An effective sample size for predicting plant disease incidence in a spatial hierarchy

ABSTRACT For aggregated or heterogeneous disease incidence, one can predict the proportion of sampling units diseased at a higher scale (e.g., plants) based on the proportion of diseased individuals and heterogeneity of diseased individuals at a lower scale (e.g., leaves) using a function derived from the beta-binomial distribution. Here, a simple approximation for the beta-binomial-based function is derived. This approximation has a functional form based on the binomial distribution, but with the number of individuals per sampling unit (n) replaced by a parameter (v) that has similar interpretation as, but is not the same as, the effective sample size (n(deff) ) often used in survey sampling. The value of v is inversely related to the degree of heterogeneity of disease and generally is intermediate between n(deff) and n in magnitude. The choice of v was determined iteratively by finding a parameter value that allowed the zero term (probability that a sampling unit is disease free) of the binomial distribution to equal the zero term of the beta-binomial. The approximation function was successfully tested on observations of Eutypa dieback of grapes collected over several years and with simulated data. Unlike the beta-binomial-based function, the approximation can be rearranged to predict incidence at the lower scale from observed incidence data at the higher scale, making group sampling for heterogeneous data a more practical proposition.     

Spatial pattern analysis of strawberry leaf blight in perennial production systems

ABSTRACT Spatial pattern of the incidence of strawberry leaf blight, caused by Phomopsis obscurans, was quantified in commercial strawberry fields in Ohio using statistics for heterogeneity and spatial correlation. For each strawberry planting, two transects were randomly chosen and the proportion of leaflets (out of 15) and leaves (out of five) with leaf blight symptoms was determined from N = 49 to 106 (typically 75) evenly spaced sampling units, thus establishing a natural spatial hierarchy to compare patterns of disease. The beta-binomial distribution fitted the data better than the binomial in 92 and 26% of the 121 data sets over 2 years at the leaflet and leaf levels, respectively, based on a likelihood ratio test. Heterogeneity in individual data sets was measured with the index of dispersion (variance ratio), C(alpha) test, a standard normal-based test statistic, and estimated theta parameter of the beta-binomial. Using these indices, overdispersion was detected in approximately 94 and 36% of the data sets at the leaflet and leaf levels, respectively. Estimates of the slope from the binary power law were significantly (P < 0.01) greater than 1 and estimates of the intercept were significantly greater than 0 (P < 0.01) at both the leaflet and leaf levels for both years, indicating that degree of heterogeneity was a function of incidence. A covariance analysis indicated that cultivar, time, and commercial farm location of sampling had little influence on the degree of heterogeneity. The measures of heterogeneity indicated that there was a positive correlation of disease status of leaflets (or leaves) within sampling units. Measures of spatial association in disease incidence among sampling units were determined based on autocorrelation coefficients, runs analysis, and a new class of tests known as spatial analysis by distance indices (SADIE). In general, from 9 to 22% of the data sets had a significant nonrandom spatial arrangement of disease incidence among sampling units, depending on which test was used. When significant associations existed, the magnitude of the association was small but was about the same for leaflets and leaves. Comparing test results, SADIE analysis was found to be a viable alternative to spatial autocorrelation analysis and has the advantage of being an extension of heterogeneity analysis rather than a separate approach. Collectively, results showed that incidence of Phomopsis leaf blight was primarily characterized by small, loosely aggregated clusters of diseased leaflets, typically confined within the borders of the sampling units.     

Spatial pattern analysis of hop powdery mildew in the pacific northwest: implications for sampling

ABSTRACT The spatial pattern of hop powdery mildew was characterized using 3 years of disease incidence data collected in commercial hop yards in the Pacific Northwest. Yards were selected randomly from yards with a history of powdery mildew, and two to five rows were selected for sampling within each yard. The proportion of symptomatic leaves out of 10 was determined from each of N sampling units in a row. The binomial and the beta-binomial frequency distributions were fit to the N sampling units observed in each row and to SigmaN sampling units observed in each yard. Distributional analyses indicated that disease incidence was better characterized by the beta-binomial than the binomial distribution in 25 and 47% of the data sets at the row and yard scales, respectively, according to a log-likelihood ratio test. Median values of the beta-binomial parameter theta, a measure of small-scale aggregation, were near 0 at both sampling scales, indicating that disease incidence was close to being randomly distributed. The variability in disease incidence among rows sampled in the same yard generally increased with mean incidence at the yard scale. Spatial autocorrelation analysis, used to measure large-scale patterns of aggregation, indicated that disease incidence was not correlated between sampling units over several lag distances. Results of a covariance analysis showed that heterogeneity of disease incidence was not dependent upon cultivar, region, or time of year when sampling was conducted. A hierarchical analysis showed that disease incidence at the sampling unit scale (proportion of sampling units with one or more diseased leaves) increased as a saturation-type curve with respect to incidence at the leaf level and could be described by a binomial function modified to account for the effects of heterogeneity through an effective sample size. Use of these models permits sampling at the sampling unit scale while allowing inferences to be made at the leaf scale. Taken together, hop powdery mildew was nearly randomly distributed with no discernable foci, suggesting epidemics are initiated from a well-distributed or readily dispersible overwintering population. Implications for sampling are discussed.     

A generalized linear modeling approach for characterizing disease incidence in a spatial hierarchy

ABSTRACT Several statistical models are introduced to quantify the effect of heterogeneity on disease incidence relationships in a three-scale spatial hierarchy: the sampling unit level (highest), the leaf scale (intermediate), and the leaflet scale (lowest). The models are an extension of the theory previously developed for a two-scale hierarchy and were tested using data collected from strawberry leaf blight epidemics. Disease incidence at the sampling-unit scale (proportion of sampling units with one or more diseased leaflets) increased as a saturation-type curve with increasing leaflet or leaf disease incidence (proportion of leaflets or leaves diseased) as predicted by the good fit of the beta-binomial distribution to the leaflet and leaf data. The relationship could be accurately described, without curve fitting, by several simple nonlinear models, in which the aggregation of disease was represented by a modified binomial function incorporating an effective sample size that was either constant or dependent on mean incidence. The relationship between incidence at the leaflet and leaf scales could be modeled based on the combined sampling-unit models for leaflets and leaves. By taking the complementary log-log (CLL) transformation of incidence, the equations could be expressed as generalized linear models, and curve fitting used to estimate the parameters. Generally, curve fitting gave slight to no improvement in the accuracy of the predictions of incidence. These models have broad applicability in sampling for disease incidence, and results can be used to interpret how diseased individuals at the lowest level in a hierarchy are arranged within sampling units.     

Relationships between several quadrat-based statistical measures used to characterize spatial aspects of disease incidence data

ABSTRACT This article investigates the relationships between various statistical measures that are used to summarize spatial aspects of disease incidence data. The focus is on quadrat data in which each plant in a quadrat is classified as diseased or healthy. We show that spatial autocorrelation plays a central role via the mean intraclass correlation, rho, which is defined as the average correlation of the disease status of all pairs of plants within the quadrat. The value of rho determines the variance of the number of infected plants in the quadrat and, if this variable follows a beta-binomial distribution, the heterogeneity parameter of the beta-binomial distribution is directly related to the mean intraclass correlation. We consider in detail a model in which the spatial autocorrelation depends only on the distance between the plants. For illustration, we consider a specific autocorrelation model that was derived from simulated data. We show that this model leads, approximately, to the binary form of the power law relating the variance of the number of infected plants per quadrat to the mean. Using an approximation technique, we then show how the index of dispersion is related to quadrat size and shape. The index of dispersion increases with quadrat size. The rate of increase is dependent on quadrat shape, but the effect of quadrat shape is small in comparison to the effect of quadrat size. Finally, we note that if the spatial autocorrelation depends on the relative orientation of the plants, as well as the distance between them, there are connections with distance class methods.     

Sequential sampling for incidence of phomopsis leaf blight of strawberry

ABSTRACT Sequential sampling models for estimation and classification were developed for the incidence of strawberry leaflets infected by Phomopsis obscurans. Sampling protocols were based on a binary power law analysis of the spatial heterogeneity of Phomopsis leaf blight in commercial fields in Ohio. For sequential estimation, samples were collected until mean disease incidence could be estimated with a preselected coefficient of variation of the mean (C). For sequential classification, samples were collected until there was sufficient evidence to classify mean incidence as being below or above a threshold (p(t)) based on the sequential probability ratio test. Monte-Carlo simulations were used to determine the theoretical average sample number (ASN) and probability of classifying mean incidence as less than p(t) (operating characteristic) for any true value of incidence. Estimation and classification sampling models were both tested with bootstrap simulations of randomly selected data sets and validated by data sets from another year that were not utilized in developing the models. In general, achieved (or calculated) C after sequentially sampling for estimation was close to the preselected C of 0.2, and mean incidence was estimated with little bias. Achieving a C of 0.1 with less than 75 sampling units (the nominal value for many original data sets) was more problematical, especially with true incidence less than 0.2. ASN for classification was only 9 to 18 at disease incidence values near p(t), and approximately five or less at incidence values far from p(t). Correct classification decisions were made in over 88% of the validation data sets. Results indicated that it is possible to estimate Phomopsis leaf blight with high precision and with high correct classification probabilities.     

Analysis of the association between the incidence of two spatially aggregated foliar diseases of strawberry

ABSTRACT Association of the incidence of leaf blight (caused by Phomopsis obscurans) and leaf spot of strawberry (caused by Mycosphaerella fragariae) was assessed at multiple scales in perennial plantings at several commercial farms over 3 years (1996 to 1998). For each field, the presence or absence of each disease was recorded from n = 15 leaflets in each of N approximately 70 evenly spaced sampling units, and the proportion of leaflets with blight, spot, and total disease (blight or spot) was determined. Individual diseases and total disease incidence were all well described by the beta-binomial distribution but not by the binomial distribution, indicating overdispersion of disease. The Jaccard similarity index was used to measure disease co-occurrence at the leaflet, sampling-unit, and field scales. Standard errors of this index for the lower two scales were obtained using the jackknife (resampling) procedure, and data randomizations were used to determine the expected Jaccard index for an independent arrangement of the two diseases, conditioned on the incidence and spatial heterogeneity of the observed disease data. Results based on these statistics showed that only 4 of 52 data sets at the leaflet level and no data sets at the sampling-unit level had Jaccard index values significantly different from that expected under an independent rearrangement of the two diseases. Rank correlation and cross-correlation statistics were calculated to determine the degree of covariation in incidence between the two diseases. Additionally, covariation between diseases was tested using a new procedure in the Spatial Analysis by Distance IndicEs (SADIE) class of tests. Covariation was detected in 21% of the data sets using rank correlation methods and in 15% of the data sets using the SADIE-based approach. The discrepancy between these two methods may be due to the rank correlation procedure not taking into account the effects of spatial pattern of disease incidence. There was no relationship between mean disease incidence per field of spot and blight or between degree of heterogeneity of the two diseases (as measured by theta of the beta-binomial distribution), demonstrating lack of covariation at the field scale. Incidence of leaflets with either disease (total disease incidence) could be well predicted using a linear combination of the estimated probabilities of leaf blight and leaf spot incidence based on independence of the two diseases. Heterogeneity of total disease incidence, measured with the estimated theta parameter of the beta-binomial distribution, could also be well predicted using a linear combination of the weighted theta values for leaf blight and leaf spot, with weights proportional to incidence of the individual diseases.     

Spatial characteristics of white mould epidemics and the development of sequential sampling plans in Australian bean fields

To improve sampling efficiency and precision in the assessment of white mould (caused by Sclerotinia sclerotiorum) disease incidence on bean (Phaseolus vulgaris), the spatial characteristics of epidemics were characterized in 54 linear transects in 18 bean fields during 2008–2010 in northern Tasmania, Australia. The incidence of diseased pods and plants was assessed prior to harvest. Distributional and correlation‐based analyses indicated the incidence of diseased pods was characterized by a largely random pattern at the individual plant scale, with some patches of similar disease levels on pods occurring at a scale of 1·5 m or greater. Collectively, these results suggested epidemics may be dominated by localized sources of inoculum. Sequential sampling approaches were developed to estimate or classify disease incidence above or below provisional thresholds of 3, 5 and 15% incidence on pods near harvest. Achieving prespecified levels of precision by sequential estimation was possible only when disease incidence on pods was greater than approximately 4% and sampling was relatively intense (i.e. 10 pods evaluated on each of at least 64 plants). Using sequential classification, correct decisions on disease status were made in at least 95% of independent validation datasets after assessment of only 10·1–15 plants, depending on classification threshold and error rates. Outcomes of this research provide the basis for implementing more efficient sampling and management strategies for this disease in Australian fields.     

Spatial heterogeneity,incidence-incidence and incidence-lesion density relationship of apple scab (<Emphasis Type="Italic">Venturia inaequalis</Emphasis>) in managed orchards

The spatial pattern of apple scab was characterized using 10 years of disease incidence and lesion density data collected in managed orchards located in Quebec, Canada. Distributional analyses indicated that scab incidence was better characterized by the beta-binomial than the binomial distribution in 53 and 65% of the data sets at the leaf and shoot scales, respectively. Median values of the beta-binomial parameter θ, a measure of small-scale aggregation, were near 0 (0.003 and 0.028) at both sampling scales, indicating that disease incidence was close to being randomly distributed (low degree of aggregation). For lesion density, the negative binomial distribution fitted the data better than the Poisson distribution in 86% of the data sets at the leaf scale. The median value of the index of dispersion k was 0.068, indicating that aggregation was present. For all apple scab measurements, the power law models provided a good fit to the data. The estimated slope and intercept parameters were significantly greater than 1 and 0, respectively, suggesting that spatial heterogeneity changed systematically with disease incidence. Results of a covariance analysis showed that spatial heterogeneity of scab incidence at both scales and lesion density was not dependent upon shoot type but that spatial heterogeneity of scab incidence and lesion density at the leaf scale was influenced by the sampling period. A hierarchical analysis showed that scab incidence at the tree scale increased as a saturation-type curve with respect to incidence at the leaf or shoot scales. A similar relationship was observed for incidences at the shoot and leaf scales. An effective sample size model based on the binary power law parameters (Madden and Hughes, Phytopathology 89:770–781, 1999) gave the best fit to the leaf and shoot data, respectively. The incidence-lesion density relationship at both scales was well described by a complementary log-log (CLL) and log transformation model ( R_adj² = 0.97 and R_adj² = 0.94 ) \left( {R_{{adj}}^2 = 0.97\,and\,R_{{adj}}^2 = 0.94} \right) , however, the models tended to underestimate lesion density. The information of the spatial relations of apple scab within and between hierarchical scales acquired from this study can be used in developing and evaluating practical disease management strategies and to improve apple scab assessments for fungicide or cultivar susceptibility trials.     

Effects of initial epidemic conditions, sporulation rate, and spore dispersal gradient on the spatio-temporal dynamics of plant disease epidemics

ABSTRACT A stochastic model that simulates the spread of disease over space and time was developed to study the effects of initial epidemic conditions (number of initial inocula and their spatial pattern), sporulation rate, and spore dispersal gradient on the spatio-temporal dynamics of plant disease epidemics. The spatial spread of disease was simulated using a half-Cauchy distribution with median dispersal distance mu (units of distance). The rate of temporal increase in disease incidence (beta(I), per day) was influenced jointly by mu and by the sporulation rate lambda (spores per lesion per day). The relationship between beta(I) and mu was nonlinear: the increase in beta(I) with increasing mu was greatest when mu was small (i.e., when the dispersal gradient was steep). The rate of temporal increase in disease severity of diseased plants (beta(S)) was affected mainly by lambda: beta(S) increased directly with increasing lambda. Intraclass correlation (kappa(t)), the correlation of disease status of plants within quadrats, increased initially with disease incidence, reached a peak, and then declined as disease incidence approached 1.0. This relationship was well described by a power-law model that is consistent with the binary form of the variance power law. The amplitude of the model relating kappa(t) to disease incidence was affected mainly by mu: kappa(t) decreased with increasing mu. The shape of the curve was affected mainly by initial conditions, especially the spatial pattern of the initial inocula. Generally, the relationship of spatial autocorrelation (rho(t,k)), the correlation of disease status of plants at various distances apart, to disease incidence and distance was well described by a four-parameter power-law model. rho(t,k) increased with disease incidence to a maximum and then declined at higher values of disease incidence, in agreement with a power-law relationship. The amplitude of rho(t,k) was determined mainly by initial conditions and by mu: rho(t,k) decreased with increasing mu and was lower for regular patterns of initial inocula. The shape of the rho(t,k) curve was affected mainly by initial conditions, especially the spatial pattern of the initial inocula. At any level of disease incidence, autocorrelation declined exponentially with spatial lag; the degree of this decline was determined mainly by mu: it was steeper with decreasing mu.     

Size, shape and intensity of aggregation of take-all disease during natural epidemics in second wheat crops

Point pattern analysis (fitting of the beta-binomial distribution and binary form of power law) was used to describe the spatial pattern of natural take-all epidemics (caused by Gaeumannomyces graminis var. tritici ) on a second consecutive crop of winter wheat in plots under different cropping practices that could have an impact on the quantity and spatial distribution of primary inoculum, and on the spread of the disease. The spatial pattern of take-all was aggregated in 48% of the datasets when disease incidence was assessed at the plant level and in 83% when it was assessed at the root level. Clusters of diseased roots were in general less than 1 m in diameter for crown roots and 1–1·5 m for seminal roots; when present, clusters of diseased plants were 2–2·5 m in diameter. Anisotropy of the spatial pattern was detected and could be linked to soil cultivation. Clusters did not increase in size over the cropping season, but increased spatial heterogeneity of the disease level was observed, corresponding to local disease amplification within clusters. The relative influences of autonomous spread and inoculum dispersal on the size and shape of clusters are discussed.     

Measuring spatial aggregation in binary epidemics: correlative analysis and the advantage of fractal-based sampling

ABSTRACT The incomplete sampling of a binary epidemic is nothing more than the overlap of two spatial patterns: the pattern of diseased plants and the pattern of sampled points. Thus, the information on the spatial arrangement of diseased plants obtained from such a sampling explicitly depends on the geometric locations of the sampled points. A number of procedures for sampling disease incidence are examined. These include samples placed on a regular grid, spatially clustered samples, randomly selected samples, and samples specified by a nested fractal design. The performance of these various sampling schemes was examined using simulated binary epidemics with varying degrees of spatial aggregation over different length scales, generated using a Neyman-Scott cluster process. A modification of spatial correlation analysis specifically geared to binary epidemics is derived and shown to be equivalent to a X(2) test comparing the number of infected plant pairs to that expected from a spatially random epidemic. This analysis was applied to the data obtained using the various sampling schemes and the results are compared and contrasted. For the same number of sampling points, the fractal design is most efficient in the detection of contagion and provides spatial information over a larger range of distance scales than other sampling schemes. However, the regular grid sampling scheme consistently yielded an estimate of average disease incidence that had the smallest variance. Sampling patterns consisting of randomly selected points were intermediate in behavior between the two extremes.     

Relationships between disease incidence at two levels in a spatial hierarchy

ABSTRACT Relationships between disease incidence measured at two levels in a spatial hierarchy are derived. These relationships are based on the properties of the binomial distribution, the beta-binomial distribution, and an empirical power-law relationship that relates observed variance to theoretical binomial variance of disease incidence. Data sets for demonstrating and testing these relationships are based on observations of the incidence of grape downy mildew, citrus tristeza, and citrus scab. Disease incidence at the higher of the two scales is shown to be an asymptotic function of incidence at the lower scale, the degree of aggregation at that scale, and the size of the sampling unit. For a random pattern, the relationship between incidence measured at two spatial scales does not depend on any unknown parameters. In that case, an equation for estimating an approximate variance of disease incidence at the lower of the two scales from incidence measurements made at the higher scale is derived for use in the context of sampling. It is further shown that the effect of aggregation of incidence at the lower of the two scales is to reduce the rate of increase of disease incidence at the higher scale.     

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.     

The explicit dependence of quadrat variance on the ratio of clump size to quadrat size

ABSTRACT 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.     

A Rainfall-Based Model for Predicting the Regional Incidence of Wheat Seed Infection by Stagonospora nodorum in New York

ABSTRACT Our goal was to develop a simple model for predicting the incidence of wheat seed infection by Stagonospora nodorum across western and central New York in any given year. The distribution of the incidence of seed infection by S. nodorum across the region was well described by the beta-binomial probability distribution (parameters p and theta). Mean monthly rainfalls in May and in June across western and central New York were used to predict p. The binary power law was used to predict theta. The model was validated with independent data collected from New York. The predicted distribution of seed infection incidence was not statistically different from the actual distribution of the incidence of seed infection.     

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.     

Sampling plans for aphids and their parasitoids in blueberry fields in Argentina

Densities of aphids (Aphis gossypii and A. spiraecola) and mummified aphids at different phenological stages of a blueberry crop were estimated for the purpose of developing sampling plans. Our data set comprised 99 samples taken during the period 2006–2008 in four fields in Buenos Aires Province, Argentina. Estimation of population density based on the proportion of sample units infested by individuals was investigated. We also calculated the minimum number of sample units to estimate the density of individuals on buds and buds + flowers using enumerative sampling. The relative precision of both methods was compared. Moreover, an enumerative sequential sampling protocol was developed. The presence–absence sampling plan gave density estimates with large variances (as measured by confidence intervals and large standard errors). The aggregation of mummies was similar on buds and buds + flowers, so the required number of sample units for density estimates was the same. Relative precision of estimates was much lower for the presence–absence sampling than the enumerative sampling, even at intermediate densities. An enumerative sequential plan would be the most appropriate and useful method in management plans for aphids and mummified aphids in blueberries.     

Spatial analysis and incidence–density relationships for downy mildew on hop

The spatial pattern of downy mildew (Pseudoperonospora humuli) on hop (Humulus lupulus) was characterized over 4 years to aid in deriving an appropriate incidence–density relationship. From 472 disease assessments (datasets), discrete distributions were fitted to the datasets to determine aggregation of disease density. Where distributions were able to be fitted, the Poisson distribution fitted 4% of the datasets and the negative binomial distribution fitted 87% of the datasets. Larger‐scale patterns of disease were assessed by autocorrelation and runs analysis; both indicated aggregation of diseased plants was less common than aggregation of disease within plants. Taylor’s power law indicated disease density was aggregated and related to mean disease density in all years. Disease incidence and density were linked by saturation‐type relationships based on the zero term of the negative binomial distribution or an empirical regression. Certain individual datasets were not described well by any incidence–density model, particularly when disease density was greater than about 0·8 diseased shoots per plant with the cultivar Cascade. When applied to 56 validation datasets, 88% of the variation in observed disease incidence was explained by the incidence–density models. Under conditions where sampling would be implemented for disease management, the requisite conditions appear to be in place for a binomial sampling plan for downy mildew.