Appendix F

Rare-Event Species: Inverse Sampling

Haldane (1945) (see also Cochran, 1977, Section 4.5) developed the inverse sampling method to estimate the proportion of individuals with a rare characteristic in a population. In a fisheries context, fish that are members of a so-called “rare-event” fish species could be considered members of a population (the population of all fish) with a rare characteristic (namely, belonging to the rare species). This appendix presents Handane’s method with applications to the management of rare fish species.

METHODOLOGY

Handane (1945) considers a situation in which there are two classes of sample elements, such as a rare-event species of fish and other, common species of fish. Fish are sampled (harvested) until m rare-event species fish are caught. Suppose that n fish (including both rare-event and common species together) must be caught to obtain m rare-event fish. Haldane (1945) shows that:

is an unbiased estimate of the true proportion P of rare-event fish in the population. Furthermore, by way of an infinite series expansion, Haldane shows to a very good approximation that the variance of p is given by:

which Cochran (1977, Section 4.5) shows is equivalent to:

which, unfortunately, depends on the unknown value of p.

However, the percent standard error (PSE) of p is given by:

which approaches the upper limit as p approaches zero.

Hence, by choosing m in advance, an upper limit on the value of PSE(p) can be obtained, and that value will be a very good approximation of PSE(p) for small values of p, which is the case for rare-event species. Examples of values of m and corresponding PSE(p) are presented in Table F.1.

Thus, to obtain an estimate of p with PSE(p) < 50 percent, one must wait for six rare-event species fish to be caught in a given fishery, and then p = (m – 1)/(n – 1) can be used to estimate the proportion of rare-event species in the total fish population. For PSE(p) < 30 percent, one must wait for 20 rare-event species to be caught.

APPLIED EXAMPLE: ESTIMATING THE PROPORTION OF A RARE-EVENT SPECIES IN A POPULATION

For example, suppose, at the beginning of the fishing season, m = 20 is chosen to achieve PSE(p) = 23.54 percent. Then, it is necessary to wait until 20 rare-event fish are caught in the fishery, and at that point in time, the total catch of the fishery, n, is noted. Suppose n = 10,000 fish. Then the unbiased estimate of the proportion of rare-event fish in the population is:

with a PSE(p) of 23.54 percent.

EXTENSION: ESTIMATING THE TOTAL POPULATION OF A RARE-EVENT SPECIES

In cases in which there is an unbiased estimate t (e.g., from a stock assessment) of the total number of fish (both rare-event and common species together) T in the population exploited by the

TABLE F.1 Examples of Values of m and Corresponding PSE(p)

fishery, and the variance of the estimate is var(t), then an unbiased estimate r of the total number of rare-event fish r in the population is given by:

with variance var(r) obtained from Goodman’s (1960) formula:

and

EXTENSION: MULTIPLE TYPES OF RARE-EVENT SPECIES

The formulas above can be generalized (Haldane, 1945) to the case of multiple types of rare-event species within the same population (e.g., multiple types of rare Grouper or Snapper species within a Grouper-Snapper complex).

EXTENSION: CLUSTER SAMPLING

For application to MRIP data, it may be necessary to extend Haldane’s method to situations involving cluster sampling (Cochran, 1977, Section 3.12). For example, a fishing trip may be considered a sample unit, and individual fish caught on the trip may be considered sample elements within that sample unit. Each fishing unit (trip) is a cluster of elements (fish). The elements (fish) are classified into two cases—rare-event and common species. Elements are likely to be clustered by unit in cases in which the spatial distribution of rare-event species is patchy, such that some units (trips) collect elements from locations where rare-event species are present, while other units (trips) collect elements from locations where such species are absent.

EXTENSION: SAMPLING WITHOUT REPLACEMENT

For cases in which the population of the rare-event species is thought to be so small that sampling without replacement may be appropriate, Espejo et al. (2008) provide some initial results toward extending Haldane’s method in this direction.

REFERENCES

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