Sequential Sampling for
5.8 Summary
Sequential sampling plans were constructed using Iwao, SPRT and Converging Line boundaries. The minimum and maximum sample sizes were the same for all three boundary types; 15 and 50 respectively. Other parameters for each sampling plan were adjusted until OC functions for plans based on the three boundary types were approximately the same. This allowed for comparison of sampling costs required to achieve comparable levels of classification precision.
Shown in Fig. 5.16 are the stop boundaries and OC and ASN functions for the Iwao and Converging Line sampling plans. Recall that the minimum sample size was set to 15, so the big difference in stop boundaries is that those for Converging Lines become narrower with increasing sample size, while those for Iwao become wider until the maximum sample size is reached. The OC functions for the two plans are nearly identical, but the ASN function of the Converging Line plan is either close to or less than the ASN function for the Iwao plan.
Fig. 5.16. Stop boundaries (a) and OC (b) and ASN (c) functions for an Iwao (___) and a Converging Lines ( … ) sequential classification sampling plan and the negative binomial distribution. Boundary parameters: cd=1, minn=15, maxn= 50, TPL a=3.05 and TPL b=1.02. Iwao plan, α=0.15. Converging Lines plan, αL
=0.05 and αU=0.1. Simulation parameters: negative binomial with kestimated by TPL, 1000 simulations.
Sequential Sampling for Classification 125
A similar, although less extreme, pattern is evident when the Converging Line and SPRT plans are compared (Fig. 5.17). Here also, the OC functions are nearly identical, but the ASN function for the Converging Line is again either close to or less than the ASN function for the SPRT.
Fig. 5.17. Stop boundaries (a) and OC (b) and ASN (c) functions for an SPRT (___) and a Converging Lines ( … ) sequential classification sampling plan and the negative binomial distribution. Boundary parameters: cd=1, minn=15, maxn=50, TPL a=3.05 and TPL b=1.02. SPRT plan, µ0=0.8, µ1=1.2 and α= β=0.25. Converging Lines plan αL=0.05 and αU=0.1. Simulation parameters:
negative binomial with kestimated by TPL, 1000 simulations.
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References and Suggested Reading
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(1996) Sequential sampling plan for scheduling control of lepidopteran pests of fresh market sweet corn. Journal of Economic Entomology89, 386–395.
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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