BBD—Computer Software for Fitting the Beta-Binomial Distribution to Disease Incidence Data

The relationship between strawberry powdery mildew and airborne conidium concentration (ACC) of Podosphaera aphanis was studied using data collected from 2006 to 2009 in 15 fields, and spatial pattern was described using 2 years of airborne inoculum and disease incidence data collected in fields planted with the June-bearing strawberry (Fragaria × ananassa) cultivar Jewel. Disease incidence, expressed as the proportion of diseased leaflets, and ACC were monitored in fields divided into 3 × 8 grids containing 24 100 m2 quadrats. Variance-to-mean ratio, index of dispersion, negative binomial distribution, Poisson distribution, and binomial and beta-binomial distributions were used to characterize the level of spatial heterogeneity. The relationship between percent leaf area diseased and daily ACC was linear, while the relationship between ACC and disease incidence followed an exponential growth curve. The V/M ratios were significantly greater than 1 for 100 and 96% of the sampling dates for ACC sampled at 0.35 m from the ground (ACC0.35m) and for ACC sampled at 1.0 m from the ground (ACC1.0m), respectively. For disease incidence, the index of dispersion D was significantly greater than 1 for 79% of the sampling dates. The negative binomial distribution fitted 86% of the data sets for both ACC1.0m and ACC0.35m. For disease incidence data, the beta-binomial distribution provided a good fit of 75% of the data sets. Taylor's power law indicated that, for ACC at both sampling heights, heterogeneity increased with increasing mean ACC, whereas the binary form of the power law suggested that heterogeneity was not dependent on the mean for disease incidence. When the spatial location of each sampling location was taken into account, Spatial Analysis by Distance Indices showed low aggregation indices for both ACCs and disease incidence, and weak association between ACC and disease incidence. Based on these analyses, it was found that the distribution of strawberry powdery mildew was weakly aggregated. Although a higher level of heterogeneity was observed for airborne inoculum, the heterogeneity was low with no distinct foci, suggesting that epidemics are induced by well-distributed inoculum. This low level of heterogeneity allows mean airborne inoculum concentration to be estimated using only one sampler per field with an overall accuracy of at least 0.841. The results obtained in this study could be used to develop a sampling scheme that will improve strawberry powdery mildew risk estimation.

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Relationships Between Several Quadrat-Based Statistical Measures Used to Characterize Spatial Aspects of Disease Incidence Data

Phytopathology ◽

10.1094/phyto.2000.90.6.568 ◽

2000 ◽

Vol 90 (6) ◽

pp. 568-575 ◽

Cited By ~ 10

Author(s):

M. S. Ridout ◽

X.-M. Xu

Keyword(s):

Spatial Autocorrelation ◽

Binomial Distribution ◽

Disease Incidence ◽

Intraclass Correlation ◽

Quadrat Size ◽

Incidence Data ◽

Statistical Measures ◽

Index Of Dispersion ◽

Rate Of Increase ◽

The Mean

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, ρ, which is defined as the average correlation of the disease status of all pairs of plants within the quadrat. The value of ρ 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.

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An Effective Sample Size for Predicting Plant Disease Incidence in a Spatial Hierarchy

Phytopathology ◽

10.1094/phyto.1999.89.9.770 ◽

1999 ◽

Vol 89 (9) ◽

pp. 770-781 ◽

Cited By ~ 42

Author(s):

L. V. Madden ◽

G. Hughes

Keyword(s):

Sample Size ◽

Binomial Distribution ◽

Disease Incidence ◽

Simulated Data ◽

Heterogeneous Data ◽

Sampling Unit ◽

Effective Sample Size ◽

Incidence Data ◽

Eutypa Dieback ◽

Number Of Individuals

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 (ndeff ) often used in survey sampling. The value of v is inversely related to the degree of heterogeneity of disease and generally is intermediate between ndeff 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.

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Quantitative and Molecular Epidemiology of Bacterial Blight of Onion in Seed Production Fields

Phytopathology ◽

10.1094/phyto-96-1345 ◽

2006 ◽

Vol 96 (12) ◽

pp. 1345-1354 ◽

Cited By ~ 15

Author(s):

L. Humeau ◽

P. Roumagnac ◽

Y. Picard ◽

I. Robène-Soustrade ◽

F. Chiroleu ◽

...

Keyword(s):

Seed Production ◽

Binomial Distribution ◽

Bacterial Blight ◽

Disease Incidence ◽

Aflp Analysis ◽

Data Sets ◽

Spatial Dependency ◽

Nested Polymerase Chain Reaction ◽

Incidence Data ◽

Polymerase Chain

Onion, a biennial plant species, is threatened by the emerging, seed-borne, and seed-transmitted Xanthomonas axonopodis pv. allii. Bacterial blight epidemics were monitored in seed production fields over two seasons. Temporal disease progress was different between the two seasons, with final incidence ranging from 0.04 to 0.06 in 2003 and from 0.44 to 0.61 in 2004. The number of hours with temperatures above 24°C was the best descriptor for predicting the number of days after inoculation for bacterial blight development on inoculated plants. Fitting the β-binomial distribution and binary power law analysis indicated aggregated patterns of disease incidence data. The β-binomial distribution was superior to the binomial distribution for 97% of the examined data sets. Spatial dependency ranged from 5.9 to 15.2 m, as determined by semivariance analysis. Based on amplified fragment length polymorphism (AFLP) analysis, it was concluded that plots predominantly were infected by the inoculated haplotype. A single other haplotype was identified by AFLP in all plots over the 2 years, and its detection in the field always followed wind-driven rains. X. axonopodis pv. allii-contaminated seed were detected by semiselective isolation and a nested polymerase chain reaction assay at levels up to 0.05% when final disease incidence was 0.61. Contaminated seed originated from both diseased and asymptomatic plants.

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Relationships Between Disease Incidence at Two Levels in a Spatial Hierarchy

Phytopathology ◽

10.1094/phyto.1997.87.5.542 ◽

1997 ◽

Vol 87 (5) ◽

pp. 542-550 ◽

Cited By ~ 74

Author(s):

G. Hughes ◽

N. McRoberts ◽

L. V. Madden ◽

T. R. Gottwald

Keyword(s):

Binomial Distribution ◽

Spatial Scales ◽

Disease Incidence ◽

Random Pattern ◽

Data Sets ◽

Unknown Parameters ◽

Incidence Data ◽

Rate Of Increase ◽

Grape Downy Mildew ◽

Spatial Hierarchy

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.

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Spatial Heterogeneity of the Incidence of Powdery Mildew on Hop Cones

Plant Disease ◽

10.1094/pd-90-1433 ◽

2006 ◽

Vol 90 (11) ◽

pp. 1433-1440 ◽

Cited By ~ 17

Author(s):

David H. Gent ◽

Walter F. Mahaffee ◽

William W. Turechek

Keyword(s):

Powdery Mildew ◽

Spatial Heterogeneity ◽

Power Law ◽

Binomial Distribution ◽

Disease Incidence ◽

Disease Assessment ◽

Cluster Sampling ◽

Data Sets ◽

Ratio Test ◽

Frequency Distributions

The spatial heterogeneity of the incidence of hop cones with powdery mildew (Podosphaera macularis) was characterized from transect surveys of 41 commercial hop yards in Oregon and Washington from 2000 to 2005. The proportion of sampled cones with powdery mildew ( p) was recorded for each of 221 transects, where N = 60 sampling units of n = 25 cones assessed in each transect according to a cluster sampling strategy. Disease incidence ranged from 0 to 0.92 among all yards and dates. The binomial and beta-binomial frequency distributions were fit to the N sampling units in a transect using maximum likelihood. The estimation procedure converged for 74% of the data sets where p > 0, and a loglikelihood ratio test indicated that the beta-binomial distribution provided a better fit to the data than the binomial distribution for 46% of the data sets, indicating an aggregated pattern of disease. Similarly, the C(α) test indicated that 54% could be described by the beta-binomial distribution. The heterogeneity parameter of the beta-binomial distribution, θ, a measure of variation among sampling units, ranged from 0.01 to 0.20, with a mean of 0.037 and a median of 0.015. Estimates of the index of dispersion ranged from 0.79 to 7.78, with a mean of 1.81 and a median of 1.37, and were significantly greater than 1 for 54% of the data sets. The binary power law provided an excellent fit to the data, with slope and intercept parameters significantly greater than 1, which indicated that heterogeneity varied systematically with the incidence of infected cones. A covariance analysis indicated that the geographic location (region) of the yards and the type of hop cultivar had little effect on heterogeneity; however, the year of sampling significantly influenced the intercept and slope parameters of the binary power law. Significant spatial autocorrelation was detected in only 11% of the data sets, with estimates of first-order autocorrelation, r1, ranging from -0.30 to 0.70, with a mean of 0.06 and a median of 0.04; however, correlation was detected in only 20 and 16% of the data sets by median and ordinary runs analysis, respectively. Together, these analyses suggest that the incidence of powdery mildew on cones was slightly aggregated among plants, but patterns of aggregation larger than the sampling unit were rare (20% or less of data sets). Knowledge of the heterogeneity of diseased cones was used to construct fixed sampling curves to precisely estimate the incidence of powdery mildew on cones at varying disease intensities. Use of the sampling curves developed in this research should help to improve sampling methods for disease assessment and management decisions.

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Sequential Sampling for Estimation and Classification of the Incidence of Hop Powdery Mildew II: Cone Sampling

Plant Disease ◽

10.1094/pdis-91-8-1013 ◽

2007 ◽

Vol 91 (8) ◽

pp. 1013-1020 ◽

Cited By ~ 8

Author(s):

David H. Gent ◽

William W. Turechek ◽

Walter F. Mahaffee

Keyword(s):

Powdery Mildew ◽

Binomial Distribution ◽

Disease Incidence ◽

Sequential Sampling ◽

Model Construction ◽

Data Sets ◽

Data Set ◽

Sampling Plans ◽

Simulated Sampling

Sequential sampling models for estimation and classification of the incidence of powdery mildew (caused by Podosphaera macularis) on hop (Humulus lupulus) cones were developed using parameter estimates of the binary power law derived from the analysis of 221 transect data sets (model construction data set) collected from 41 hop yards sampled in Oregon and Washington from 2000 to 2005. Stop lines, models that determine when sufficient information has been collected to estimate mean disease incidence and stop sampling, for sequential estimation were validated by bootstrap simulation using a subset of 21 model construction data sets and simulated sampling of an additional 13 model construction data sets. Achieved coefficient of variation (C) approached the prespecified C as the estimated disease incidence, [Formula: see text], increased, although achieving a C of 0.1 was not possible for data sets in which [Formula: see text] < 0.03 with the number of sampling units evaluated in this study. The 95% confidence interval of the median difference between [Formula: see text] of each yard (achieved by sequential sampling) and the true p of the original data set included 0 for all 21 data sets evaluated at levels of C of 0.1 and 0.2. For sequential classification, operating characteristic (OC) and average sample number (ASN) curves of the sequential sampling plans obtained by bootstrap analysis and simulated sampling were similar to the OC and ASN values determined by Monte Carlo simulation. Correct decisions of whether disease incidence was above or below prespecified thresholds (pt) were made for 84.6 or 100% of the data sets during simulated sampling when stop lines were determined assuming a binomial or beta-binomial distribution of disease incidence, respectively. However, the higher proportion of correct decisions obtained by assuming a beta-binomial distribution of disease incidence required, on average, sampling 3.9 more plants per sampling round to classify disease incidence compared with the binomial distribution. Use of these sequential sampling plans may aid growers in deciding the order in which to harvest hop yards to minimize the risk of a condition called “cone early maturity” caused by late-season infection of cones by P. macularis. Also, sequential sampling could aid in research efforts, such as efficacy trials, where many hop cones are assessed to determine disease incidence.

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Thymic involution and rising disease incidence with age

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1714478115 ◽

2018 ◽

Vol 115 (8) ◽

pp. 1883-1888 ◽

Cited By ~ 90

Author(s):

Sam Palmer ◽

Luca Albergante ◽

Clare C. Blackburn ◽

T. J. Newman

Keyword(s):

T Cell ◽

Power Law ◽

Wide Spectrum ◽

Disease Incidence ◽

Thymic Involution ◽

Incidence Data ◽

Age Related ◽

Exponential Decline ◽

Combining Data ◽

Cancer Types

For many cancer types, incidence rises rapidly with age as an apparent power law, supporting the idea that cancer is caused by a gradual accumulation of genetic mutations. Similarly, the incidence of many infectious diseases strongly increases with age. Here, combining data from immunology and epidemiology, we show that many of these dramatic age-related increases in incidence can be modeled based on immune system decline, rather than mutation accumulation. In humans, the thymus atrophies from infancy, resulting in an exponential decline in T cell production with a half-life of ∼16 years, which we use as the basis for a minimal mathematical model of disease incidence. Our model outperforms the power law model with the same number of fitting parameters in describing cancer incidence data across a wide spectrum of different cancers, and provides excellent fits to infectious disease data. This framework provides mechanistic insight into cancer emergence, suggesting that age-related decline in T cell output is a major risk factor.

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