Sequential Sampling for
5.5 Converging Lines
Both Iwao’s and Wald’s methods are based in some way or other on statistical theory. But it may be possible to improve the efficiency of classification by escaping from established theoretical frameworks and concentrating on practical goals.
Iwao’s stop boundaries consist of divergent curvilinear lines and those for the SPRT are parallel lines (for most distributions of interest to pest managers). To force a decision at a maximum sample size, both sets of lines are abruptly brought together to a point. An alternative approach would be to bring the upper and lower boundaries together more gradually, to meet at the maximum sample size. This makes intuitive sense, because precision of a classification should improve as the sample size increases, so the stop boundaries should be allowed to converge. In fact, this type of boundary proved popular with the medical researchers, who found that Wald’s SPRT was not entirely ideal for their purposes (Armitage, 1975).
Converging Line stop boundaries have not to our knowledge been applied in pest management.
Converging Line stop boundaries could be developed in several ways. We use the following rationale: the boundaries consist of two straight lines, one of which meets the Sn-axis (the total count axis) above zero, and the other of which meets the n-axis (the sample size axis) above zero. The two lines converge and meet at the point (maxn, cd× maxn) (see Fig. 5.10). The question to be answered when specifying one of these boundaries is: Where should the stop boundaries meet the horizontal (n) and vertical (Sn) axes? In general, the further apart the lines are, the greater the sample size and consequently the greater the precision in classification.
An intuitive way to quantify the spread in the stop boundaries is to specify the minimum sample size, minn, and calculate classification intervals there about the critical density (as in Equation 3.11, with minnreplacing nB). This allows for sym- metric stop boundaries as well as boundaries that differentially guard against the two types of misclassification.
The upper and lower boundary points corresponding to the minimum sample size are calculated exactly as for Iwao’s method:
(5.6)
where zαLand zαUare standard normal deviates, and V is the variance when the mean is equal to cd.
The complete stop boundary consists of straight lines joining the point (maxn, cd×maxn) to each of the points (minn, Lminn) and (minn, Uminn). These lines can be extended to meet the axes, but of course the boundary is not strictly relevant below minn(Fig. 5.10).
L cd z V
U cd z V
L
U minn
minn
minn minn
minn minn
= −
= +
α
α
The quantities used to specify the width of the stop boundaries can be inter- preted as classification intervals (Section 3.2.2), but this is not particularly mean- ingful. It is best to do as we have suggested with other sequential plan parameters:
αUand αLshould be viewed as tools for manipulating the stop boundaries so that desirable OC and ASN functions are achieved (Table 5.2). Additional parameters that can be used for this purpose are minn, maxnand cd (or cp). In general, decreas- ing αUand/or αL will increase the width of the boundaries, make the OC more steep and increase ASN. Decreasing αUbut keeping αLfixed raises the upper arm of the stop boundary and makes the upper part of the OC steeper, and similarly for αL. Increasing minnand maxnwill increase the width of the boundaries, make the OC more steep and increase ASN. As for Iwao’s and Wald’s plans, increasing the variance of the sample observations above the variance used to construct the stop boundary (which applies only to the negative binomial and beta-binomial
Sequential Sampling for Classification 117
Fig. 5.10. Converging Line sequential classification stop boundaries. The boundaries consist of Converging Lines that meet at the point maxn, cd×maxn, where maxnis the maximum sample size and cdis the critical density. The upper and lower boundary points corresponding to the minimum sample size (minn) are determined as
,
where zαUis a standard normal deviate such that P(Z > zαU) = αU, zαLis a standard normal deviate such that P(Z> zαL) =aL, and Vis the variance when the mean =cd.
The complete stop boundary consists of straight lines joining the boundary points at minnto the point (maxn, cd×nmax) and extending to the ‘Total count’ and ‘Sample size’ axes.
U cd z V
L cd z V
U L
minn minn minn
minn minn
= + minn
= −
α and α
Exhibit 5.3. Converging Lines plan: the effect of altering parameters
In this example we illustrate how some of the parameters for Converging Line stop boundaries influence the OC and ASN functions. The example is based on sam- pling lepidopteran pests of fresh market sweetcorn in New York State. Hoffman et al.(1996) showed that the sampling distribution of plants infested with caterpillars could be described by a beta-binomial distribution when plants were sampled in groups of 5 (‘clusters of five’, to use the terminology of Chapter 4). These authors parameterized the beta-binomial distribution using θ, which is a measure of aggre- gation that ranges from 1 to infinity, rather than ρ. However, ρ = θ/(1 + θ). These authors reported the median value for ρ to be 0.1, and the 90th and 95th per- centiles to be 0.23 and 0.44. The critical proportion of plants infested with larvae was 0.15 prior to silking of the maize and 0.05 after silking. In this example, we use cp=0.15.
The influence of αUand αLwas first studied by setting both of these parame- ters equal to one of three values: 0.05, 0.1 or 0.2. The remaining parameters for these plans were the same: cp = 0.15, the number of plants examined at each sample location in the field (R) =5, ρ=0.1, minn=15 and maxn=50. One thou- sand simulation replicates were made to estimate each OC and ASN value. The stop boundaries and OC and ASN functions are shown in Fig. 5.11. Increasing αU and αL caused the stop boundaries to move closer together. This resulted in decreasing ASN functions, but only a small reduction in the precision of the classi- fications.
The influence of the maximum sample size was examined by setting αUand αLto 0.05 and allowing maxnto have one of three values: 25, 50 or 75. All other parameters were as described above. The stop boundaries and OC and ASN func- tions are shown in Fig. 5.12. Increasing maxn resulted in steeper OC functions (increased precision of classification) and increased ASN functions. Note that the maximum ASN value is much less than maxn when maxn=75. This is because the stop boundaries are very narrow for this value of maxn.
The influence of the minimum sample size was examined by setting maxn=50 and allowing minnto have one of three values: 10, 20 or 30. All other parameters were as described above. The stop boundaries and OC and ASN functions are shown in Fig. 5.13. Increasing the minimum sample size over the range 10, 20, 30 caused the stop boundaries to lie further apart and resulted in an increase in the ASN functions by nearly 50% (from 10 to 20) and by nearly 25% (from 20 to 30).
The corresponding changes in the OC functions were small and might not be justi- fied by the extra sampling costs.
The parameter ρof the beta-binomial distribution measures the correlation in incidence rate among plants within a cluster of plants. As noted in Section 4.8, as ρ increases, the variability of p, the proportion of plants infested, also increases. This in turn increases the variance in the estimated proportion of infested plants throughout the field. Shown in Fig. 5.14 are OC and ASN functions when stop distributions) flattens the OC function, while near cd(or cp) the ASN tends to be lower; otherwise, it tends to be higher. These patterns are illustrated in Exhibit 5.3.
The OC and ASN functions can only be obtained by simulation.
Sequential Sampling for Classification 119
boundaries were calculated using ρ = 0.1 and sampling was simulated using ρ=0.1, 0.23 and 0.44. As might be expected, increasing the ρused in the simula- tions above the value used to calculate the stop boundaries caused the ASN func- tion to decrease and the OC function to become more shallow (less precision). This is the same type of effect as we found in Exhibits 5.1 (Iwao) and 5.2 (SPRT).
Whether or not these changes are important depends on the frequency with which different values of ρmight occur in practice.
Continued Fig. 5.11. Stop boundaries (a) and OC (b) and ASN (c) functions for a Converging Lines sequential classification sampling plan and the beta-binomial distribution.
The effect of changing the boundary parameters αUand αL: αU= αL=0.05 (___), 0.1 ( … ) and 0.2 (- – -). Other boundary parameters: cp=0.15, R=5, ρ=0.1, minn=15 and maxn=50. Simulation parameters: beta-binomial with ρ=0.1, 1000 simulations.
Continued Fig. 5.12. Stop boundaries (a) and OC (b) and ASN (c) functions for a Converging Lines sequential classification sampling plan and the beta-binomial distribution. The effect of changing the boundary parameter maxn: maxn=25 (___), 50 ( … ) and 75 (- – -). Other boundary parameters: cp=0.15, αU= αL=0.05, R=5, ρ=0.1 and minn=15. Simulation parameters: beta-binomial with ρ=0.1, 1000 simulations.
In this chapter, we have not (so far) mentioned estimation. As noted in Chapter 3, the goals of estimation and classification are different, so stop criteria for sequential plans for estimation should probably be different from those for classification.
Sequential sampling plans for estimation do exist. Some discussions are highly mathematical (see, e.g. Siegmund, 1985) and others are less so (see, e.g. Green, 1970). In contrast to stop boundaries for classification, stop boundaries for estima- tion have no critical density and they tend to bend in towards the starting point (n