Usefulness of Sampling Plans 6 6
7.6 Incorporating a Variance–Mean Relationship
Up to this point, we have assumed that while kmay not remain constant, it does not change systematically with density. However, as noted in previous chapters, k often tends to increase with density and can be modelled using a variance–mean relationship. If a variance–mean relationship, such as Taylor’s Power Law (TPL) (Equation 3.14) holds for the pest in question, and the negative binomial distribu- tion describes sample counts, the value of kcan be regarded as a function of µ:
Fig. 7.11. The OC functions for binomial count SPRT plans with T=3(a), 5(b), 8(c) and 11(d) used to classify the density with respect to cd=5. Sample counts were described by a negative binomial distribution. Parameters for all sampling plans were k=0.8, µ0=4, µ1=6, α= β =0.1, minn=5 and maxn=50. The parameter kof the negative binomial distribution used to describe the populations sampled was 0.8 (___), 0.4 ( … ), and 1.5 (- – -).
(7.10)
This is fine if we can believe that σ2 is perfectly estimated from µby TPL.
That would be too much to ask, but we should be able to assume that σ2can be estimated by TPL with a little variability. We showed in Chapter 5 (Appendix) how we could include variability around TPL in the simulations to estimate OC and ASN functions for full count sampling plans. The extension to binomial count sampling plans is reasonably straightforward: the value of k generated with or without variability around TPL (Equation 5A.4 or 5A.5) is used, along with µ, to estimate the probability of at least Tpests on a sample unit.
The OC functions obtained in this way may be regarded as averages of the OC functions that would be obtained for each possible value of k. They are therefore quite different from the OC functions obtained using a constant k, as in Exhibit 7.2. So which approach should be used? Examination of OC functions using con- stant values of k can provide insight into ‘worst case’ scenarios. In contrast, OC functions obtained using TPL with variability yield an OC that would be expected on average. Ideally, both types of OC functions should be used to judge the accept- ability of a proposed sample plan.
k= a b
− =
− µ
σ µ
µ
µ µ
2 2
2
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Exhibit 7.3. Binomial classification sampling with the negative binomial distribu- tion and TPL.
This example is a continuation of Exhibit 7.2, sampling European red mite. Now we use TPL to estimate k. Nyrop and Binns (1992) estimated a = 4.3, b= 1.4 and mse = 0.286 ( ). We can use 0.54 as an estimate of σε in Equation 5A.4. Three SPRT sample plans (µ1=6, µ0=4, α= β =0.1, minn=5 and maxn= 50) were compared: full count sampling, binomial sampling with T=0 and no vari- ation about TPL, binomial sampling with T=0 including variation about TPL (σε= 0.54). The OC and ASN functions are shown in Fig. 7.12. Without variation about TPL, the OC functions for the complete count and binomial count models are nearly the same. Adding variation about TPL caused the OC function for the bino- mial count plan to become significantly flatter.
The effect of variability in TPL on the OC functions for binomial count plans can be lessened by increasing the tally number. The third plan above was com- pared with two others with different tally numbers, 2 and 5. The results are shown in Fig. 7.13. The OC function became steeper as Twas increased. In Figs 7.9 and 7.10, we can see that for T=5, the OC functions for binomial count sample plans with and without variation about TPL are nearly the same.
Continued mse =0 54.
Fig. 7.12. The OC (a) and ASN (b) functions for three SPRT plans used to classify the density with respect to cd=5. Sample counts were described by a negative binomial distribution and the variance modelled using TPL with a=4.3 and b=1.4. The first plan (___) was based on complete counts. The second and third plans were based on binomial counts with T=0. SPRT parameters for all three sampling plans were µ0=4, µ1=6, α= β =0.1, minn=5 and maxn=50. The standard deviation of the variance predicted from TPL was: 0 (___), 0 ( … ), and 0.55 (- – -).
Fig. 7.13. The OC (a) and ASN (b) functions for three binomial count SPRT plans used to classify the density with respect to cd=5. Sample counts were described by a negative binomial distribution and the variances modelled using TPL with a=4.3 and b=1.4. SPRT parameters for all three sampling plans were µ0=4, µ1=6, α= β =0.1, minn=5 and maxn=50. The standard deviation of the variance predicted from TPL was 0.55. Tally numbers (T) were 0 (___), 2 ( … ) 5 (- – -).
A probability model is not a necessary prerequisite for developing a binomial count sample plan or for estimating their OC and ASN functions. A probability model for the p–µrelationship can be replaced by an empirical model. Several models have been proposed (see, e.g. Ward et al., 1986), but one that has been found most useful and can easily be fitted using linear regression is as follows:
ln(ln(1 p)) =cT+ dTln(µ) (7.11)
where µis the mean and pis the proportion of sample observations with more than T organisms. Note that this model has been shown to fit the relationship for values of Tgreater than as well as equal to 0 (see, e.g. Gerrard and Chiang, 1970), so the parameters have a subscript Twhich signifies that they depend on the tally number used. In a strict sense, there are problems with fitting Equation 7.11 using linear regression, because both p and µ are estimated and so dT will tend to be underestimated (see, e.g. Schaalje and Butts, 1993). However, provided that the data cover a wide range of µaround cd(or, alternatively, if the range 0–1 is nearly covered by p), and the estimates of pand µare relatively precise, the underestima- tion should be ignorable. It is more important to check that the data really are linear before fitting the model. Upon fitting the model, the critical proportion, cpT (note the suffix T) is calculated as
(7.12) There will always be variability about the relationship expressed by Equation 7.12.
This variability is not used to determine the critical proportion, but it is of critical importance for estimating the OC and ASN functions.
Once cpThas been determined, OC and ASN functions for any sample plan that classifies binomial counts with respect to cpTcan be estimated using a bino- mial distribution, relating the range of true means (µi) to a range of probabilities (pi) using Equation 7.11. Here, we must consider variability in Equation 7.11 because, while this model may provide a good average description of the data, not all data points fall exactly on the line. Corresponding to any true mean (µi) there will be a range of probability values around pi = 1 exp(ecTµidT), and the OC function at µiis an average of points, each of which corresponds to one of these probability values.
The procedure is similar to that followed for assessing the consequences of variation about TPL (Chapter 5, Appendix). We assume that actual values of ln(ln(1 p)) are approximately normally distributed about the regression line, as determined from the equation
ln(ln(1 p)) =cT+ dTln(µ) + z(0,σε) (7.13) where z(0,σε) is a normally distributed random variable with mean 0 and standard deviation σε. The OC and ASN functions are determined using simulation. Each time a sample plan is used to classify a particular density during the simulation,
cpT= −1 e−
(
ecTcddT)
= −1 exp(
−e cdcT dT)
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