Effective Sample Size for Line Transect Sampling Models with an Application to Marine Macroalgae

  • Jonathan Acosta
  • Felipe Osorio
  • Ronny Vallejos


This paper provides a framework for estimating the effective sample size in a spatial regression model context when the data have been sampled using a line transect scheme and there is an evident serial correlation due to the chronological order in which the observations were collected. We propose a linear regression model with a partially linear covariance structure to address the computation of the effective sample size when spatial and serial correlations are present. A recursive algorithm is described to separately estimate the linear and nonlinear parameters involved in the covariance structure. The kriging equations are also presented to explore the kriging variance between our proposal and a typical spatial regression model. An application in the context of marine macroalgae, which motivated the present work, is also presented.


Effective sample size Line transects Spatial association ARMA processes Marine macroalgae 



Ronny Vallejos was partially supported by Fondecyt Grant 1120048, Chile, AC3E Grant FB-0008, and USM Grant 12.15.09. Felipe Osorio was partially supported by FONDECYT Grant 1140580. Jonathan Acosta was partially supported by PIIC at UTFSM, Chile. The authors are indebted to Luis Aris, Luis Figueroa, and Carlos Cortés from IFOP for providing the macroalgae dataset and for helpful discussions. The authors are also grateful to Diego Alvarez from UTFSM for providing preliminary computational results regarding the macroalgae dataset. In addition, the authors would like to thank Dr. Emilio Porcu at UTFSM for his constant support. The authors acknowledge the suggestions from two anonymous referees and an associate editor and the editor of JABES that improved the manuscript.


  1. Anderson, T. W. (1973), Asymptotically efficient Estimation of the covariance matrices with linear structure, The Annals of Statistics 1: 135–141.Google Scholar
  2. Banerjee, S., Carlin, B., and Gelfand, A. (2004), Hierarchical Modeling and Analysis for Spatial Data, Chapman & Hall, London.Google Scholar
  3. Barndorff-Nielsen, O., Kent, J., Sørensen, M. (1982). Normal variance-mean mixtures and \(z\) distributions. International Statistical Review 50: 145–159.Google Scholar
  4. Box, G. (1954a), Some Theorems on quadratic forms applied in the study of analysis of variance problems. I. Effect of inequality of variance in the one–way classification, Annals of Mathematical Statistics 25: 290–302.Google Scholar
  5. ——– (1954b), Some theorems on quadratic forms applied in the study of analysis of variance problems. II. Effects of inequality of variance and of correlation between errors in the two–way classification, Annals of Mathematical Statistics 25: 484–98.Google Scholar
  6. Box, G. E. P., and Cox, D. R. (1964), An analysis of transformations, Journal of the Royal Statistical Society. Series B 26: 211–252.Google Scholar
  7. Brockwell, P., and Davis, R. (2006), Time Series: Theory and Methods, Springer, New York.Google Scholar
  8. Clifford, P., Richardson, S., and Hémon, D. (1989), Assessing the significance of the correlation between two spatial processes, Biometrics 45: 123–34.Google Scholar
  9. Cressie, N. (1993), Statistics for Spatial Data, Wiley, New York.Google Scholar
  10. Cressie, N., and Lahiri, S. N. (1996), Asymptotics for REML estimation of spatial covariance parameters, Journal of Statistical Planning and Inference 50: 327–341.Google Scholar
  11. Crujeiras, R. M., and Van Keilegom I. (2010), Least squares estimation of non linear spatial trends, Computational Statistics and Data Analysis 54: 452–465.Google Scholar
  12. Dale, M. R. T., and Fortin M-J. (2009), Spatial autocorrelation and statistical tests: some solutions, Journal of the Agricultural, Biological, and Environmental Statistics 14: 188–206.Google Scholar
  13. Dutilleul, P. (1993), Modifying the \(t\) test for assessing the correlation between two spatial processes, Biometrics 49: 305–314.Google Scholar
  14. Faes, C., Molenberghs, G., Aerpts, M., Verbeke, G., and Kenward, M. (2009), Effective sample size and an alternative small-sample degrees-of-freedom method, The American Statistician 63: 389–399.Google Scholar
  15. Field, C., and Genton, M. G. (2006). The multivariate \(g\) distribution, Technometrics 48: 104–111.Google Scholar
  16. Griffith, D. (2005), Effective geographic sample size in the presence of spatial autocorrelation, Annals of the Association of American Geographers 95: 740–760.Google Scholar
  17. Headrick, T. C., Kowalchuk, R. K., and Sheng, Y. (2008). Parametric probability densities and distribution functions for Tukey \(g\) transformations and their use for fitting data, Applied Mathematical Sciences 2: 449–462.Google Scholar
  18. Hedley, S. L., and Buckland, S. T. (2004), Spatial Models for line transect sampling, Journal of the Agricultural, Biological, and Environmental Statistics 9: 181–199.Google Scholar
  19. Kutner, M., Nachtsheim, C., Neter, J., and Li, W. (2004), Applied Linear Statistical Models, McGraw-Hill/Irwin, Homewood, IL.Google Scholar
  20. Mardia, K. V., and Marshall, R. J. (1984), Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika 71: 135–146.Google Scholar
  21. Protassov, R. S. (2004). EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed \(\lambda \), Statistics and Computing 14: 67–77.Google Scholar
  22. R Core Team (2015). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.
  23. Rasmussen, C. E., and William, C. K. I. (2006), Gaussian Processes for Machine Learning, The MIT Press, Massachusetts.Google Scholar
  24. Ribeiro, P. J., Diggle, P. J. (2015). geoR: Analysis of Geostatistical Data. R package version 1.7-5.1.
  25. Rubin, D. B., Szatrowski, T. H. (1982), Finding maximum likelihood estimates of patterned covariance matrices by the EM algorithm, Biometrika 69: 657–660.Google Scholar
  26. Szatrowski, T. H. (1980), Necessary and sufficient conditions for explicit solutions in the multivariate normal estimation problem for patterned means and covariances, The Annals of Statistics 8: 802–810.Google Scholar
  27. Vallejos, R., and Osorio, F. (2014), Effective sample size of spatial process models, Spatial Statistics 9: 66–92.Google Scholar
  28. Vásquez, J., B., and Santelices, J. A. (1990), Ecological effects of harvesting Lessonia (Laminariales, Phaeophyta) in central Chile, Hydrobiología 204/205: 41–47.Google Scholar

Copyright information

© International Biometric Society 2016

Authors and Affiliations

  • Jonathan Acosta
    • 1
  • Felipe Osorio
    • 2
  • Ronny Vallejos
    • 1
  1. 1. Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.Instituto de EstadísticaPontificia Universidad Católica de ValparaísoValparaísoChile

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