Adaptive rectangular sampling: An easy, incomplete, neighbourhood-free adaptive cluster sampling design Section 3. A real case study and simulationsAdaptive rectangular sampling: An easy, incomplete, neighbourhood-free adaptive cluster sampling design Section 3. A real case study and simulations

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In this section, adaptive rectangular sampling (ARS) is evaluated and compared with adaptive two-stage sequential sampling (ATS) and two-stage simple random sampling without replacement (TSS). Here, ARS is not compared with adaptive cluster sampling (ACS) for two reasons: first, Salehi and Smith (2005) compared ATS with two-stage ACS, and, second, it is not fair to compare ARS with ACS or even incomplete ACS because ACS needs to define, use and follow the neighbourhood, while ARS does not.

If ATS is a design free of neighbourhood, then ARS satisfies this condition too, because, if a sampler can recognize the border of the cells, or, in other words, can distinguish secondary sampling units (SSUs), the sampler can also recognize an SSU with its radius area. In addition, the area to be surveyed may be specified before samples are taken. Based on a map of the area, it is possible to use SRSWOR for the SSUs and the area around them if the SSUs satisfy the condition. Because the sampler need not return to the area to take the second phase of the sample (according to the ATS process), ARS seems to be easier and less costly than ATS. For a better comparison, the cost factor should be taken into consideration.

The comparison is done using two kinds of data: a real case study and simulation cases.

Here, efficiency is defined as

$$\text{eff}\left(\mathrm{.}\right)\mathrm{<mo>=</mo>}\frac{\text{MSE}\left({\overline{y}}_{\text{tss}}\right)}{\text{MSE}\left(\mathrm{.}\right)},$$

where ${\overline{y}}_{\text{tss}}$ is the conventional mean estimator in TSS, MSE stands for mean square error and “.” stands for one of the following:

• ${\widehat{\mu }}_{\text{ATS}}:$ Murthy’s estimator in ATS, which is unbiased, and which will be referred to as “ATS”. This estimator, for the mean of the $h^{\text{th}}$ primary sampling unit (PSU), would be presented as

$${\widehat{\mu }}_{\text{ATS}}\mathrm{<mo> = </mo>}{\widehat{q}}_1{\overline{y}}_{hc}+\left(1-{\widehat{q}}_1\right){\overline{y}}_{h{c}^{\prime }},(3.1)$$

• where ${\widehat{q}}_1$ is the proportion of the units that satisfy condition $C$ in the initial sample, and ${\overline{y}}_{hc}$ and ${\overline{y}}_{h{c}^{\prime }}$ are the means of the final-sample units satisfying condition $C$ and not satisfying condition $C,$ respectively, in the $h^{\text{th}}$ PSU. In this estimator, the first portion of the unit satisfying condition $C$ is estimated from the initial sample, and the respective value is adapted to the mean of the sample satisfying condition $C$ to construct the estimator.
• ${\widehat{\mu }}_{\text{ARS}}:$ the $\pi ‑$estimator in ARS, which is unbiased.
• ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}:$ the $\pi ‑$estimator in ARS, with $\pi$ estimated from the final sample using equation 2.3. It is not unbiased, and its relative bias is defined as $\text{Rbias}\mathrm{.}{\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}\mathrm{<mo>=</mo>}\left|\left({\mu }_y-{\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}\right)/{\mu }_y\right|.$

From now on, the acronym $“\text{ARS}”$ refers to both ${\widehat{\mu }}_{\text{ARS}}$ and ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}.$

Furthermore, two formulas are used for the error in estimating inclusion probabilities in ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}:$

• ${\widehat{\mu }}_{D\mathrm{.}Inclu}:$ this shows the mean of the difference between real inclusion probabilities and the respective estimation (i.e., the mean of $e_{hj}\mathrm{<mo>=</mo>}{\pi }_{hj}-{\widehat{\pi }}_{hj}$ for all the sample units)
• ${\widehat{\mu }}_{AD\mathrm{.}Inclu}:$ this shows the mean of the absolute difference between real inclusion probabilities and the respective estimation (i.e., the mean of $\left|e_{hj}\right|\mathrm{<mo>=</mo>}\left|{\pi }_{hj}-{\widehat{\pi }}_{hj}\right|$ for all the sample units).

A non-overlapping scheme is used in this section.

A real case study on a blue-winged teal population

Smith, Conroy and Brakhage (1995) used a population of blue-winged teal to evaluate ACS. The population comes from comprehensive counts, which were made from helicopters from December 13 to 15, 1992, in central Florida. The blue-winged teal population is extremely clustered, with a total of $N\mathrm{<mo>=</mo>\; 200}$ units (Figure 3.1). A simulation study found ACS to be efficient for this population, in the sense that the variance of the estimator is smaller than in simple random sampling (Smith, Conroy and Brakhage 1995).

The population was partitioned into $M\mathrm{<mo>=</mo>\; 8,4,2}$ PSUs. ARS, ATS and TSS were performed in the population with different values for $m,$ $n_{1h},$ $R$ and $d$ (a multiple in ATS that indicates the size of the additional sample in the second phase for units satisfying condition $C),$ with 25,000 simulations for each combination of values. For a fair comparison, $d$ was chosen in such a way that the expected final sample sizes for ATS and ARS were almost the same. For TSS, the sample size in each simulation was the same as that for ARS. The expected sample sizes were calculated using Monte Carlo simulations. It is notable that in ARS, if two or more adaptively added samples overlapped, the overlap was measured once. Practically, if there is overlap in the sample, the relevant cells must be sampled and measured only once.

Results are presented in Tables 3.1 and 3.2. For information about the MSEs of the estimators, $\text{MSE}\left({\overline{y}}_{\text{tss}}\right)$ is presented in the results. With this MSE and the efficiency of the estimators, the MSEs of the other estimators are easy to calculate. The results are noteworthy: ARS was better than ATS in all situations. ARS, unlike ATS, was also always more efficient than TSS. The efficiency of ARS was sometimes seven or eight times that of TSS, whereas this number was at most around two and a half for ATS. For more than 55% of the cases, the efficiency of ARS was greater than 2, whereas this was true of less than 5% of the cases for ATS.

The relative bias of ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}$ is acceptable for most of the cases; it may be unacceptable for a few cases with a little sample size. For around 61% of the cases, the relative bias was less than 0.03, and, for around 92% of them, the relative bias was less than 0.07.

Efficiency improved by increasing the radius $R,$ and a larger radius $R$ was proper for larger PSUs. In this population, there are two important clusters at the top of the population plot. With $R\mathrm{<mo>=</mo>\; 2},$ selecting one of the cells in a large PSU as the initial sample led to the selection of all of them. That is why $R\mathrm{<mo>=</mo>\; 2},$ with a large enough initial sample size, showed such significant efficiency.

In addition, the number of PSUs in the first stage was important, and the results indicate that more PSUs lead to efficiency improvements. As discussed before, the efficiency of ATS depends on the size, shape and location of the PSUs. When the population could not be partitioned into some empty and full PSUs, ATS was not as efficient (see populations 2 and 4). But as ARS uses the cluster form of the population, it is not as dependent on PSUs and could even perform in a population with one PSU, which would be meaningless for ATS.

In addition, for Population 1, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.025}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}=\text{0}\text{.002};$ for Population 2, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.023}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}=\text{0}\text{.005};$ for Population 3, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.024}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}\mathrm{<mo>=</mo>}-0.004;$ and, for Population 4, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.025}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}\mathrm{<mo>=</mo>}-\mathrm{0.008.}$ The mean of the inclusion probabilities for these simulations was around 0.22. According to ${\widehat{\mu }}_{D\mathrm{.}Inclu}$ and ${\widehat{\mu }}_{AD\mathrm{.}Inclu},$ the errors in estimating the inclusion probabilities seem to be almost negligible. This is why ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}$ was almost unbiased. The relative bias of ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}$ showed acceptable precision for $“{\widehat{\pi }}_j”\mathrm{s}.$

Artificial populations

The spatial pattern was generated with an R code following the Poisson cluster process (Brown 2003). The number of clusters was selected from a Poisson distribution, and cluster centres were randomly located throughout the site. Individuals within the cluster were located around the cluster centre at a random distance, following an exponential distribution, and in a random direction, following a uniform distribution. The parameters of the code were changed to generate three different populations. With the addition of the population in the example subsection of the paper, this subsection uses four artificial populations to evaluate ARS (see Figure 3.2):

• Population 5: rare and not clustered,
• Population 6: not rare, but clustered,
• Population 7: not rare and not clustered,
• Population 8: rare and clustered.

Results are presented in Tables 3.3 and 3.4. The relative bias of ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}$ was acceptable for all cases.

In Population 5, ATS was not efficient at all (except in one situation), but ARS performed well and was always more efficient than both ATS and TSS. ARS was better with $R\mathrm{<mo>=</mo>\; 1},$ relative to $R\mathrm{<mo>=</mo>\; 2},$ because the population was not clustered and a large radius could have wasted the sample. However, because of a large cluster in the lower-right PSU, ARS was efficient for $R\mathrm{<mo>=</mo>\; 2.}$

In Population 6, which was highly clustered but not rare, ATS was not as efficient as TSS in almost half of the cases, especially when the sample size was not very large. The performance of ARS was very good. Because of the large size of the clusters, ARS was better with $R\mathrm{<mo>=</mo>\; 2}$ than $R\mathrm{<mo>=</mo>\; 1},$ and, in one case, it was 12 times as efficient as TSS, while this number was 1.18 for ATS. With $R\mathrm{<mo>=</mo>\; 2},$ if a nonempty cell was selected as the initial sample, many of the other nonempty cells would also be selected. With a large enough initial sample size, almost all of them would be selected, providing almost complete information on the population and higher efficiency in comparison with other designs.

In Population 7, which was an almost-ordinary population, ARS and ATS were not efficient. In this case, the population was not clustered, and ARS wasted the sample searching around cells with a response satisfying condition $C$ that were almost all empty. In such situations, ATS is more efficient than ARS (this happened in some cases), since ATS spreads the additional sample size equally across all the PSUs.

Finally, in Population 8, which was rare and completely clustered, ATS was almost not efficient at all. Again, ARS performed very well; it was sometimes six times as efficient as TSS and ATS.

In addition, for Population 5, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.025}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}=-0.004;$ for Population 6, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.020}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}=-0.010;$ for Population 7, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.027}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.006};$ and, for Population 8, ${\widehat{\mu }}_{AD\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.016}$ and ${\widehat{\mu }}_{D\mathrm{.}Inclu}\mathrm{<mo>=</mo>\; 0.004.}$ The means of the inclusion probabilities for the four populations were, respectively, 0.21, 0.31, 0.24 and 0.34. Again, the errors in estimating the inclusion probabilities were almost negligible.

Lastly, ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}$ showed significant efficiency, even higher than ${\widehat{\mu }}_{\text{ARS}}$ (sometimes for a larger sample size). Since ${\widehat{\mu }}_{\text{ARS}}$ is unbiased, ${\widehat{\mu }}_{\text{ARS}}$ is preferred when there is enough information to calculate it. When information for calculating $“{\pi }_j”\mathrm{s}$ is lacking, ${\widehat{\mu }}_{\text{ARS}\mathrm{.}\widehat{\pi }}$ is a very good alternative for estimating the population mean with almost no bias.

Costs are not discussed, because this factor favours ARS, which is a cheaper design in comparison with ATS and TSS. Since ARS is more efficient than the other designs without considering costs, it is obvious that with costs factored in, the efficiency of ARS would be higher again. On the other hand, if the costs of much travelling under ATS and TSS are almost the same as the cost of searching more cells to find whether they satisfy condition $C$ (not measuring them exactly) to calculate the unbiased $\pi ‑$estimator, then the comparison is fair.

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