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A gamma-shaped detection function for line-transect surveys with mark-recapture and covariate data

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Abstract

We have developed a procedure for estimating animal population size from aerial survey data collected simultaneously by two observers on the same sighting platform. We used a line transect sample design where transects follow elevation contours in mountainous terrain. Because our 10 data sets from aerial line transect surveys, conducted over a terrestrial environment, consistently show unimodal detection shapes, we chose a gamma-shaped detection function that is unimodal and admits covariates. We fit models separately to data from each observer, and then used all of the data to estimate the probabilities at the apex of the detection curves. We used a Horvitz-Thompson estimator to estimate the population size. We illustrate our procedure on a recently collected brown bear data set.

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References

  • Akaike, H. (1985), “Prediction and Entrophy,” in A Celebration of Statistics, eds. A. C. Atkinson and S. E. Finberg, Berlin: Springer-Verlag, pp. 1–24.

    Google Scholar 

  • Aldredge, J. R., and Gates, C. E. (1985), “Line Transect Estimators for Left-Truncated Distributions,” Biometrics, 41, 273–280.

    Article  Google Scholar 

  • Alpiza-Jara, R., and Pollock, K. H. (1996), “A Combination of Line Transect and Capture-Recapture Sampling Model for Multiple Observers in Aerial Surveys,” Journal of Ecological and Environmental Statistics, 3, 311–327.

    Article  Google Scholar 

  • Borchers, D. L., and Burnham, K. P. (2004), “General Formulation for Distance Sampling,” in Advanced Distance Sampling, Estimating Abundance of Biological Populations, eds. S. T. Buckland, D. R. Anderson, K. P. Burnham, J. L. Laake, D. L. Borchers, and L. Thomas, Oxford: Oxford University Press, pp. 6–30.

    Google Scholar 

  • Borchers, D. L., Buckland, S. T., Goedhart, P. W., Clarke, E. D., and Hedley, S. L. (1998a), “Horwitz-Thompson Estimators for Double Platform Line Transect Surveys,” Biometrics, 54, 1221–1237.

    Article  MATH  Google Scholar 

  • Borchers, D. L., Laake, J. L., Southwell, C., and Paxton, C. G. M. (2006), “Accommodating Unmodeled Heterogeneity in Double-Observer Distance Sampling Surveys,” Biometrics, 62, 372–378.

    Article  MathSciNet  Google Scholar 

  • Borchers, D. L., Zucchini, W., and Fewster, R. M. (1998b), “Mark Recapture Models for Line Transect Surveys,” Biometrics, 54, 1207–1220.

    Article  MATH  Google Scholar 

  • Buckland, S. T., and Turnock, B. J. (1992), “A Robust Line Transect Method,” Biometrics, 48, 901–909.

    Article  Google Scholar 

  • Buckland, S. T., Anderson, D. R., Burnham, K. P., Laake, J. L., Borchers, D. L., and Thomas, L. (2001), Introduction to Distance Sampling, Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Burnham, K. P., Anderson, D. R., and Laake, J. L. (1980), “Estimation of Density From Line Transect Sampling of Biological Populations,” Wildlife Monograph 72, supplement to Journal of Wildlife Management, 44.

  • Burnham, K. P., Buckland, S. T., Laake, J. L., Borchers, D. L., Marques, T. A., Bishop, J. R. B., and Thomas, L. (2004), “Further Topics in Distance Sampling,” in Advanced Distance Sampling, Estimating Abundance of Biological Populations, eds. S. T. Buckland, D. R. Anderson, K. P. Burnham, J. L. Laake, D. L. Borchers, and L. Thomas, Oxford: Oxford University Press, pp. 307–392.

    Google Scholar 

  • Chen, S. X. (1999), “Estimation in Independent Observer Line Transect Surveys for Clustered Populations,” Biometrics, 55, 754–759.

    Article  MATH  Google Scholar 

  • Conover, W. J. (1999), Practical Nonparametric Statistics (3rd ed.), New York: Wiley.

    Google Scholar 

  • Drummer, T. D., and McDonald, L. L. (1987), “Size Bias in Line Transect Sampling,” Biometrics, 43, 13–21.

    Article  MATH  Google Scholar 

  • Drummer, T. D., DeGrange, A. R., Pank, L. L., and McDonald, L. L. (1990), “Adjusting for Group Size Influence in Line Transect Sampling,” Journal of Wildlife Management, 54, 511–514.

    Article  Google Scholar 

  • Efron, B., and Tibshirani, R. J. (1993), An Introduction to the Bootstrap, New York: Chapman & Hall.

    MATH  Google Scholar 

  • Hiby, A. R., and Lovell, P. (1998), “Using Aircraft in Tandem Formation to Estimate the Abundance of Harbour Porpoise,” Biometrics, 54, 1280–1289.

    Article  MATH  Google Scholar 

  • Hiby, L. (1999), “The Objective Identification of Duplicate Sightings in Aerial Survey for Porpoise,” in Marine Mammal Survey and Assessment Methods, eds. G. W. Garner, S. C. Amstrup, J. L. Laake, B. F. J. Manly, L. L. McDonald, and D. G. Robertson, Rotterdam: A. A. Balkema, pp. 179–189.

    Google Scholar 

  • Hiby, L., and Krishna, M. B. (2001), “Line Transect Sampling From a Curved Path,” Biometrics, 57, 727–731.

    Article  MathSciNet  Google Scholar 

  • 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, 663–685.

    Article  MATH  MathSciNet  Google Scholar 

  • Innes, S., Heide-Jorgensen, M. P., Laake, J. L., Laidre, K. L., Cleator, H. P., Richard, P., and Stewart, R. E. A. (2002), “Surveys of Belugas and Narwhals in the Canadian High Arctic in 1996,” NAMMCO Scientific Publications, 4, 169–190.

    Google Scholar 

  • Johnson, B. K., Lindzey, F. G., and Guenzel, R. J. (1991), “Use of Aerial Line Transect Surveys to Estimate Pronghorn Populations in Wyoming,” Wildlife Society Bulletin, 19, 315–321.

    Google Scholar 

  • Laake, J. L. (1999), “Distance Sampling With Independent Observers: Reducing Bias From Heterogeneity by Weakening the Conditional Independence Assumption,” in Marine Mammal Survey and Assessment Methods, eds. G. W. Garner, S. C. Amstrup, J. L. Laake, B. F. J. Manly, L. L. McDonald, and D. G. Robertson, Rotterdam: Balkema Press, pp. 137–148.

    Google Scholar 

  • Laake, J. L., and Borchers, D. L. (2004), “Methods for Incomplete Detection at Distance Zero,” in Advanced Distance Sampling, Estimating Abundance of Biological Populations, eds. S. T. Buckland, D. R. Anderson, K. P. Burnham, J. L. Laake, D. L. Borchers, and L. Thomas, Oxford: Oxford University Press, pp. 108–189.

    Google Scholar 

  • Manly, B. F., McDonald, L. L., and Garner, G. W. (1996), “The Double Count Method With Two Independent Observers,” Journal of Ecological and Environmental Statistics, 1, 170–189.

    MathSciNet  Google Scholar 

  • Marques, F. F. C., and Buckland, S. T. (2003), “Incorporating Covariates Into Standard Line Transect Analyses,” Biometrics, 35, 597–604.

    MathSciNet  Google Scholar 

  • Patil, G. P., Taillie, C., and Wigley, R. L. (1979a), “Transect Sampling Methods and Their Application to Deep-Sea Red Crab,” in Environmental Biomonitoring, Assessment, Prediction, and Management#x2014;Certain Case Studies and Related Quantitative Issues, eds. J. Cairns, Jr., G. P. Patil, and W. E. Waters, Fairland, MD: International Co-Operative Publishing House, pp. 51–75.

    Google Scholar 

  • Patil, S. A., Burnham, K. P., and Kovner, J. L. (1979b), “Nonparametric Estimation of Plant Density by the Distance Method,” Biometrics, 35, 597–604.

    Article  Google Scholar 

  • Quang, P. X. (1991), “A Nonparametric Approach to Size-Biased Line Transect Sampling,” Biometrics, 47, 269–279.

    Article  MathSciNet  Google Scholar 

  • Quang, P. X., and Becker, E. F. (1996), “Line Transect Sampling Under Varying Conditions With Application to Aerial Surveys,” Ecology, 77, 1297–1302.

    Article  Google Scholar 

  • — (1997), “Combining Line Transect and Double Count Sampling Techniques for Aerial Surveys,” Journal of Agricultural, Biological, and Environmental Statistics, 2, 230–242.

    Article  MathSciNet  Google Scholar 

  • — (1999), “Aerial Survey Sampling of Contour Transects Using Double-Count and Covariate Data,” in Marine Mammal Survey and Assessment Methods, eds. G. W. Garner, S. C. Amstrup, J. L. Laake, B. F. J. Manly, L. L. McDonald, and D. G. Robertson, Rotterdam: Balkema Press, pp. 87–97.

    Google Scholar 

  • Quang, P. X., and Lanctot, R. B. (1991), “A Line Transect Model for Aerial Surveys,” Biometrics, 47, 1089–1102.

    Article  Google Scholar 

  • R Development Core Team (2008), R: A Language and Environment for Statistical Computing, Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0. Available at http://www.R-project.org.

    Google Scholar 

  • Ramsey, F. L., Wildman, V., and Engbring, J. (1987), “Covariate Adjustments to Effective Area in Variable-Area Wildlife Surveys,” Biometrics, 43, 1–11.

    Article  Google Scholar 

  • S-PLUS Version 6.2 (2003), Insightful Corporation, Seattle, WA.

  • Schwartz, C. C., Miller, S. D., and Haroldson, M. A. (2003), “Grizzly Bear (Ursus arctos),” in Wild Mammals of North America; Biology, Management, and Conservation, eds. G. A. Feldmamer, B. C. Thompson, and J. A. Chapman, Baltimore, MD: John Hopkins University Press, pp. 556–586.

    Google Scholar 

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Authors and Affiliations

  1. Alaska Department of Fish and Game, Division of Wildlife Conservation, 99518, Anchorage, AK

    E. F. Becker (Biometrician)

  2. Department of Mathematical Sciences, University of Alaska, 99775, Fairbanks, AK

    P. X. Quang (Professor Emeritus)

Authors
  1. E. F. Becker
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  2. P. X. Quang
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Correspondence to E. F. Becker.

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Becker, E.F., Quang, P.X. A gamma-shaped detection function for line-transect surveys with mark-recapture and covariate data. JABES 14, 207–223 (2009). https://doi.org/10.1198/jabes.2009.0013

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  • DOI: https://doi.org/10.1198/jabes.2009.0013

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