Model- vs. design-based sampling and variance estimation on continuous domains

1. Aquatic Resource Surveys Designs and Models for DAMARS R82-9096-01 Model- vs. design-based sampling and variance estimationon continuous domains Cynthia Cooper OSU Statistics September 11, 2004

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. 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. 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. Comparison of approaches - Design-based • 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

6. Comparison of approaches - Design-based • Inclusion probability • Element-wise – Sum of probabilities of all samples which include the ith element • i • Pair-wise -- Sum of … which include ith & jth elements • ij • For continuous domains • Inclusion probability densities (IPD) (Cordy (1993))

7. Comparison of approaches - Model-based • 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

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 (πij-πiπj) (Stevens (1997))

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. 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. 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(si)-z(so))2] = g(||si-so||) = g(h) • Typically, h   E[…] 

12. Variance estimators - Model-based • Variance • Quantifies stochastic variability of expected value of response • Vanishes as ||si-so|| → 0 • Mean-square prediction error (MSPE) • a.k.a. MSE • Variance + bias2 • Sample process variability of BLUP • Weighted averages vary less • Varies more as sample range increases relative to resolution

13. Proposed model-assisted variance (VMA) Use covariance structure of response to model variability due to sampling process • 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

14. Precursors of and precedence formodeling 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. Methods – part 1 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

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. Methods – part 2 Repeat 1000 times per realization (continued) • For the design-based context • Estimate total (zhat) • HT estimator for continuous domain • Compute VHT, VYG and VMA • Compare estimated variances with empirical variance (V[zhat]) • For the model-based context example (Kriging) • Randomly selected zo at fixed location over 1000 trials • Obtain zhat, VOK, VMA

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

20. Results – Design-based application Exponential covariance with range= 2 and sill= 4 Avg Med Avg Med Avg Med 200 200 Observed V[zhat] Observed V[zhat] Observed V[zhat] 0 0 0 -6000 -4000 -2000 0 2000 4000 6000 1000 1500 2000 2500 3000 3500 4000 4500 1000 2000 3000 4000 5000 Yates-Grundy Variance Horvitz-Thompson Variance Model-assisted Variance

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

22. Results – Model-based application Exponential covariance with range= 1 and sill= 1 (stratified sample with randomized origin) Avg Avg 100 Observed V[zhat] Observed V[zhat] 0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Kriging variance (MSPE) Model-assisted variance

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 πij

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

25. R82-9096-01 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