Environmental and Ecological Statistics

, Volume 22, Issue 4, pp 725–737 | Cite as

Distance sampling with a random scale detection function

  • Cornelia S. Oedekoven
  • Jeffrey L. Laake
  • Hans J. Skaug


Distance sampling was developed to estimate wildlife abundance from observational surveys with uncertain detection in the search area. We present novel analysis methods for estimating detection probabilities that make use of random effects models to allow for unmodeled heterogeneity in detection. The scale parameter of the half-normal detection function is modeled by means of an intercept plus an error term varying with detections, normally distributed with zero mean and unknown variance. In contrast to conventional distance sampling methods, our approach can deal with long-tailed detection functions without truncation. Compared to a fixed effect covariate approach, we think of the random effect as a covariate with unknown values and integrate over the random effect. We expand the random scale to a mixed scale model by adding fixed effect covariates. We analyzed simulated data with large sample sizes to demonstrate that the code performs correctly for random and mixed effect models. We also generated replicate simulations with more practical sample sizes (\(^{\sim }100\)) and compared the random scale half-normal with the hazard rate detection function. As expected each estimation model was best for different simulation models. We illustrate the mixed effect modeling approach using harbor porpoise vessel survey data where the mixed effect model provided an improved model fit in comparison to a fixed effect model with the same covariates. We propose that a random or mixed effect model of the detection function scale be adopted as one of the standard approaches for fitting detection functions in distance sampling.


Abundance estimation AD Model Builder Half-normal  Harbor porpoise detections Heterogeneity in detection probabilities  Mixed effects 



We thank Steve Buckland for reviewing a nearly final draft of the paper. Cornelia Oedekoven was supported by a studentship jointly funded by the University of St Andrews and EP-SRC, through the National Centre for Statistical Ecology (EP-SRC Grant EP/C522702/1). Hans Skaug thanks the Center for Stock Assessment Research for facilitating his visit to University of California, Santa Cruz.

Supplementary material

10651_2015_316_MOESM1_ESM.pdf (178 kb)
Supplementary material 1 (pdf 178 KB)


  1. Borchers D, Burnham K (2004) Advanced distance sampling. In: Buckland ST, Anderson DR, Burnham KP, Laake JL, Borchers DL, Thomas L (eds) General formulation for distance sampling. Oxford University Press, OxfordGoogle Scholar
  2. Buckland ST, Anderson DR, Burnham KP, Laake JL, Borchers DL, Thomas L (2001) Introduction to distance sampling. Oxford University Press, LondonGoogle Scholar
  3. Fournier DA, Skaug HJ, Ancheta J, Ianelli J, Magnusson A, Maunder MN, Nielsen A, Sibert J (2012) AD Model Builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models. Optim Methods Softw 27:233–249CrossRefGoogle Scholar
  4. Hedley SL, Buckland ST (2004) Spatial models for line transect sampling. J Agric Biol Environ Stat 9:181–199CrossRefGoogle Scholar
  5. Laake J, Borchers D, Thomas L, Miller D, Bishop J (2013) mrds: Mark-recapture distance sampling. R package version 2(1):4Google Scholar
  6. Marques FFC, Buckland ST (2003) Incorporating covariates into standard line transect analyses. Biometrics 53:924–935CrossRefGoogle Scholar
  7. Marques TA, Buckland ST, Borchers DL, Tosh D, McDonald RA (2010) Point transect sampling along linear features. Biometrics 66:1247–1255CrossRefPubMedGoogle Scholar
  8. Marques TA, Thomas L, Fancy SG, Buckland ST (2007) Improving estimates of bird densities using multiple covariate distance sampling. Auk 124:1229–1243CrossRefGoogle Scholar
  9. Miller DL, Thomas L (in press) Mixture models for distance sampling detection functions. PLoS OneGoogle Scholar
  10. Pledger (2000) Unified maximum likelihood estimates for closed capture–recapture models using mixtures. Biometrics 56:434–442CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Cornelia S. Oedekoven
    • 1
  • Jeffrey L. Laake
    • 2
  • Hans J. Skaug
    • 3
  1. 1.CREEM, School of Mathematics and StatisticsUniversity of St AndrewsSt AndrewsUK
  2. 2.National Marine Mammal Laboratory Alaska Fisheries Science CenterNOAASeattleUSA
  3. 3.Department of MathematicsUniversity of BergenBergenNorway

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