New interval estimating procedures for the disease transmission probability in multiplevector transf

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New interval estimating procedures for the
disease transmission probability in
multiple-vector transfer designs

• Joshua M. Tebbs and Christopher R. Bilder
• Department of Statistics
• Oklahoma State University
• tebbs_at_okstate.edu and chris_at_chrisbilder.com

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Introduction

• Plant disease is responsible for major losses in
  agricultural throughout the world
• Diseases are often spread by insect vectors
  (e.g., aphids, leafhoppers, planthoppers, etc.)
• Example www.knowledgebank.irri.org/ricedoctor_mx/
  Fact_Sheets/Pests/Planthopper.htm

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Example

• Ornaghi et al. (1999) study the effects of the
  Mal Rio Cuarto (MRC) virus and its spread by
  the Delphacodes kuscheli planthopper
• The MRC virus is most-damaging maize virus in
  Argentina
• It was desired to estimate p, the probability of
  disease transmission for a single vector
• Vector-transfers are often used by plant
  pathologists wanting to estimate p
• In such experiments, insects are moved from an
  infected source to the test plants

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Single-vector transfers

• The most straightforward way to estimate p is by
  using a single-vector transfer
• Each test plant contains one vector, and test
  plants must be individually caged
• Under the binomial model, the proportion of
  infected test plants gives the maximum likelihood
  estimate of p
• Disadvantages with a single-vector transfer
• Requires a large amount of space (since insects
  must be individually isolated)
• Is a costly design since one needs a large number
  of test plants and individual cages

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Multiple-vector transfers

• A group of s gt 1 insect vectors is allocated to
  each test plant.
• Even though test plants are occupied by multiple
  insects, the goal is still to estimate p, the
  probability of disease transmission for a single
  vector

?
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Multiple-vector transfers

• Advantages of a multiple-vector versus
  single-vector transfer
• Potential savings in time, cost, and space
• Statistical properties of estimators are much
  better (for a fixed number of test plants)
• A multiple-vector transfer is an application of
  the group-testing experimental design
• Other applications of group testing
• Infectious disease seroprevalence estimation in
  human populations
• Disease-transmission in animal studies
• Drug discovery applications

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Notation and assumptions

• Define
• n number of test plants
• s number of insects per plant (group size)
• Y1 infected test plant plant for which at
  least one vector (out of s) infects
• Y0 uninfected test plant plant for which no
  vectors (out of s) infect
• Assumptions
• Common group size s
• The statuses of individual vectors are iid
  Bernoulli random variables with mean p
• The statuses of test plants are independent
• Test plants are not misclassified

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Maximum likelihood estimator for p

• Let T ?Y denote the number of infected test
  plants. Under our design assumptions, T has a
  binomial distribution with parameters n and
• The maximum likelihood estimator of p is given by
• where (the proportion of infected
  test plants)
• Estimates of p are computed by only examining the
  test plants (and not the individual vectors
  themselves)
• The binomial model is only appropriate if test
  plants do not differ materially in their
  resistance to pathogen transmission

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Properties of the MLE and the Wald CI

• The statistic has the following properties
• Consistent as n gets large
• Approximately normally distributed more
  precisely, where
• A 100(1-?) percent Wald confidence interval is
  given by
• where

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Variance stabilizing interval (VSI)

• Goal Find whose variance is free of the
  parameter p
• Solve the following differential equation
• With c0 1, a solution is given by
• It follows that
• is a 100(1-?) percent confidence interval for p.
  Here,

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Modified Clopper-Pearson (CP) interval

• The number of infected test plants, T, has a
  binomial distribution with parameters n and
  
• One can obtain an exact Clopper-Pearson interval
  for ? and then transform back to the p scale
  (Chiang and Reeves, 1962)
• Exact 100(1-?) percent confidence limits for p
  are given by
• 
  and
• where F1-?,a,b denotes the 1-? quantile of the
  central F distribution with a (numerator) and b
  (denominator) degrees of freedom

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Comparing the Wald, VSI, and CP

• The Wald interval is simple and easy to compute.
  However, it has three main drawbacks
• Provides symmetric confidence intervals even
  though the distribution of may be very skewed
• Often produces negative lower limits when p is
  small!
• The VSI handles each of these drawbacks
• Not symmetric
• Always produces lower limits within the parameter
  space (i.e., strictly larger than zero)
• The CP intervals main advantage is that its
  coverage probability is always greater than or
  equal to 1-?. However, such intervals can be
  wastefully wide, especially if n is small.

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Bayesian estimation

• Prior distribution for p
• One parameter Beta distribution
• for a known value of ?
• Takes into account p is small
• Example when ? 52.4

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Bayesian estimation

• Prior distribution for p
• Why use one parameter instead of two parameter
  Beta?
• Sensible model acknowledging p is small
• Bayes and empirical Bayes estimators are simpler
• Resulting estimator using squared error loss with
  a two parameter beta is ratio of complicated
  alternating sums
• See Chaubey and Li (Journal of Official
  Statistics, 1995) for Bayes estimators

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Bayesian estimation

• Posterior distribution for 0 lt p lt 1
• Note U 1 - (1 - P)s beta(t 1, n - t ?/s)

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Empirical Bayesian estimation

• Use the marginal distribution for T to derive an
  estimate for ?
• Why?
• Avoid possible poor choice for ?
• n is often small in multiple-vector transfer
  experiments
• Posterior may be adversely affected by the prior
• Marginal distribution of T for t 0, 1, , n
• Maximize fT(t?) as a function of ? to obtain the
  marginal maximum likelihood estimate,
• Iteratively solve for ? inwhere ?( ) is the
  digamma function

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Credible intervals

• (1 - ?)100 Equal-tail
• pL, pU satisfy
  and
• Use relationship with Beta distribution, U 1 -
  (1 - p)s beta(t 1, n - t /s)
• Interval
• where B?,a,b is the ? quantile of a Beta(a,b)
  distribution

Remember that ? 1 - (1 - p)s implies p 1 - (1
- ?)1/s
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Credible intervals

• (1 - ?)100 highest posterior density (HPD)
  regions
• Posterior is unimodal and right skewed
• Find pL, pU such that (1 - ?)100 area of
  posterior density is included and pU - pL is as
  small as possible
• See Tanner (1996, p. 103-4)
• Key is to sample from posterior distribution
• Use U 1 - (1 - p)s beta(t 1, n - t /s)
  relationship

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Example - Ornaghi et al. (1999)

• Data
• s 7 planthoppers per plant
• n 24 plants
• t 3 infected plants observed
• 95 interval estimates for p

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Interval comparisons

• Coverage where I(n,t,s) 1 if the interval
  contains 1 and I(n,t,s) 0 otherwise.
• Do not consider the t 0 and t n cases
• Poor multiple-vector transfer experimental design
• See Swallow (1985, Phytopathology) for guidance
  in choosing s
• Brown, Cai, and DasGupta (2001, Statistical
  Science)
• Frequentist evaluation similar to how Carlin and
  Louis (2000) approach evaluating confidence and
  credible intervals

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Interval comparisons

• ? 0.05, n40, and s10
• Black line denotes Wald bold line denotes plot
  title

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Summary

• Best interval VSI or modified Clopper-Pearson
• Credible intervals may be improved by taking into
  account variability of the ? estimators
• Bootstrap intervals mentioned in abstract VSI
  and Clopper-Pearson perform better
• Many other intervals could be investigated!
• Website
• www.chrisbilder.com/bilder_tebbs
• Contains R programs for examining the interval
  estimation properties
• Different values of p, n, and s can be used
• Also calculates empirical Bayes estimators
• Program for Ornaghi et al. (1999) data example

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New interval estimating procedures for the
disease transmission probability in
multiple-vector transfer designs

• Joshua M. Tebbs and Christopher R. Bilder
• Department of Statistics
• Oklahoma State University
• tebbs_at_okstate.edu and chris_at_chrisbilder.com

Contact address starting Fall 2003 Joshua M.
TebbsDepartment of StatisticsKansas State
University
Christopher R. BilderDepartment of
StatisticsUniversity of Nebraska-Lincolnchris_at_ch
risbilder.com