1. San Diego Chapter ASA # 1 USES OF POWER IN DESIGNING LONG-TERM ENVIRONMENTAL SURVEYS N. Scott Urquhart Department of Statistics Colorado State University Fort Collins, CO 80523-1877

2. San Diego Chapter ASA # 2 OUTLINE FOR TONIGHT  Long-Term Environmental Surveys  Agencies involved  Sorts of Summaries of Interest  Sources of Variation – Major ones  A Statistical Model  Superimposed on an Adapted Classical Sampling Model  Calculation of Power Using this Model  Illustrations  General  Specific  Generalizations - as Time Allows

3. San Diego Chapter ASA # 3 LONG-TERM ENVIRONMENTAL SURVEYS  Objective: To Establish  The Current Status  Detect Long-Term Trends  Evaluate “Extent” of Various Classes  Of the Resource(s) of Interest  Usually Ecological or Living Resources  Agencies = Who  US Environmental Protection Agency (EPA)*  States and Tribes, and Local Jurisdictions  Response to Legislation Like the Clean Water Act  Forest Service – “Forest Health”  National Park Service*  Soil Conservation Service (not the current name)  National Marine Fisheries Service ( “ )  National Wetlands Inventory

4. San Diego Chapter ASA # 4 RESPONSES of INTEREST  EPA  Variety of Chemical Measures of Water Quality  Nitrogen to Heavy Metals to Pesticides  Acid Neutralizing Capacity (ANC)  Important in Evaluating the Effect of “Acid Rain”  Composition of “Bugs” in the Aquatic Community  Thought to Contain Better Info on total Effects than Individual Chemicals  Fish Populations – Composition, not size  Clean Water Act Includes Reporting on Temperature Pollution

5. San Diego Chapter ASA # 5 RESPONSES of INTEREST (continued)  National Park Service (Eg: Olympic NP in WA)  Vegetation  Bird Populations  Composition  Size of Various Species  Streams/Rivers  Fish Populations  Macroinvertebrate Communities  Extent of Intermittent Streams  Health of Glaciers  Extent – Shrinking with Global Warming?  Composition

6. San Diego Chapter ASA # 6 RESPONSES of INTEREST (continued II)  Grand Canyon National Park  Erosion Around Archeological Resources  Near-river Terrestrial Environment (GCMRC)

7. San Diego Chapter ASA # 7 SPATIAL EXTENT  Generally Large Areas  This is the Way Congress Writes Laws  Regions can be very large  12 Western States  ND, SC, MT, WY, CO, ID, UT, NV, AZ, WA, OR, CA  Midatlantic Highlands  parts of PA, VA, WV, DE, MD  Individual States  Lands of Several related Tribes, or Even Only One  Groups of National Parks  Groups of Sanitation Districts, or even  Individual Sanitation Districts*

8. San Diego Chapter ASA # 8 SUMMARIES of INTEREST  Extent by Classes  Track Changes Between Classes  National Wetlands Inventory  Major focus  Has Very Good Graphic Depiction of Class Changes  “Status”  Often is summarized as an Estimated Cumulative DistributionFunction (cdf)  Pose some Interesting Statistical Inference Problems Due to  Variable Probability Sampling – Almost Always Needed  Spatially Continuous Resources – No List Can Exist

9. San Diego Chapter ASA # 9 EXAMPLE OF STATUS, SUMMARIZED BY A cdf

10. San Diego Chapter ASA # 10 ESTIMATED CUMULATIVE DISTRIBUTION FUNCTION OF SECCHI DEPTH, EMAP AND “DIP-IN”

11. San Diego Chapter ASA # 11 SUMMARIES of INTEREST (continued)  Trends  Directional Changes in Responses  Reality: Detection of Short-Term Cycles is Beyond the Resources for the Foreseeable Future  Great Big Changes Don’t Require Surveys  So Interest Lies in Modest-Sized Long-Term Changes in One Direction  This means Changes the Scale of 1% to 2% Per Year  Usually a Trend for a Region  Regional Summaries of Individual Site Trends  Sometimes how trend varies in relation to other things

12. San Diego Chapter ASA # 12  POPULATION VARIANCE:  YEAR VARIANCE:  RESIDUAL VARIANCE: IMPORTANT COMPONENTS OF VARIANCE

13. San Diego Chapter ASA # 13  POPULATION VARIANCE:  VARIATION AMONG VALUES OF AN INDICATOR (RESPONSE) ACROSS ALL LAKES IN A REGIONAL POPULATION OR SUBPOPULATION IMPORTANT COMPONENTS OF VARIANCE ( - CONTINUED)

14. San Diego Chapter ASA # 14  YEAR VARIANCE:  CONCORDANT VARIATION AMONG VALUES OF AN INDICATOR (RESPONSE) ACROSS YEARS FOR ALL LAKES IN A REGIONAL POPULATION OR SUBPOPULATION  NOT VARIATION IN AN INDICATOR ACROSS YEARS AT A LAKE  DETRENDED REMAINDER, IF TREND IS PRESENT  EFFECTIVELY THE DEVIATION AWAY FROM THE TREND LINE (OR OTHER CURVE) IMPORTANT COMPONENTS OF VARIANCE ( - CONTINUED II)

15. San Diego Chapter ASA # 15  RESIDUAL COMPONENT OF VARIANCE  HAS SEVERAL SUBCOMPONENTS  YEAR*LAKE INTERACTION  THIS CONTAINS MOST OF WHAT MOST ECOLOGISTS WOULD CALL YEAR TO YEAR VARIATION, I.E. THE LAKE SPECIFIC PART  INDEX VARIATION  MEASUREMENT ERROR  CREW-TO-CREW VARIATION  LOCAL SPATIAL = PROTOCOL  SHORT TERM TEMPORAL IMPORTANT COMPONENTS OF VARIANCE ( - CONTINUED - III)

16. San Diego Chapter ASA # 16 BIOLOGICAL INDICATORS HAVE SOMEWHAT MORE VARIABILITY THAN PHYSICAL INDICATORS – BUT THIS VARIES, TOO  Subsequent slides show the relative amount of variability  Ordered by the amount of residual variability: least to most (aquatic responses)  Acid Neutralizing Capacity  Ln(Conductance)  Ln(Chloride)  pH(Closed system)  Secchi Depth  Ln(Total Nitrogen)  Ln(Total Phosphorus)  Ln(Chlorophyll A)  Ln( # zooplankton taxa)  Ln( # rotifer taxa)  Maximum Temperature And others, both aquatic and terrestrial

17. San Diego Chapter ASA # 17

18. San Diego Chapter ASA # 18 SOURCE OF COMPONENTS OF VARIANCE FROM GRAND CANYON  Grand Canyon Monitoring and Research Center  Effects of Glen Canyon Dam on the Near-River Habitat in the Grand Canyon  At Various Heights Above the River  Height Is Measured as the Height of the River’s Water at Various Flow Rates  Eg: 15K cfs, 25K cfs, 35K cfs, 45K cfs & 60K cfs  Using First Two Years’ Data  Mike Kearsley – UNA  Design = Spatially Balanced  With about 1/3 revisited

19. San Diego Chapter ASA # 19

20. San Diego Chapter ASA # 20 ALL VARIABILITY IS OF INTEREST  The Site Component of Variance is One of the Major Descriptors of the Regional Population  The Year Component of Variance Often is Small, too Small to Estimate. If Present, it is a Major Enemy for Detecting Trend Over Time.  If it has even a moderate size, “sample size” reverts to the number of years.  In this case, the number of visits and/or number of sites has no practical effect.

21. San Diego Chapter ASA # 21 ALL VARIABILITY IS OF INTEREST ( - CONTINUED)  Residual Variance Characterizes the Inherent Variation in the Response or Indicator.  But Some of its Subcomponents May Contain Useful Management Information  CREW EFFECTS ===> training  VISIT EFFECTS ===> need to reexamine definition of index (time) window or evaluation protocol  MEASUREMENT ERROR ===> work on laboratory/measurement problems

22. San Diego Chapter ASA # 22 DESIGN TRADE-OFFS: TREND vs STATUS  How do we Detect Trend in Spite of All of This Variation?  Recall Two Old Statistical “Friends.”  Variance of a mean, and  Blocking

23. San Diego Chapter ASA # 23 DESIGN TRADE-OFFS: TREND vs STATUS ( - CONTINUED)  VARIANCE OF A MEAN:  Where m members of the associated population have been randomly selected and their response values averaged.  Here the “mean” is a regional average slope, so "  2 " refers to the variance of an estimated slope ---

24. San Diego Chapter ASA # 24 DESIGN TRADE-OFFS: TREND vs STATUS ( - CONTINUED - II)  Consequently  Becomes  Note that the regional averaging of slopes has the same effect as continuing to monitor at one site for a much longer time period.

25. San Diego Chapter ASA # 25 DESIGN TRADE-OFFS: TREND vs STATUS ( - CONTINUED - III)  Now,  2, in total, is large.  If we take one regional sample of sites at one time, and another at a subsequent time, the site component of variance is included in  2.  Enter the concept of blocking, familiar from experimental design.  Regard a site like a block  Periodically revisit a site  The site component of variance vanishes from the variance of a slope.

26. San Diego Chapter ASA # 26 STATISTICAL MODEL  CONSIDER A FINITE POPULATION OF SITES  {S 1, S 2, …, S N }  and A TIME SERIES OF RESPONSE VALUES AT EACH SITE:  A FINITE POPULATION OF TIME SERIES  TIME IS CONTINUOUS, BUT SUPPOSE  ONLY A SAMPLE CAN BE OBSERVED IN ANY YEAR, and  ONLY DURING AN INDEX WINDOW OF, SAY, 10% OF A YEAR

27. San Diego Chapter ASA # 27 STATISTICAL MODEL -- II

28. San Diego Chapter ASA # 28 STATISTICAL MODEL -- III

29. San Diego Chapter ASA # 29 STATISTICAL MODEL -- IV  IF p INDEXES PANELS, THEN  Sites are nested in panels: p ( i ) and  Years of visit are indicated by panel with n pj = 0 or n pj > 0 for panels visited in year j.  The vector of cell means (of visited cells) has a covariance matrix 

30. San Diego Chapter ASA # 30 STATISTICAL MODEL -- V  Now let X denote a regressor matrix containing a column of 1s and a column of the numbers of the time periods corresponding to the filled cells. The second elements of contain an estimate of the regional trend and its variance.

31. San Diego Chapter ASA # 31  Ability of a panel plan to detect trend can be expressed as power.  We will evaluate power in terms of these ratios of variance components  Power depends on the ratios of variance components, the panel plan, and on TOWARD POWER

32. San Diego Chapter ASA # 32 NOW PUT IT ALL TOGETHER  Question: “ What kind of temporal design should you use for Northwest National Parks?  We’ll investigate two (families) of recommended designs.  All illustrations will be based on 30 site visits per year, a reasonable number given resources.  General relations are uninfluenced by number of sites visited per year, but specific performance is.  We’ll use the panel notation Trent McDonald published.

33. San Diego Chapter ASA # 33 RECOMMENDATION OF FULLER and BREIDT  Based on the Natural Resources Inventory (NRI)  Iowa State & US Department of Agriculture  Oriented toward soil erosion &  Changes in land use  Their recommendation  Pure panel =[1-0] =“Always Revisit”  Independent =[1-n]=“Never Revisit”  Evaluation context  No trampling effect – remotely sensed data  No year effects  Administrative reality of potential variation in funding from year to year MATH RECOME 100% 50% 0% 50%

34. San Diego Chapter ASA # 34 TEMPORAL LAYOUT OF [(1-0), (1-n)] YEAR1234567891011121314151617181920 [1-0]XXXXXXXXXXXXXXXXXXXX [1-n]X X X X X X X X X X X X X X X X X X X X

35. San Diego Chapter ASA # 35 FIRST TEMPORAL DESIGN FAMILY  30 site visits per year [1-0]3020100 [1-n]0102030 ALWAYSREVISITNEVERREVISIT

36. San Diego Chapter ASA # 36 POWER TO DETECT TREND FIRST TEMPORAL DESIGN FAMILY NO YEAR EFFECT Always Revisit Never Revisit

37. San Diego Chapter ASA # 37 POWER TO DETECT TREND FIRST TEMPORAL DESIGN FAMILY, MODEST (= SOME) YEAR EFFECT

38. San Diego Chapter ASA # 38 POWER TO DETECT TREND FIRST TEMPORAL DESIGN FAMILY BIG (= LOTS) YEAR EFFECT

39. San Diego Chapter ASA # 39 SERIALLY ALTERNATING TEMPORAL DESIGN [(1-3) 4 ] SOMETIMES USED BY EMAP YEAR123456789101112131415161718192021 FIAXXX [(1-3) 4 ] XXXXXX XXXXX XXXXX XXXXX

40. San Diego Chapter ASA # 40 SERIALLY ALTERNATING TEMPORAL DESIGN [(1-3) 4 ] SOMETIMES USED BY EMAP YEAR1234567891011… FIAXX [(1-3) 4 ] XXX… XXX… XXX… XX…  Unconnected in an experimental design sense  Very weak design for estimating year effects, if present

41. San Diego Chapter ASA # 41 SPLIT PANEL [(1-4) 5, --- ] YEAR123456789101112131415161718192021 FIAXXX [(1-4) 5 ] XXXXX XXXX XXXX XXXX XXXX  AGAIN, Unconnected in an experimental design sense  Matches better with FIA  Still a very weak design for estimating year effects, if present

42. San Diego Chapter ASA # 42 SPLIT PANEL [(1-4) 5,(2-3) 5 ]  This Temporal Design IS connected  Has three panels which match up with FIA YEAR123456789101112131415161718192021 FIAXXX [(1-4) 5 ] XXXXX XXXX XXXX XXXX XXXX [(2-3) 5 ] XXXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX

43. San Diego Chapter ASA # 43 SECOND TEMPORAL DESIGN FAMILY  30 site visits per year [1-4]3020100 [2-3]051015

44. San Diego Chapter ASA # 44 POWER TO DETECT TREND SECOND TEMPORAL DESIGN FAMILY NO YEAR EFFECT

45. San Diego Chapter ASA # 45 POWER TO DETECT TREND SECOND TEMPORAL DESIGN FAMILY SOME YEAR EFFECT

46. San Diego Chapter ASA # 46 POWER TO DETECT TREND SECOND TEMPORAL DESIGN FAMILY LOTS OF YEAR EFFECT

47. San Diego Chapter ASA # 47 COMPARISON OF POWER TO DETECT TREND DESIGN 1 & 2 = ROWS YEAR EFFECT NONE SOME LOTS

48. San Diego Chapter ASA # 48 POWER TO DETECT TREND VARYING YEAR EFFECT AND TEMPORAL DESIGN

49. San Diego Chapter ASA # 49 STANDARD ERROR OF STATUS TEMPORAL DESIGN 1, NO YEAR EFFECT TOTAL OF 30 SITES 110 SITES VISITED BY YEAR 5 410 SITES VISITED BY YEAR 20

50. San Diego Chapter ASA # 50 STANDARD ERROR OF STATUS TEMPORAL DESIGN 2, NO YEAR EFFECT TOTAL OF 150 SITES TOTAL OF 75 SITES

51. San Diego Chapter ASA # 51 GENERALIZATIONS  Each site can have its own trend  These very likely differ  How should we approach this reality?  There is a cdf of trends across the region  Variation in trends can be partitioned  Components are very similar to those used for responses:  Years  Rivers  Sites within rivers

52. San Diego Chapter ASA # 52 ILLUSTRATION   Stoddard, J.L., Kahl, J.S., Deviney, F.A., DeWalle, D.R., Driscoll, C.T., Herlihy, A.T., Kellogg, J.H., Murdoch, J.R. Webb, J.R., and Webster, K.E. (2003). Response of Surface Water Chemistry to the Clean Air Act Amendments of 1990. EPA/620/R-02/004. US Environmental Protection Agency, Washington, DC.

53. San Diego Chapter ASA # 53

54. San Diego Chapter ASA # 54 This research is funded by U.S.EPA – Science To Achieve Results (STAR) Program Cooperative Agreement # CR - 829095 The work reported here today was developed under the STAR Research Assistance Agreement CR-829095 awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This presentation has not been formally reviewed by EPA. The views expressed here are solely those of presenter and STARMAP, the Program he represents. EPA does not endorse any products or commercial services mentioned in this presentation. FUNDING ACKNOWLEDGEMENT