1. 1 Evaluation of Standards data collected from probabilistic sampling programs Eric P. Smith Y. Duan, Z. Li, K. Ye Statistics Dept., Virginia Tech Presented at the Monitoring Science and Technology Symposium, Denver, CO Sept 20-24.

2. 2 Outline  Background Standards assessments  Single site analysis  Regional analysis Mixed model approach Bayesian approach  Upshot: need models that allow for additional information to be used in assessments

3. 3 320

4. 4 Standards assessment – 303d  Clean Water Act section 303d mandates states in US to monitor and assess condition of streams  Site impaired – list site, start TMDL process (Total Max Daily Loading)  Impaired means site does not meet usability criteria

5. 5 Linkages in 303(d) Set goals and WQS Implement strategies [NPDES, 319, SRF, etc] Conduct monitoring Develop strategies [TMDLs] Yes No Meeting WQS? Apply Antidegradation 303(d) List standards tests Local to regional Sampling plan

6. 6 Impaired sites  Site impaired if standards not met  Standards – defined through numerical criteria Involve frequency, duration, magnitude  –Old method Site impaired if >10% of samples exceed criteria Implicit statistical decision process- error rates

7. 7 Test of impairment

8. 8 Some newer approaches  Frequency: Binomial method Test p<0.1  Magnitude Acceptance sampling by variables Tolerance interval on percentile Test criteria by computing mean for the distribution of measurements and comparing with what is expected given the percentile criteria

9. 9 Problems  Approach is local Limited sampling budget; many stations means small sample sizes per station Impairment may occur over a region Modeling must be relatively simple (hard to account for seasonality, temporal effects) Does not complement current approaches to sampling Site history is ignored Not linked to TMDL analysis (regional) and 305 reporting

10. 10 Probabilistic sampling schemes  Randomly selected sites  Rotating panel surveys Some sites sampled at all possible times Other sites sampled on rotational basis Sites in second group may be randomly selected

11. 11 Making the assessment regional  fixed effects (time, covariates)  random ones (site, location) Y = mean + site Y = mean + time + site General model Y = X + Z = fixed effect model + random effects

12. 12 Regional Mixed Model  Allows for covariates  Allows for a variety of error structures Temporal, spatial, both  Does not require equal sample sizes etc  Allows estimation of means for sites with small sample sizes Improves estimation by borrowing information from other sites

13. 13 Simple model  Testing is based on estimate and variance of mean for site i ( i )  Can also test for regional impairment using distribution of grand mean Random site effect Error term allows for modeling of temporal or spatial correlation

14. 14 Error and stochastic components  Covariance Structure without correlation (one random effect model)  Spatial Covariance Structure Random site effect Error term allows for modeling of temporal or spatial correlation

15. 15 Test based on OLS estimations for each site i  Baseline is the numeric criterion. For DO, we use 5, and for PH 6.  Model based: same idea but mean and variance may be estimated from model

16. 16 Simulation results: different means, variance=1, normal 3 sites-12 obs – 6.28 is the mean for the boundary 56.281,2-5 3-6.28 1-5 2,3-6.28 1-6.28 2,3-7.28 7.281,2-6.28 3-8.28 Site 10.990.0620.9870.9710.02700.038 Site 20.990.0640.9920.084000.039 Site 30.990.0640.1520.073000 Expect.05 Two bad sites Pull third site One badAll good Two border One good

17. 17 Located in SW Virginia Good bass fishing

18. 18 DO data collected at four stations of PHILPOTT RESERVOIR (years 2000, 2001 & 2002)

19. 19 Evaluation based on Do data of PHILPOTT RESERVIOR (2000-2002) 4ASRE046.90Model based4ASRE052.314ASRE056.06 n28 3132 Sample mean7.556.666.67 Sample variance % excceding Binomial p-value 5.81 11.5406 9.56 26.0096 16.15 28.0033 Test statistic5.64.272.992.35 critical value4.755.055.195.2 conclusion Fail to rejectreject Single site analysis

20. 20 Bayesian approach  a is a random site effect  Error term may include temporal correlation or spatial  Priors on parameters Mean –uniform a is normal (random effect) variance has prior Produces results similar to first approach

21. 21 Alternative: Using historical data  Power prior – Chen, Ibrahim, Shao 2000  Use likelihood from the previous assessment (D 0 ). Basic idea: weight new data by prior data  Power term,, determines influence of historical data.  Modification to work with Winbugs

22. 22 Incorporate Historical Data using Power Priors  Make random, and assign a prior on it. The joint posterior of becomes where D is current data and D 0 is past data  Advantage: Improve the precision of estimates.

23. 23

24. 24 PH data collected at four stations: use past information to build prior

25. 25 Evaluate site impairment based on PH data with power priors Note – log transformation applied to improve normality

26. 26  If multiple historical data sets are available, assign a different for each historical data set. where  Data collected at adjacent stations could be used as “historical” data. Power Priors with Multiple Historical Data Sets

27. 27 DO data collected at four stations of PHILPOTT RESERVOIR (years 2000, 2001 & 2002)

28. 28 Evaluate site impairment based on DO data collected at four stations of PHILPOTT RESERVOIR (years 2000, 2001 & 2002)

29. 29 DO data collected at four stations of MOOMAW RESERVOIR (years 2000 & 2001)

30. 30 Evaluate site impairment based on DO data collected at four stations of MOOMAW RESERVOIR (years 2000 & 2001)

31. 31 Comments  Advantages Greater flexibility in modeling Allows for site history to be included Can include spatial and temporal components Can better connect to TMDL analysis and probabilistic sampling  Disadvantage Requires more commitment to the modeling process Greater emphasis on the distributional assumptions  http://www.stat.vt.edu/facstaff/epsmith.h tml

32. 32 Needs  More applications to evaluate  Temporal/spatial modeling  Evaluation of error rates  Bayesian modeling and null and alternative hypotheses

33. 33 Sponsor RD-83136801-0 This talk was not subjected to USEPA review. The conclusion and opinions are soley those of the authors and not the views of the Agency.