Journal of Agricultural, Biological, and Environmental Statistics

, Volume 12, Issue 4, pp 450–469

Spatial designs and properties of spatial correlation: Effects on covariance estimation

  • Kathryn M. Irvine
  • Alix I. Gitelman
  • Jennifer A. Hoeting

DOI: 10.1198/108571107X249799

Cite this article as:
Irvine, K.M., Gitelman, A.I. & Hoeting, J.A. JABES (2007) 12: 450. doi:10.1198/108571107X249799


In a spatial regression context, scientists are often interested in a physical interpretation of components of the parametric covariance function. For example, spatial covariance parameter estimates in ecological settings have been interpreted to describe spatial heterogeneity or “patchiness” in a landscape that cannot be explained by measured covariates. In this article, we investigate the influence of the strength of spatial dependence on maximum likelihood (ML) and restricted maximum likelihood (REML) estimates of covariance parameters in an exponential-with-nugget model, and we also examine these influences under different sampling designs—specifically, lattice designs and more realistic random and cluster designs—at differing intensities of sampling (n=144 and 361). We find that neither ML nor REML estimates perform well when the range parameter and/or the nugget-to-sill ratio is large—ML tends to underestimate the autocorrelation function and REML produces highly variable estimates of the autocorrelation function. The best estimates of both the covariance parameters and the autocorrelation function come under the cluster sampling design and large sample sizes. As a motivating example, we consider a spatial model for stream sulfate concentration.

Key Words

Effective rangeExponential covarianceIn-fill asymptoticsNugget-to-sill ratio

Copyright information

© International Biometric Society 2007

Authors and Affiliations

  • Kathryn M. Irvine
    • 1
  • Alix I. Gitelman
    • 2
  • Jennifer A. Hoeting
    • 3
  1. 1.Department of Mathematical SciencesMontana State UniversityBozeman
  2. 2.Department of StatisticsOregon State UniversityCorvallis
  3. 3.Department of StatisticsColorado State UniversityFort Collins