Abstract
Consider a survey of a plant or animal species in which abundance or presence/absence will be recorded. Further assume that the presence of the plant or animal is rare and tends to cluster. A sampling design will be implemented to determine which units to sample within the study region. Adaptive cluster sampling designs Thompson (1990) are sampling designs that are implemented by first selecting a sample of units according to some conventional probability sampling design. Then, whenever a specified criterion is satisfied upon measuring the variable of interest, additional units are adaptively sampled in neighborhoods of those units satisfying the criterion. The success of these adaptive designs depends on the probabilities of finding the rare clustered events, called networks. This research uses combinatorial generating functions to calculate network inclusion probabilities associated with a simple Latin square sample. It will be shown that, in general, adaptive simple Latin square sampling when compared to adaptive simple random sampling will (i) yield higher network inclusion probabilities and (ii) provide Horvitz-Thompson estimators with smaller variability.
Similar content being viewed by others
References
Bellhouse, D.R. (1977) Some optimal designs for sampling in two dimensions. Biometrika, 64, 605–11.
Borkowski, J.J. (1996) Simple Latin Square Sampling + k Designs. Technical Report No. 6–1–96, Dept. of Math. Sci., Montana State University.
Buckland, S.T., Anderson, D.R., Burnham, K.P., and Laake, J.L. (1993) Distance Sampling: Estimating Abundance of Biological Populations, Chapman and Hall, London.
Cochran, W.G. (1977) Sampling Techniques, Wiley, New York.
Gilbert, R.O. (1987) Statistical Methods for Environmental Pollution Monitoring, Van Nostrand Reinhold, New York.
Hansen, M.H., Hurwitz, W.N., and Madow, W.G. (1953) Sample Survey Methods and Theory—Volume 1, Wiley, New York.
Heyadat, A.S. and Sinha, B.K. (1991) Design and Inference in Finite Population Sampling, Wiley, New York.
Horvitz, D.G. and Thompson, D.J. (1952) A Generalization of Sampling Without Replacement From a Finite Universe. Journal of the American Statistical Association, 47, 663–85.
Jessen, R.J. (1975) Square and Cubic Lattice Sampling. Biometrics, 31, 449–71.
Kish, L. (1965) Survey Sampling, Wiley, New York.
MAPLE V Software, Release 2 (1993) © by the University of Waterloo. Waterloo, Ontario.
Munholland, P.L. and Borkowski, J.J. (1993) Adaptive Latin Square Sampling + 1 Designs. Technical Report No. 3–23–93, Dept. of Math. Sci., Montana State University.
Munholland, P.L. and Borkowski, J.J. (1996) Latin Square Sampling + 1 Designs. To appear in Biometrics, March 1996.
Patterson, H.D. (1954) The errors of lattice sampling. Journal of the Royal Statistical Society, Ser. B, 16, 140–9.
Raj, D. (1968) Sampling Theory, McGraw-Hill, New York.
Riordan, J. (1964) Generating Functions. In Applied Combinatorial Mathematics, E.F. Beckenbach (Ed.) 67–93, Wiley, New York.
Schaeffer, R.L., Mendenhall, W., and Ott, L. (1986) Elementary Survey Sampling, Duxbury Press, Boston.
Seber, G.A.F. (1982) The Estimation of Animal Abundance, Macmillan, New York.
Seber, G.A.F. (1986) A Review of Estimating Animal Abundance. Biometrics, 42, 267–92.
Seber, G.A.F. (1992) A Review of Estimating Animal Abundance II. International Statistical Review, 60, 129–66.
Thompson, S.K. (1990) Adaptive Cluster Sampling. Journal of the American Statistical Association, 85, 1050–9.
Thompson, S.K. (1991a) Adaptive Cluster Sampling: Designs with Primary and Secondary Units. Biometrics, 47, 1103–15.
Thompson, S.K. (1991b) Stratified Adaptive Cluster Sampling. Biometrika, 78, 389–97.
Thompson, S.K. (1992) Sampling, Wiley, New York.
Thompson, S.K., Ramsey, F.L., and Seber, G.A. (1992) An Adaptive Procedure for Sampling Animal Populations. Biometrics, 48, 1195–9.
Thompson, S.K. and Seber, G.A.F. (1994) Detectability in Conventional and Adaptive Sampling. Biometrics, 50, 712–24.
Thompson, S.K. and Seber, G.A.F. (1996) Adaptive Sampling, Wiley, New York.
Townsend, M. (1987) Discrete Mathematics: Applied Combinatorics and Graph Theory, Benjamin/Cummings, Menlo Park, CA.
Tucker, A. (1980) Applied Combinatorics, Wiley, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Borkowski, J.J. Network inclusion probabilities and Horvitz-Thompson estimation for adaptive simple Latin square sampling. Environmental and Ecological Statistics 6, 291–311 (1999). https://doi.org/10.1023/A:1009635530700
Issue Date:
DOI: https://doi.org/10.1023/A:1009635530700