1. Comparison of Variance Estimators for Two-dimensional, Spatially-structured Sample Designs. Don L. Stevens, Jr. Susan F. Hornsby* Department of Statistics Oregon State University

2. The research described in this presentation has been funded by the U.S. Environmental Protection Agency through the STAR Cooperative Agreement CR82-9096-01 Program on Designs and Models for Aquatic Resource Surveys 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 R82-9096-01

3. Preview Widely accepted that a more or less regular pattern of points (e.g., systematic sampling) is more efficient than SRS A variety of variance estimators for estimated mean are available for 1-dimensional systematic sampling We will examine the behavior of some variance estimators for 2-dimensional systematic and spatially-balanced (not necessarily regular) designs

4. Variance Estimators Wolter (1985) identified eight 1- dimensional variance estimators for 1- dimensional systematic sampling D’Orazio (2003) extended three of these to 2-dimensional systematic sampling Stevens & Olsen (2003) developed an estimator for 2-dimensional spatially- balanced samples

5. Simulation Study D’Orazio used simulation to compare estimators on a lattice generated from a Gaussian random field using several covariance functions –32 x 32 lattice –Calculated variance estimator for all 16 possible 8 x 8 samples –Generated the random field using the Gaussian Random Field package in R

8. Simulation Study Replicate D’Orazio’s study for the exponential covariance model, with the addition of the NBH estimator Check the behavior of the estimators on a spatially-patterned surface that is not stationary.

9. Variance Estimators Simplest approach: assume SRS:

10. Variance Estimators Horizontal stratification

11. Variance Estimators Vertical stratification

12. Variance Estimators 1 st Order autocorrelation correction –1-dimension, the Durbin- Watson statistic

13. Variance Estimators 1 st Order autocorrelation correction –2-dimension, Geary’s c index of spatial autocorrelation

14. Variance Estimators Cochran’s Autocorrelation Correction –1-dimension

15. Variance Estimators Cochran’s Autocorrelation Correction –2-dimension –Use Moran’s I in place of in formula for w

16. Stevens & Olsen Neighborhood Estimator General form for variable probability, continuous population D i is the set of neighbors for point i

17. Stevens & Olsen Neighborhood Estimator Weights are chosen so that Weights are a decreasing function of distance, and vanish outside of local neighborhood and w ij =0 for j  D i

18. Stevens & Olsen Neighborhood Estimator For constant probability, finite population

20. Gaussian Random Fields

24. Patchy Surfaces

25. Result GRF cv=1-exp(-2x) MeanBias 95% Coverage V SRS 0.0118340.008010.99943 V sh 0.0057220.001900.97706 V sv 0.0057280.001900.978625 V ac 0.0057090.001890.982000 V cac 0.001436-0.00230.788187 V NBH 0.0049490.001130.973062

26. Result GRF cv=1-exp(-0.5x) MeanBias 95% Coverage V SRS 0.014480.006930.99356 V sh 0.012750.005210.98719 V sv 0.012700.005150.98731 V ac 0.012700.005150.98881 V cac 0.00626 -0.00130.89950 V NBH 0.009920.002380.97681

28. Results Patchy Surface MeanBias 95% Coverage V SRS 0.000870.000580.99950 V sh 0.000400.000110.95796 V sv 0.000320.000020.93487 V ac 0.00036 0.000060.96290 V cac 0.00009-0.00020.74199 V NBH 0.000330.000030.95408

29. Conclusions The hs, vs, ac, and nbh estimators all seem to work reasonably well for both the GRF and patchy surfaces The nbh estimator seems to give coverages that are a bit closer to nominal than the hs, vs, or ac estimators The nbh works for variable probability, spatially constrained designs for which the other estimators do not.