Sampling inspection to prevent the invasion of alien pests: statistical theory of import plant quarantine systems in Japan

Abstract

The establishment of appropriate import quarantine systems is the best known method for preventing the unintentional introduction of invasive alien pests. However, quarantine systems are sometimes judged as non-tariff barriers against trade by the World Trade Organization. The construction of a common scientific theory for quarantine systems is thus extremely important to prevent invasion without causing international conflict. We explain several statistical theories that have been adopted in import plant quarantine systems in Japan. Quarantine systems include three major components: (1) import sampling inspection, (2) early detection procedures, and (3) emergency control. We first explain the principle of risk management that was commonly adopted in these components. Then, we explain the method for calculating the required sample size in the import sampling inspection. We then explain hierarchical sampling inspection for detecting alien pests inside Japan. We further explain the theory for declaring the eradication of invasive alien pests as an emergency control. Actual examples of quarantine actions against the invasion of plum pox virus disease and citrus huanglongbing are discussed.

Appendix: Derivation of the required sample size (Eq. 2) for the hypergeometric distribution

Let p be the proportion of infested units in a consignment of a size n. Let s be the number of units drawn from the consignment at random. Let Y be the number of infested units in the sample. The probability that the sample contains no infested units for a given p is given by the zero-term of a hypergeometric distribution.

$$\Pr (\left. {Y = 0} \right|p) = \frac{{\left( {\begin{array}{*{20}c} {np} \\ 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {n - np} \\ s \\ \end{array} } \right)}}{{\left( {\begin{array}{*{20}c} n \\ s \\ \end{array} } \right)}} = \frac{(n - np)!(n - s)!}{n!(n - s - np)!} .$$

(22)

We can rearrange the equation to yield Eq. 23 which is given by the multiplication of np components.

$$\begin{aligned} \Pr ( {Y = 0} |p) &= \frac{(n - np)!}{n!} \cdot \frac{(n - s)!}{(n - s - np)!} \\ &= \frac{1}{n(n - 1)(n - 2)\ldots(n - np + 1)}\frac{(n - s)(n - s - 1)(n - s - 2)\ldots(n - s - np + 1)}{1} \\ &= \left( {1 - \frac{s}{n}} \right)\left( {1 - \frac{s}{n - 1}} \right)\left( {1 - \frac{s}{n - 2}} \right)\ldots \left( {1 - \frac{s}{n - np + 1}} \right) \end{aligned}$$

(23)

We can interpret this expression by an intuitive manner. Let us attach a number on each infested unit sequentially. The first parenthesis in the right-hand side of Eq. 23 indicates the probability that the first infested unit is not included in the sample. The second parenthesis indicates the probability that the second infested unit is not included in the sample under the condition that the first infested unit is not included in the sample. The third parenthesis indicates the probability that the third infested unit is not included in the sample under the condition that the first and the second infested units are not included in the sample. Thus, Eq. 23 indicates the probability that none of the np units is included in the sample. The rearrangement in Eq. 23 may be called the f-binomial type rearrangement, because it is given by $\prod_{i = 1}^{np} \left( { 1 - f_{i} } \right)$, where f _i is the frequency of sample at the ith infested unit. If we replace the denominator in each parenthesis by the mean between the first denominator n and the last denominator n − (np − 1), the quantity becomes larger than or equal to the original quantity, because log _e (1 − (s/n)) is an upward-convex function of n, that is, the second derivative of log _e (1 − (s/n)) about n is negative. Thus, we obtain the following inequality.

$$\Pr (\left. {Y = 0} \right|p) \le \left( {1 - \frac{s}{{n - \frac{np - 1}{2}}}} \right)^{np} .$$

(24)

The equality holds if np = 1. We want to regulate the probability as Pr(Y = 0|p) ≤ β. Hence, we equate the right-hand side of inequality 24 to β. Then, the rearrangement yields the following sample size.

$$s = \left( {n - \frac{np - 1}{2}} \right)\left( {1 - \beta^{1/(np)} } \right) .$$

(25)

The actual number of infested units is an integer; a consignment is defined as an infested consignment if the number of infested units is equal to or larger than $\left\lceil {np} \right\rceil$, where $\lceil \; \rceil$ indicates the ceiling function. Hence we should express the equation by

$$s = \left\lceil {\left( {n - \frac{{\left\lceil {np} \right\rceil - 1}}{2}} \right)\left( {1 - \beta^{{1/(\left\lceil {np} \right\rceil )}} } \right)} \right\rceil .$$

(26)

The lot infested by the proportion p is not permitted in our risk management but it is permitted in ISO 2859-0. Hence Eq. 26 is slightly different from the corresponding formula given in ISO 2859-0 (ISO 1995, p 12).

In contrast to the f-binomial type rearrangement given by Eq. 23, we can alternatively consider another rearrangement which is given by Eq. 27.

$$\begin{aligned} \Pr ({Y = 0}|p) &= \frac{(n - s)!}{n!} \cdot \frac{(n - np)!}{(n - s - np)!} \\ &= \frac{1}{n(n - 1)(n - 2)\ldots (n - s + 1)}\frac{(n - np)(n - np - 1)(n - np - 2)\ldots (n - np - s + 1)}{1} \\ &= \left( {1 - \frac{np}{n}} \right)\left( {1 - \frac{np}{n - 1}} \right)\left( {1 - \frac{np}{n - 2}} \right)\ldots \left( {1 - \frac{np}{n - s + 1}} \right) \end{aligned}$$

(27)

This rearrangement may be called the p-binomial type rearrangement, because it is given by $\prod_{i = 1}^{s} \left( { 1 - p_{i} } \right)$, where p _i is the proportion of infested units at the ith sampled unit. Then, we obtain the inequality by a similar manner as Eq. 24.

$$\Pr (\left. {Y = 0} \right|p) \le \left( {1 - \frac{np}{{n - \frac{s - 1}{2}}}} \right)^{s}$$

(28)

This form of approximation was introduced by Cochran (1977) and later introduced by ISPM No. 31 (IPPC 2008). The approximation by the right-hand side of Eq. 28 is superior to that of Eq. 24 if the sample size is smaller than the number of infested units, i.e., if s < np. However, Eq. 28 seems not as useful as Eq. 24, because the sample size (s) is not given by an explicit form in Eq. 28.