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*Ecological Archives*
A021-026-A3

### Ilana L. Abrahamson, Cara R. Nelson, and David L. R. Affleck. 2011. Assessing the performance of sampling designs for measuring the abundance of understory plants. *Ecological Applications* 21:452–464.

Appendix C. Proof of unbiasedness of the Strip Adaptive Cluster Sampling (SACS) Estimator.

Define the population of interest as the grid of *G* equal-sized cells and let *y*_{l} denote the attribute value for the *l*^{th} cell (*l* = 1, 2, …, *G*). An adaptive cluster sampling design divides this population into *K* mutually exclusive networks *N*_{k} (*k* = 1, 2, …, *K*) each comprised of *s*_{k} cells (S_{k} *s*_{k} = *G*) with attribute totals *z*_{k} = S *l*Î*N*_{k} *y*_{l} . For the derivations below it is useful to evenly distribute the network totals among component cells by defining the variable as the mean amount of attribute (*z*_{k} /*s*_{k}) for the network to which cell *l* belongs. The population mean is:

With SACS, the primary sampling unit is a contiguous 1 × *m* strip of cells with a fixed orientation. Let the random variable d_{k} denote the number of cells from the *k*^{th} network that are contained in the sample strip (if network *k* is not encountered along the strip then d_{k} = 0, otherwise d_{k} Î {1, 2, 3, …, *m*}). Provided that the location of the strip is selected uniformly at random, an unbiased estimator of µ following SACS is

| C.1 |

Unbiasedness is established as follows. First, given that the strip is located uniformly at random, the probability that the strip will fall in a given column *c* of the grid that defines the population is

| C.2 |

where *G*_{c} is the total number of cells in column *c* (S_{c }*G*_{c} = *G*). Second, conditional on the strip being located in column *c* the expected value of d_{k} is

| C.3 |

where *s*_{kc} is the number of cells in network *k* and also in column *c* (S_{c} *s*_{kc} = *s*_{k}). Strictly speaking, Eq. C.2 holds only for networks in the interior of the population or, for example, if the strip is reflected back upon itself when a population boundary is encountered (see Gregoire and Valentine 2008). Combining results Eq. C.1 and Eq. C.2 yields

and therefore

If multiple strips are located independently at random then the dispersion among strip-wise estimates can be used to unbiasedly estimate the variance of Eq. 1 and to construct confidence intervals for µ.

LITERATURE CITED

Gregoire, T. G., and H. T. Valentine. 2008. Sampling strategies for natural resources and the environment. Taylor and Francis Group, Boca Raton, Florida, USA.

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