same total number of trees. In that case, disease incidence is given by pˆ = Σi Xi /nN. Cochran’s (1936) article was probably the first quantitative analysis of the pattern of plant disease incidence. Cochran fitted the binomial distribution to the observed frequency distribution of tomato plants infected with tomato spotted wilt virus (TSWV), noting “... we expect the observed series to deviate from the expected one by having too many groups with many diseased plants, too many with few diseased plants and too few with diseased plants about the average...”. Table 1 shows the observed and expected binomial frequencies for the Armillaria mortality data of Bruhn et al. (1996). Among the observed frequencies, there are rather too many observations in the tails of the distributions and rather too few near the means, compared with the expected binomial frequencies. Such observations are typical of aggregated (patchy, clustered, heterogeneous) disease incidence data (Madden and Hughes 1995). The poor fit of the observed and the expected binomial frequencies is confirmed by χ2 tests of goodness of fit: Bawlb [1]

Methods In reanalyzing the data of Bruhn et al. (1996) we have, of necessity, accepted their methodology for the assessment of mortality attributed to Armillaria root disease. The β-binomial analyses described below were carried out using the computer software BBD (Madden and Hughes 1994), based on the algorithm published by Smith (1983) and available from the authors on request, and EGRET (Cytel Software Corporation, Cambridge, MA 02142, U.S.A.).

The binomial distribution In their article on the formulation of management recommendations for black spruce seed orchards, Bruhn et al. (1996) required a statistical test of spatial aggregation of the incidence Received March 12, 1997. Accepted November 2, 1997. G. Hughes.2 Institute of Ecology and Resource Management, University of Edinburgh, Agriculture Building, West Mains Road, Edinburgh EH9 3JG, Scotland, U.K. L.V. Madden. Department of Plant Pathology, Ohio State University, Wooster, OH 44691, U.S.A. 1 2

Paper by J.N. Bruhn, J.D. Mihail, and T.R. Meyer. Can. J. For Res. 26: 298–305. Author to whom all correspondence should be addressed. e-mail: [email protected]

Can. J. For. Res. 28: 154–158 (1998)

© 1998 NRC Canada

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Discussion Table 1. Observed and expected frequencies of Armillaria root disease mortality in four-tree clusters in four black spruce seed orchards in northwest Ontario.

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Orchard and no. dead (of 4) Bawlb Lake 0 1 2 3 4 Ferguson 0 1 2 3 4 Melgund 0 1 2 3 4 Vermeersch 0 1 2 3 4

Observed frequencies*

Binomial expected frequencies

β-binomial expected frequencies

3683 320 66 9 2

3614.4 444.7 20.5 0.4 0.0

3682.9 321.0 63.5 11.4 1.3

3506 376 96 18 4

3399.2 564.7 35.2 1.0 0.0

3505.9 377.4 92.5 21.1 3.1

2173 1141 439 200 47

1846.8 1574.5 503.4 71.5 3.8

2184.7 1089.3 498.8 184.6 42.6

3117 497 165 47 14

2913.3 833.1 89.3 4.3 0.1

3117.4 495.3 163.8 52.1 11.4

*See Table 2 of Bruhn et al. (1996).

Lake, χ2 = 186.3, 1 df, P < 0.001; Ferguson, χ2 = 251.6, 1 df, P < 0.001; Melgund, χ2 = 576.4, 2 df, P < 0.001; Vermeersch, χ2 = 336.8, 1 df, P < 0.001.

The b-binomial distribution The binomial distribution is a useful model for describing random disease incidence data and for detecting aggregation as a deviation from randomness. A distributional model that can be used to describe aggregated disease incidence data is the β-binomial, a discrete probability distribution derived by regarding the binomial parameter π as a variable that has the β density Γ(α + β) α−1 π (1 − π)β−1 Γ(α)Γ(β) in which 0 ≤ π ≤ 1, and α and β are constants >0. Writing p = α/(α + β) and θ = 1/(α + β), the β-binomial distribution is then x−1

n−x−1

Π(p + iθ) Π (1 − p + iθ)

[2]

n i=0 Prob(X = x) =   x 

i=0 n−1

Π(1 + iθ) i=0

in which x takes the values 0, 1, 2, ..., n. The parameter p is the expected disease incidence (the mean value of the (now vari-

able) binomial parameter π) and θ is an aggregation parameter that characterizes the variation in π. When θ > 0, the variance of the β-binomial is larger than that of the binomial distribution with the same mean; the binomial distribution is obtained when θ = 0. The resulting probabilities can be multiplied by N, the number of sampling units, to give the expected β-binomial frequencies. The first application of the β-binomial distribution in plant disease epidemiology seems to have been in the description of the frequency distribution of rice plants infected with yellow dwarf disease (caused by a mycoplasma-like organism) (Shiyomi and Takai 1979; Takai and Shiyomi 1980). Unfortunately, it is clear that neither of these articles, both published in Japanese, nor a subsequent one published in English (Shiyomi 1981), made much impact on the mainstream epidemiological literature. Subsequently, Qu et al. (1990) fitted the β-binomial to the TSWV data originally published by Cochran (1936). More recently, Hughes and Madden (1993) described the use of the β-binomial distribution to characterize the frequency distribution of diseased plants per sampling unit, using data from a number of studies of plant virus disease incidence as examples. These studies, and a number of others in which plant disease incidence was recorded (Madden and Hughes 1994; Madden et al. 1995a, 1995b; Zarnoch et al. 1995; Tanne et al. 1996), showed that in cases where the binomial distribution provides an inadequate description of the frequency distribution of diseased plants per sampling unit because the data are aggregated, the β-binomial usually provides a significant improvement. The expected β-binomial frequencies for the Armillaria mortality data of Bruhn et al. (1996) are shown in Table 1. The improved fit (by comparison with expected binomial frequencies) of the observed and the expected β-binomial frequencies for each orchard is confirmed by χ2 tests of goodness of fit: Bawlb Lake, χ2 = 0.31, 1 df, P = 0.58; Ferguson, χ2 = 0.33, 1 df, P = 0.57; Melgund, χ2 = 11.44, 2 df, P = 0.003; Vermeersch, χ2 = 1.14, 2 df, P = 0.56. An alternative, and more direct, method of making the comparison between the fits of the binomial and β-binomial distributions to observed data is shown in Table 2. The binomial distribution is first fitted to the data. Then, on subsequently fitting the β-binomial, the resulting reduction in residual deviance (referred to as the likelihood ratio statistic, LRS) provides an indication of the extent to which an improvement in fit has been achieved (e.g., Qu et al. 1990). As expected, the β-binomial offers a significant improvement for each of the four Armillaria root disease mortality data sets of Bruhn et al. (1996).

A power law for grouped binary data Hughes and Madden (1992) discussed a relationship between observed variance of disease incidence, vobs, and the corresponding (theoretical) binomial variance, vbin = π(1 − π)/n (in which, as above, observed disease incidence provides an estimate of π). The relationship [3]

vobs = A(vbin)b

in which A and b are parameters to be estimated, often provides a good empirical description of observed data (e.g., Madden and Hughes 1995; Madden et al. 1995a, 1995b). Logarithmic transformation of eq. 3 leads to a straight line with slope b and intercept log(A). Figure 1 shows the Armillaria root disease mortality data of Bruhn et al. (1996) plotted in this way. Since © 1998 NRC Canada

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Can. J. For. Res. Vol. 28, 1998 Table 2. Maximum likelihood parameter estimates for the β-binomial distribution fitted to observed data (see Table 1) for Armillaria root disease mortality in four-tree clusters in four black spruce seed orchards in northwest Ontario. Orchard

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Bawlb Lake Ferguson Melgund Vermeersch

^ (p)* Logit

SE

θˆ

SE

LRS (1 df)

P†

–3.48 –3.18 –1.55 –2.64

0.054 0.049 0.025 0.041

0.133 0.174 0.195 0.249

0.0196 0.0209 0.0134 0.0222

162.0 255.9 487.7 491.1