1. Model- vs. design-based sampling and variance estimation on continuous domains Cynthia Cooper OSU Statistics September 11, 2004 R82-9096-01

2. 2 Introduction Research on model- and design-based sampling and estimation on continuous domains Compare... Basis of inference of each Sampling concepts Interpretation of variance Variance estimation

3. 3 Duality in Environmental Monitoring Design-based Estimates –Status and trend –No model of underlying stochastic process Defensible –Probability sample Avoid selection bias Control sample process variance Model-based predictions –Stochastic behavior of response –Forecasting/prediction conditional on the observed data

4. 4 General Outline Introduction Summary comparison of approaches Summary characterization of variance estimators Proposed model-assisted variance estimator Simulation methods Design-based context results Model-based (kriging) results Conclusion

5. 5 Probability samples – unbiased estimates Basis for long-run frequency properties –Design-induced randomness – sample process variance Basic linear estimator scales up sample responses to extrapolate to population –Inclusion probabilities Examples –EPA EMAP –ODFW Monitoring Plan Augmented Rotating Panel –USFS Forest Inventory and Analysis Comparison of approaches - Design-based

6. 6 Inclusion probability –Element-wise – Sum of probabilities of all samples which include the i th element  i –Pair-wise -- Sum of … which include i th & j th elements  ij For continuous domains –Inclusion probability densities (IPD) (Cordy (1993))

7. 7 Response generated by a stochastic process Likelihood-based approaches to estimating parameters of model BLUP –Conditional on values observed in sample Examples –Mining surveys –Soil and hydrology surveys Comparison of approaches - Model-based

8. 8 Variance estimators - Design-based Quantifies variability induced by sampling process Variance of linear estimators –Scale up square and cross-product terms with inverse marginal and pair-wise inclusion probability densities (IPDs) For continuous domains –Congruent tessellation stratified samples w/ one observation per stratum Require randomized grid origin to achieve non-zero cross-product terms (π ij -π i π j ) (Stevens (1997))

9. 9 Variance estimators - Design-based Horvitz-Thompson (HT) Can be negative –Especially samples with a point pair in close proximity Requires randomly-located tessellation grid

10. 10 Variance estimators - Design-based Yates-Grundy (YG) Assumes fixed effective sample size Point pairs with close proximity can destabilize (Stevens (2003)) Requires randomly-located tessellation grid

11. 11 Variance estimators - Model-based Estimating MSPE of BLUP –Involves variances and covariances associated with square and cross-product terms of error Assume form of covariance that describes rate of decay of covariance Exponential Spherical Must result in positive-definite covariance matrix Incremental stationarity –E[(z(s i ) -z(s o )) 2 ] = g(||s i -s o ||) = g(h) –Typically, h   E[…] 

12. 12 Variance estimators - Model-based Variance –Quantifies stochastic variability of expected value of response –Vanishes as ||s i -s o || → 0 Mean-square prediction error (MSPE) –a.k.a. MSE –Variance + bias 2 Sample process variability of BLUP –Weighted averages vary less –Varies more as sample range increases relative to resolution

13. 13 Proposed model-assisted variance (V MA ) Predict variance within a stratum Variance is reduced by mean covariance (assuming positively correlated elements) –Similar to error variance computations (Ripley (1981)) Within-stratum estimated as –Sill reduced by within-stratum average covariance Linear estimator variance estimated as sum of squared coefficients times within-stratum variance Use covariance structure of response to model variability due to sampling process

14. 14 Precursors of and precedence for modeling covariance Cochran (1946) –Finite population –Serial correlation w/ discrete lags Bellhouse (1977) –Continued extension of Cochran’s work to finite populations ordered on two dimensions Small-area estimation model-assisted approaches –J.N.K Rao (2003)

15. 15 Random field (background) generated in R M. Schlather's GaussRF() of R package RandomFields Exponential covariance structure b*exp(-h/r) –(e.g. 4*exp(-h/2)) h is distance; b and r are "sill" and "range" parameters Methods – part 1

16. 16 Methods – part 1a Repeat 1000 times per realization Stratified sample –n=100; one observation per stratum; stratum size 2x2 –Simple square-grid tessellation Randomized origin Constant origin REML estimate of covariance parameters (b,r)

17. 17 Methods – part 2 Repeat 1000 times per realization (continued) For the design-based context –Estimate total (z hat ) HT estimator for continuous domain –Compute V HT, V YG and V MA –Compare estimated variances with empirical variance (V[z hat ]) For the model-based context example (Kriging) –Randomly selected z o at fixed location over 1000 trials –Obtain z hat, V OK, V MA

18. 18

19. 19 Results – Design-based application Empirical median relative error Compares estimated variances with empirical variance of estimate of total (V[z hat ]) (Stratified sample with randomized origin)

20. 20 Results – Design-based application Exponential covariance with range= 2 and sill= 4 10002000300040005000 0 200 Model-assisted Variance Observed V[zhat] 10001500200025003000350040004500 0 Yates-Grundy Variance Observed V[zhat] -6000-4000-20000200040006000 0 200 Horvitz-Thompson Variance Observed V[zhat] Avg Med Avg Med Avg Med

21. 21 Results – Design-based application Ratios of empirical standard deviations (Stratified sample with randomized origin)

22. 22 Results – Model-based application 0.00.20.40.60.81.01.2 0 100 Kriging variance (MSPE) Observed V[zhat] Avg 0.00.20.40.60.81.01.2 0 Model-assisted variance Avg Exponential covariance with range= 1 and sill= 1 (stratified sample with randomized origin) Observed V[zhat]

23. 23 Concluding - Model-assisted approach Small-area precedence Application to systematic and one-observation- per-stratum samples Effective alternative to direct estimators of continuous-domain randomized-origin tessellation stratified samples –Empirical results – less bias, better efficiency Doesn’t require randomly-located tessellation grid on continuous domain for non-zero π ij

24. 24 Acknowledgements Thanks to Don Stevens Committee members OSU Statistics Faculty UW QERM Faculty

25. 25 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 R82-9096-01