Adaptive rectangular sampling: An easy, incomplete, neighbourhood-free adaptive cluster sampling design Section 4. DiscussionAdaptive rectangular sampling: An easy, incomplete, neighbourhood-free adaptive cluster sampling design Section 4. Discussion

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The adaptive rectangular sampling (ARS) design is an adaptive design that is easy to manage, saves on travel, is easy to calculate, is neighbourhood-free, and controls the final sample size.

The design is adaptive because the final sample size depends on observed values, and the design is able to find rare clustered events.

It is easy to manage because it is straightforward in determining the places that should be investigated for the additional sample. The design uses the intuitive behaviour of field biologists $–$ once they find a rare event, they want to search in the immediate neighbourhood. It is even easier than adaptive cluster sampling (ACS). In addition, ARS can perform in both a two-stage form and a one-stage form. With this design, unlike with adaptive two-stage sequential sampling (ATS), there is no need to worry about the size, location and shape of primary sampling units. Unlike ACS, incomplete ACS (IACS) and ATS, it is possible for ARS to indicate the entire potential sample that the samplers need to select before they start sampling.

As for the travel-saving feature of ARS, there is no difference between adaptive designs such as ACS, ARS and ATS in the first phase of the second stage, for selecting the initial sample. But, in the second phase, ARS travels between cells generally less than ACS and IACS (with its edge units) and, especially, much less than ATS and two-stage sampling, with equal sample sizes. Because of this feature, ARS would be appropriate for costly travelling surveys of clustered populations, regardless of its efficiency.

ARS is easy to calculate, because the inclusion probabilities for the final sample size are easy to calculate, and this means that the $\pi ‑$estimator can be used instead of Murthy’s estimator. Murthy’s estimator, equation 3.1, is strongly dependent on the size of the initial sample (and on estimating $q);$ as initial samples could be small in some situations, this is a weakness of Murthy’s estimator. Therefore, one of the advantages of ARS as a sequential design is its avoidance of Murthy’s estimator and its use of the $\pi ‑$estimator instead. In addition, calculating the $\pi ‑$estimator in IACS is not very easy, because it is a little complicated to estimate $“{\pi }_j”\mathrm{s}$ (see Chao and Thompson 1999). As discussed in Section 2.1, it is easy to calculate or estimate $“{\pi }_j”\mathrm{s}$ in ARS, compared with the method used by Chao and Thompson (1999).

The design is neighbourhood-free, in the sense that it does not follow the neighbourhood as in ACS and IACS; this would be complicated for the sampler after certain steps. ARS is an easy design for samplers, especially for difficult environments. ACS has not yet been used on a routine basis in field surveys for forest inventory and biodiversity monitoring, as there are also practical difficulties in field implementation (Yang et al. 2011). A design like ARS may be more appropriate in such environments.

The design controls the final sample size well with the choice of radius $R.$ This paper presents an easy version of ARS. ARS can be performed in different ways (e.g., someone could plan a design to sample around a cell instead of investigating all the cells around it). This is a suggestion for future work on ARS.

To use ARS, it is important to know that the population units are separated in a cluster form; otherwise, the design would waste the sample units. This is one of the disadvantages of ARS. An advantage of this design is its expansion of the definition of clusters. Because the designer can change the radius $R,$ a cluster in ARS consists of units that are around each other even at a distance, and there is no need for them to be adjacent.

Compared with other designs, ARS has some of the same advantages. Like ACS, it takes advantage of clustering to find rare events; like ATS, it does not need to follow the neighbourhood. And, like IACS, it controls the final sample size well.

ARS is a new design, and it should be evaluated on real populations to enable researchers to find out its abilities, advantages and disadvantages.

Appendix

For ${\pi }_{hj},$ it is easy to see that

$${\pi }_{hj}\mathrm{<mo>=</mo>}P\left(B_{hj}\right)\mathrm{<mo>= </mo>}P\left({\cup }_{k\in s_{hj}}{A}_{hk}\right)\mathrm{<mo>= </mo>\; 1}-P\left({\cap }_{k\in s_{hj}}{A}_{hk}^{{}^{\prime }}\right)\mathrm{<mo>= </mo>\; 1}-\frac{\left(\begin{array}{c}{N}_h-f_{hj}\\ n_{1h}\end{array}\right)}{\left(\begin{array}{c}{N}_h\\ n_{1h}\end{array}\right)}\mathrm{,}$$

and, for ${\pi }_{hj{j}^{\prime }},$ using the fundamental probability principle,

$$\begin{array}{ll}{\pi }_{hj{j}^{\prime }}\hfill & \mathrm{<mo>=</mo>}P\left(B_{hj}\cap B_{h{j}^{\prime }}\right)\mathrm{<mo>=</mo>\; 1}-P{\left(B_{hj}\cap B_{h{j}^{\prime }}\right)}^{\prime }\mathrm{<mo>=</mo>\; 1}-P\left(B_{hj}^{{}^{\prime }}\cup B_{h{j}^{\prime }}^{{}^{\prime }}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>\; 1}-\left[P\left(B_{hj}^{{}^{\prime }}\right)+P\left(B_{h{j}^{\prime }}^{\prime }\right)-P\left(B_{hj}^{{}^{\prime }}\cap B_{h{j}^{\prime }}^{\prime }\right)\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>\; 1}-\left[1-P\left({\cup }_{k\in s_{hj}}{A}_{hk}\right)+1-P\left({\cup }_{k\in s_{h{j}^{\prime }}}{A}_{hk}\right)-\left(1-P\left({\cup }_{k\in s_{hj{j}^{\prime }}}{A}_{hk}\right)\right)\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>\; 1}-\frac{\left(\begin{array}{c}{N}_h-f_{hj}\\ n_{1h}\end{array}\right)}{\left(\begin{array}{c}{N}_h\\ n_{1h}\end{array}\right)}-\frac{\left(\begin{array}{c}{N}_h-f_{h{j}^{\prime }}\\ n_{1h}\end{array}\right)}{\left(\begin{array}{c}{N}_h\\ n_{1h}\end{array}\right)}+\frac{\left(\begin{array}{c}{N}_h-f_{hj{j}^{\prime }}\\ n_{1h}\end{array}\right)}{\left(\begin{array}{c}{N}_h\\ n_{1h}\end{array}\right)}.\hfill \end{array}$$

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