Converging Classes of Model and Designbased Spatial Samples

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Converging Classes of Model- and Design-based
Spatial Samples

• Cynthia Cooper
• Don Stevens
• OSU Statistics

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Duality in Environmental Monitoring Hypothetical
Example OCN Coho Spawner Densities

• Design-based Estimates of status and trend
  current year
• Generalized difference estimator uses inclusion
  probability densities
• No attempt to model underlying stochastic process
  describing spawner returns through time
• Model-based predictions
• (Hypothetically) Spawner densities at OC basins
  predicted by previous years ocean and basin
  stream-flow conditions and previous generations
  densities
• Likelihood-based approaches to estimating
  parameters that specify the stochastic model
• Conditional on the observed data no attempt to
  quantify potential bias no use of sampling
  design probability density functions

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Outline

• Design- vs. model-based approaches
• Examples
• Issues
• Proposed approach to analyzing convergence
• Hypothetical illustration

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Spatial Samples

• Design-based
• Probability samples
• Basis for long-run frequency properties
  (design-induced randomness)
• Fixed Y values
• Common objective
• Unbiased minimized-variance estimate of status of
  population
• Model-based
• Y values are random variables generated by a
  stochastic process
• Common objectives
• Estimate parameters of stochastic model
• Predict values of Y for locations or region
• Conditioned on values observed in sample
• No accounting for sample design

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Examples

• Design-based
• EPA EMAP
• ODFW Monitoring Plan Augmented Rotating Panel
• USFS Forest Inventory and Analysis
• Model-based
• Mining surveys
• Soil and hydrology surveys
• Judgement Sampling
• Aquatic Resource monitoring reported in 305b NWQI
  reports

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Discussions / controversies

• Selection bias
• S.A. Peterson, N. S. Urquhart, and E. B. Welsh
  (1999)
• Paulsen, S.G., R. M. Hughes, and D. P. Larsen
  (1998)
• Thompson, S.K. (2002)
• Lack of independence
• Rao, J.N.K., Bellhouse, D.R., (1990)
• Smith, T.M.F. (1984)
• Unbalanced data
• Royall, R.M., Cumberland, W.G. (1985)
• Robustness of model assumptions
• Hansen, M.H., Madow, W.G., Tepping, B.J. (1983)
• de Gruijter, J.J., ter Braak, C.J.F., (1990)

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Question

• When/how would model-based sampling be treated as
  design-based sampling?
• What optimality/performance properties would the
  analysis have?
• What is required for a model-based or
  design-based sample to be interchangeable?
• Optimizing sample designs simultaneously for
  design- model- unbiasedness and minimized
  model-average MSE
• Godambe, V.P., Thompson, M.E. (1986)
• Bellhouse, D.R. (1977)

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Analyzing convergence of sample classes

• Design-based sampling strategies are defined by
  the joint and marginal inclusion probabilities
  (or densities), plus an estimator
• Cassel, C-M, Särndal C-E, Wretman J.H. (1977)
• Model-based restricted random sampling achieves
  balance and/or adequate point-pair distances in
  distance bins
• Given restricted sampling, what are the
  characteristics of the inclusion probability
  densities?
• What restrictions on inclusion probability (1st
  or 2nd order) give a sample that is near-optimal
  according to (arbitrary) model-based criteria,
  (e.g., minimize kriging variance)?

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Hypothetical model-based/design-based relationship

• Constrained randomization to achieve good point
  distribution for estimating spatial covariance
  function
• Warrick-Myers (1987)
• Summary of WM model-based sampling strategy
• Specify desired number ni of point-pairs for each
  distance bin
• Select a sample (from many random samples) which
  minimizes (among the samples generated) a sum of
  squared differences between achieved and desired
  distribution of point-pair distances

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Hypothetical model-based/design-based relationship

• Evaluating resultant inclusion probability
  densities
• Sample size n k distance bins (n(n-1)/2 (n1
  n2 nk) )
• Assume isotropic case
• Spatial extent described by area R (R)
• where bin number i determined by s1-s2

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Hypothetical model-based/design-based relationship

• Sampling strategy minimizes mean-square
  prediction error (Kriging variance) model-based
  result
• One can base an estimator of regional total (or
  mean) on the inclusion probability densities
• Design-based estimate
• Agreement of estimates may depend on local v.
  regional estimates
• Brus, D.J., de Gruijter, J.J. (1993)
• When the denser clusters of points of the sample
  coincide with greater variability in Y in the
  random field, the variance of the design-based
  estimate would be reduced.

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The research described in this presentation has
been funded by the U.S. Environmental Protection
Agency through the STAR Cooperative Agreement
CR82-9096-01 National Research Program on
Design-Based/Model-Assisted Survey Methodology
for Aquatic Resources at Oregon State
University. It has not been subjected to the
Agency's review and therefore does not
necessarily reflect the views of the Agency, and
no official endorsement should be inferred