2.1 Production system description

Following harvest from the orchard, picked fruit are brought into the pack house for grading and sorting (Figure 1(a)), the basis of which can include selection related to marketability, eating quality and/or the presence of physical symptoms indicative of disease. The process is becoming increasingly mechanized, using optical grading machines that are capable of classifying individual fruit on the basis of size, weight, shape and colour features. During the grading and sorting process, fruit may end up being rejected for reasons including being too big, too small, not sufficiently ripe (“green”), overly ripe (“soft”), having unwanted colour, being misshapen, blemished or visibly damaged. These fruit end up in the “reject” bin (Figure 1(b)). The proportion of fruit that is “rejected” may be influenced by the crop type, growing conditions, pest prevalence and target market, and is typically a nontrivial proportion of the harvest.

Figure 1

Schematic of the production process, including (a) fruit production and harvest, (b) grading, sorting and packing/rejection, (c) inspection of consignment of reject bin samples, (d) inference on infestation level within packed fruit, and (e) packed fruit bound for market with associated inferred infestation status. (The schematic illustration was provided by Brent Henderson.)

Figure 1 Long description

A schematic showing an orchard, then a packing house, then grading. Accepted fruit go into consignment, then standard 600 fruit consignment sampling. Rejected fruit go to the reject bin, then reject bin sampling. Both samples are used for combined inference across consignment and reject bin about infestation level. The consignment then goes to market.

We consider two inspection compliance measures, namely consignment sampling and reject bin sampling (Figure 1(c)). Consignment sampling involves the inspection of samples drawn from the consignment of packed product. Generally the inspection associated with consignment samples is restricted to being nondestructive (visual only), and hence the inspection sensitivity is typically low for internal pests (for example, fruit fly larvae). Reject bin sampling involves the inspection of samples drawn from fruit that have been “rejected” for packing and placed in the reject bin. Destructive inspection of these sample incurs no production losses, and can include slicing and brown sugar flotation, resulting in considerably higher detection sensitivities in comparison to visual inspection.

The data arising from the samples are used to make inference (Figure 1(d)) on the infestation level of fruit that is potentially bound for market (Figure 1(e)).

2.2 Model description

To undertake detection probability calculations on the basis of inspection samples, we need to estimate the prevalence of infested fruit within both the reject bin and the associated consignment of packed product: this will be affected by the visual properties of infested and uninfested fruit in combination with the selectivity of grading with respect to fruit infestation status. We then estimate the sensitivity and specificity of the inspection procedures as applied to samples from the consignment and the reject bin. To do this, we define the following binary random variables (see Figure 2):

• • $I=1$ when a piece of fruit is infested/infected by a pest/pathogen, and $I=0$ otherwise;

• • $D=1$ when a piece of fruit has defective features (too big, too small, blemished, wrong colour, …), noting that defective features are considered as being potentially detectable during the grading process, and $D=0$ otherwise; and

• • $R=1$ when a piece of fruit is rejected for packing during grading and placed in the reject bin, and $R=0$ otherwise.

Figure 2

Model description, with I = infested, D = defective and R = rejected. Solid arrows between binary states represent parameters defined by the conditional probability of the state at the arrow head, given the state at the arrow tail. For example, $\phi = P(D=1|I=0)$ . Solid arrows pointing away from a state represent probabilities of infection ( $I=1$ ) given the state at the arrow tail. Dashed arrows represent exceptions: $\pi _h$ is simply the unconditional probability $P(I=1)$ and $\psi $ is related to $P(D=1\mid I=1)$ via equation (2.1).

Figure 2 Long description

A diagram representing events I=Infested, D=Defective and R=Rejected as boxes, each with a 0 and a 1 representing the binary states each can have. There are solid arrows between some of these, representing a conditional probability, labelled with the relevant parameter. These are: phi as the probability of D=1 given I=0, rho as that of R=1 given D=1, and sigma as that of R=0 given D=0. There are two solid arrows pointing away from a state; these are pi_r as the probability of infection (I=1) given R=1, and pi_c as that of I=1 given R=0. Two dashed arrows represent exceptions, where pi_h is the unconditional probability of infection, and psi is given by equation 1.

Next, we define the conditional probabilities relating to screening sensitivity and specificity:

• • $\phi =P(D=1\mid I = 0)=P(\text {Defective}\mid \text {Not infested})=$ base rate of defective fruit in absence of infestation;

• • $\rho =P(R=1\mid D=1)=P(\text {Rejected}\mid \text {Defective})=$ grading sensitivity; and

• • $\sigma =P(R = 0\mid D = 0)=P(\text {Not rejected} \mid \text {Not defective})=$ grading specificity.

Let $\psi =P(I \rightarrow D)=P(\text {Infestation generates physical defects})$ —that is, the propensity of infestation to generate physical symptoms that could be considered defective. For example, in the case of codling moth, infestation can cause defects in the form of tunnelling holes. It is defined such that

(2.1) $$ \begin{align} P(D=1\mid I=1) &= \phi + \psi (1 - \phi) , \end{align} $$

where we assume independence between the innate propensity of fruit to exhibit defects (that is, $\phi $ ) and the state of being infested. When $\psi = 0$ , infestation and defects are independent, that is, $P(D=1\mid I=1) = \phi = P(D=1\mid I=0)$ . When $\psi = 1$ , the presence of infestation guarantees defects, that is, $P(D=1\mid I=1) = 1$ . Then the probability of harvested fruit being defective is

$$ \begin{align*} P(D=1) &= P(D=1\mid I=1)P(I=1) + P(D=1\mid I = 0)P(I = 0) \\ &= (1 - (1-\phi)(1-\psi) ) \pi_h + \phi(1 - \pi_h), \end{align*} $$

where $\pi _h= P(I=1)$ , the prevalence of infestation in harvested fruit entering the pack house (prior to grading).

Further, define the underlying prevalence of infestation following grading as follows:  $\pi _r$ is the prevalence of infestation among rejected fruit; and $\pi _c$ is the prevalence of infestation among the nonrejected (that is, packed) consignment of fruit.

The prevalence of infestation in the packed consignment is the measure of primary interest from a risk perspective, and is the prevalence within fruit conditional on it not being rejected during grading. That is, $\pi _c = P(I=1\mid R = 0)$ . The prevalence of fruit in the reject bin is $\pi _r = P(I=1\mid R=1)$ . These probabilities are computed using Bayes’ theorem, but first some conditional probabilities of rejection given infestation state are needed. We have

$$ \begin{align*} P(R=1\mid I=1) &= P(R=1\mid D=1)P(D=1\mid I=1)\\ &\qquad + P(R=1\mid D = 0)P(D = 0\mid I=1) \\ &= \rho [1 - (1-\phi)(1-\psi) ] + (1-\sigma)(1-\phi)(1-\psi). \end{align*} $$

Likewise,

$$ \begin{align*} P(R=1\mid I = 0) &= P(R=1\mid D=1)P(D=1\mid I = 0)\\ &\qquad + P(R=1\mid D = 0)P(D = 0\mid I = 0) \\ &= \rho \phi + (1 - \sigma)(1-\phi). \end{align*} $$

For the prevalence within the packed consignment (that is, the event “Not rejected”), applying Bayes’ theorem gives

(2.2) $$ \begin{align} \pi_c &= P(I=1\mid R = 0) \nonumber \\ &= \frac{P(R = 0\mid I=1)P(I=1)}{P(R = 0\mid I=1)P(I=1) + P(R = 0\mid I = 0)P(I = 0)} \nonumber \\ &= \frac{(1-P(R=1\mid I=1))P(I=1)}{(1-P(R=1\mid I=1))P(I=1) + (1-P(R=1\mid I = 0))P(I = 0)} \nonumber\\ &= \frac{(1 - [\rho \psi + (1 - \sigma)(1-\psi)]) \pi_h}{(1 - [\rho \psi + (1 - \sigma)(1-\psi)]) \pi_h + (1-[\rho \phi + (1-\sigma)(1-\phi)])(1-\pi_h )}. \end{align} $$

Likewise, the prevalence within the reject bin (that is, the event “Rejected”) is calculated as

(2.3) $$ \begin{align} \pi_r &= P(I=1\mid R=1) \nonumber \\ &= \frac{P(R=1\mid I=1)P(I=1)}{P(R=1\mid I=1)P(I=1) + P(R=1\mid I = 0)P(I = 0)} \nonumber \\ &= \frac{A}{B}, \end{align} $$

where

$$ \begin{align*} A = (\rho [1 - (1-\phi)(1-\psi) ] + (1-\sigma)(1-\phi)(1-\psi)) \pi_h \end{align*} $$

and

$$ \begin{align*} B &= (\rho [1 - (1-\phi)(1-\psi) ] + (1-\sigma)(1-\phi)(1-\psi)) \pi_h \\ & \qquad +(\rho \phi + (1 - \sigma)(1-\phi) ) (1 - \pi_h). \end{align*} $$

2.3 Extending the ISPM31 approach

The ISPM31 sampling protocol and associated calculations focus on the risk (“confidence level”) of a user-specified design prevalence (“level of detection”) being exceeded, given no detections from within a consignment sample of known size and detection efficacy. This frequentist approach for undertaking compliance sampling can be extended to having no detections in either a consignment sample or a sample from the associated reject bin. To facilitate this, let:

• • $n_{rs}$ be the number of fruit sampled from the reject bin; and

• • $n_{cs}$ be the number of fruit sampled from the consignment.

For the sensitivity of inspection for the different measures, we have:

• • $\delta _{cs}$ is the detection sensitivity applied to samples from the packed consignment, combining both the probability of an infested piece exhibiting symptoms of infestation being detected (here we are assuming by the human eye) and the subsequent diagnostic method confirming infestation; and

• • $\delta _{rs}$ is the sensitivity of diagnostic method applied to samples from the reject bin sample, applied regardless of physical symptoms.

The values of both $\delta _{cs}$ and $\delta _{rs}$ depend on the method of inspection and associated diagnostic procedure. We assume here that the inspection of the consignment sample is a two-step process starting with visual screening, followed by a diagnostic examination (for example, slicing) for pieces detected as showing symptoms (defects). Hence, to apply $\delta _{cs}$ within calculations we need to estimate the proportion of infested fruit within the consignment (not rejected during grading) that exhibit defects. That is,

$$ \begin{align*} P(D=1\mid R = 0,I=1) &= \frac{P(R = 0\mid D=1,I=1)P(D=1\mid I=1)}{P(R = 0 \mid I=1)} \\ &= p_c \quad (\text{say}), \end{align*} $$

where

$$ \begin{align*} P(R = 0 \mid I=1) &= P(R = 0 \mid D=1) P(D=1\mid I=1) \\ & \qquad+ P(R = 0 \mid D = 0) P(D = 0 \mid I=1) \\ &= (1 - \rho)(1 - (1-\phi)(1-\psi)) + \sigma (1 - \phi)(1 - \psi), \end{align*} $$

hence

$$ \begin{align*} p_c &= \frac{(1-\rho) \psi}{(1 - \rho)(1 - (1-\phi)(1-\psi)) + \sigma (1 - \phi)(1 - \psi)}. \end{align*} $$

With respect to the sampling data let:

• • X be the number of detections from consignment sampling; and

• • Y be the number of detections from reject bin sample.

Assuming that the numbers of fruit within the consignment and reject bin are large in comparison to the respective samples $n_{cs}$ and $n_{rs}$ , the Poisson approximation to the binomial is appropriate [Reference Yamamura, Katsumata, Yoshioka, Yuda and Kasugai8], and the probability of detecting at least one infested fruit given the underlying prevalence, inspection and grading parameters (designated $\boldsymbol {\theta }$ ) is

(2.4) $$ \begin{align} P(\text{At least one detection} \mid \pi_c, \pi_r, \boldsymbol{\theta}) &= 1 - P(\text{No detections} | \pi_c, \pi_r, \boldsymbol{\theta}) \nonumber\\ &= 1 - P(X=0, Y=0 | \pi_c, \pi_r, \boldsymbol{\theta}) \nonumber\\ &= 1 - P(X=0| \pi_c, \boldsymbol{\theta})P(Y=0| \pi_r, \boldsymbol{\theta}) \nonumber\\ &= 1 - \exp(-\delta_{cs} n_{cs} \pi_c p_c )\exp(-\delta_{rs} n_{rs} \pi_r)\nonumber\\ &= 1 - \exp(-\delta_{cs} n_{cs} \pi_c p_c -\delta_{rs} n_{rs} \pi_r), \end{align} $$

where $\pi _c$ and $\pi _r$ are calculated as in equations (2.2) and (2.3). Note that we have not included the proportion of infested fruit within the reject bin that exhibit symptoms in equation (2.4), as it is assumed that the fruit sampled for inspection are chosen without regard to defect status, and are all subjected to the same diagnostic investigation (for example, slicing). Note also that X and Y are conditionally independent given prevalence. We derive the prevalence of defects amongst infested fruit consigned to the reject bin in Appendix A should readers want to modify equation (2.4) to account for only inspecting symptomatic fruit from the reject bin.

Calculating $\pi _c$ such that the power to detect at least one infested fruit is at a desired confidence level (for example, 95%) is undertaken numerically. That is, for a given value of $\pi _h$ , we calculate the corresponding values of $\pi _c$ and $\pi _r$ , and from these the probability of at least one detection occurring (using equation (2.4)), from the consignment sample ( $X>0)$ and/or the reject bin sample ( $Y>0$ ).

2.4 Measuring effectiveness of reject bin sampling

The conditional probability $P(X=0 \mid Y>0)$ gives us the probability that consignment sampling would have missed an infestation given that the reject bin sampling found an infestation. This gives us a measure of effectiveness of sampling: if the probability is low, then reject bin sampling is not worth pursuing as it is unlikely to add anything; conversely, if the probability is high, then it is.

2.5 Use cases of interest

We focus on three general scenarios to evaluate how inspecting rejected samples will affect the phytosanitary state of the packed consignment. First, in Scenario 1, we examine high symptomatic physical expression that is readily detectable by both human observation and machine grading, with visual confirmation following destructive sampling. For the other scenarios, we examine low symptomatic physical expression of infestation, where symptomatic units are still detectable by both human observation and machine grading: for Scenario 2, visual confirmation is straightforward following destructive sampling; but for Scenario 3 it is of low sensitivity as laboratory procedures are required. Details are provided in Table 1. We restrict our inference to single consignment samples of size 600, though note that combining the results from multiple consignment samples can provide stronger evidence for low levels or absent infestation [Reference Caley, Gladish, Kingham and Van Klinken2].

Table 1

Details of use case scenarios for the inspection of samples from the reject bin in terms of $\psi $ (the probability of an infested piece of fruit exhibiting a defect), $\rho $ (the probability of a piece of fruit with a defect being rejected during grading) and the inspection sensitivity for samples from consignments ( $\delta _{cs}$ ) and the reject bin ( $\delta _{rs}$ ).

Table 1 Long Description

Details of use case scenarios for the inspection of samples from the reject bin in terms of psi (the probability of an infested piece of fruit exhibiting a defect), rho (the probability of a piece of fruit with a defect being rejected during grading) and the inspection sensitivity for samples from consignments (delta_cs) and the reject bin (delta_rs). Scenario 1 is named High symptomatic physical expression with psi=1.0, rho=0.99 & 0.9, delta_cs=1.0 and delta_rs=1.0. This represents a pest that reliably generates detectable external symptoms and easily detectable by destructive inspection. Scenario 2 is named Low symptomatic physical expression (clear infestation) with psi=0.1, rho=0.99 & 0.9, delta_cs=1.0 and delta_rs=1.0. This represents a pest (e.g. internal) that generates minimal external symptoms, though easily detectable by destructive inspection. Scenario 3 is named Low symptomatic physical expression (cryptic infestation) with psi=0.1, rho=0.9, delta_cs=0.1 and delta_rs=1.0. This represents a pest (e.g. internal) that generates minimal external symptoms, and is only reliably detectable by laboratory procedure (e.g. grow-out, eDNA) following destructive inspection.