This appendix provides details on how to calculate the aggregation-related parame- ters kof the negative binomial distribution and ρof the beta-binomial distribution.
For both distributions, three cases may be distinguished:
1. The aggregation parameter is constant, and a single value can be used for all densities (negative binomial) or incidences (beta-binomial). In this situation, the methods in this appendix are not needed. In practice, however, kand ρare rarely constant.
2. The parameter varies in value with the level of density or incidence, and a model is needed to capture the relationship. For both the negative binomial and the beta-binomial distributions, good descriptions of the aggregation parameter are obtained on the basis of a relationship between the variance and the mean density or incidence.
3. The parameter varies in value with the level of density or incidence, but in addition to this, there are significant differences in variance between fields with similar densities. This variability affects expected sampling performance.
This appendix deals with situations 2 and 3.
The value of kcan be modelled as a function of the mean, using TPL:
σ2=aµb (5A.1)
Some variability always exists in variance–mean models, so actual values of σ2will vary about the value predicted by the model (Equation 5A.1). In the simulations where OC and ASN functions are estimated, allowance can be made for variability around the model as follows. If TPL is fitted by linear regression, we generate a value for σ2by
(5A.2)
where zis normally distributed with mean 0 and standard deviation σε(the square root of the mean square error for the regression used to estimate TPL can be used as an estimate of σε). By equating sample and theoretical variances for the negative binomial distribution,
(5A.3) we obtain an estimate of k:
a e
k µb z = +µ µ2
ln(σ ) ln( ) ln( )µ ( σ )
σ µ
2 ε 2
= + + 0
=
a b z
or a b ze
,
Sequential Sampling for Classification 127
(5A.4) Each time a sampling plan is used to classify a particular density during a simula- tion, Equation 5A.4 is used to generate kfor the population being sampled. If an OC determined using simulation is to be based on 500 simulation runs, then 500 different values of k would be determined, one for each simulation run at each value of µ. If variability is not to be included in the simulations, the value used for k is
(5A.5) Equating theoretical and sample variances is an example of the ‘Method of Moments’, because according to statistical terminology the variance is one of the
‘moments’ of a probability distribution (it is the second moment, the first being the mean).
The derivation for ρis very similar. Its value can be modelled as a function of the incidence, p, using the model (Hughes et al., 1996)
(5A.6) where R is the cluster size, and A and b are the parameters of the variance–
incidence model. Again, there is always some variability in incidence–mean models, so actual values of σ2 will vary about the value predicted by the model (Equation 5A.6). In the simulations where OC and ASN functions are estimated, allowance can be made for variability around the model as follows. If the model is fitted by linear regression, we generate a value for σ2by
(5A.7)
where zis normally distributed with mean 0 and standard deviation σε. By equating sample and theoretical variances,
lnσ ln ln σ
σ
2 ε
2
1 0
1
( )
=( )
+ ( )
− +
( )
=
( )
−
A b p p
R z
or A p p
R e
b z
,
σ2 1
=
( )
−
A p p
R
b
k= a b
− µ
µ µ
2
k= a b ze
− µ
µ µ
2
Beta-binomial
(5A.8)
we obtain an estimate of ρ:
(5A.9)
This formula is used exactly as described above for Equation (5A.4). If variability is not to be included in the simulations, the value used for ρis
(5A.10) ρ =
( )
−( )
− −
−
−
1
1 1 1
1
R 1
AR
p p
b
[ ]
bρ =
( )
−( )
− −
−
−
1
1 1 1
1
R 1
AR e
p p
b z
[ ]
bA p p
R e p p
R R
b
1 z 1
1 1
( )
−
=
( )
−
+
(
ρ[
−] )
Sequential Sampling for Classification 129
In this chapter we review how the usefulness of sampling plans for pest manage- ment decision-making may be maximized by proper choice of design ingredients, by suitable presentation of the decision tool to users, and by use of simulation tools and field methods, along with user interaction, for evaluation and target-oriented design of plans. Evaluation of a sampling plan involves more than just the operat- ing characteristic and average sample number functions, which were discussed in the preceding chapters. Qualitative aspects, such as the practicality and simplicity of the decision guide, and the reliability, representativeness and relevance are equally important. These aspects, although not easily expressed quantitatively, have a considerable influence on adoption in practice. It is therefore necessary to evaluate qualitative aspects of performance in direct consultation and interaction with end-users. Quantitative indicators of performance, such as the expected out- come from sampling and time spent sampling, can be readily defined and analysed using computer simulation. While field evaluations of sampling plans can generally not be used to estimate operating characteristic (OC) and average sample number (ASN) functions, they are useful in other contexts and have the advantage that there are no model-based artefacts in the evaluation results. When evaluating the results of a field or computer evaluation, interaction with end-users is again indis- pensable. The economics of sampling are an important aspect of sampling plan evaluation, and we need a decision-theoretic framework that allows a quantitative evaluation of the value of sampling.
The text covers the full range of components which go into creating and implementing a good decision guide. We look first at the nature of the design process: the process between the recognition that a decision guide based on sampling is needed and the implementation of a decision guide in practice. We follow this with an examination of the necessary design ingredients of a decision guide, expanding the introductory discussion in Chapter 2. Then we discuss the various ways decision guides can be evaluated. Finally, we show how to