2. USES OF POWER IN DESIGNING LONG-TERM ENVIRONMENTAL SURVEYS
N. Scott Urquhart Department of Statistics
Colorado State University
Fort Collins, CO

3. 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

4. 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

5. 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

6. 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?

7. RESPONSES of INTEREST (continued II)
Grand Canyon National Park
Erosion Around Archeological Resources
Near-river Terrestrial Environment (GCMRC)

8. SPATIAL EXTENT Generally Large Areas Regions can be very large
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*

9. SUMMARIES of INTEREST Extent by Classes “Status”
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 Distribution Function (cdf)
Pose some Interesting Statistical Inference Problems Due to
Variable Probability Sampling – Almost Always Needed
Spatially Continuous Resources – No List Can Exist

10. EXAMPLE OF
STATUS,
SUMMARIZED
BY A cdf

11. ESTIMATED CUMULATIVE DISTRIBUTION FUNCTION OF SECCHI DEPTH, EMAP AND “DIP-IN”

12. 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

13. IMPORTANT COMPONENTS OF VARIANCE
POPULATION VARIANCE:
YEAR VARIANCE:
RESIDUAL VARIANCE:

14. IMPORTANT COMPONENTS OF VARIANCE
( - CONTINUED)
POPULATION VARIANCE:
VARIATION AMONG VALUES OF AN INDICATOR (RESPONSE) ACROSS ALL LAKES IN A REGIONAL POPULATION OR SUBPOPULATION

15. IMPORTANT COMPONENTS OF VARIANCE
( - CONTINUED II)
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)

16. IMPORTANT COMPONENTS OF VARIANCE
( - CONTINUED - III)
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

17. Subsequent slides show the relative amount of variability
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

19. 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

21. 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.

22. 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

23. 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

24. 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 "s2" refers to the variance of an estimated slope ---

25. 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.

26. DESIGN TRADE-OFFS: TREND vs STATUS ( - CONTINUED - III)
Now, s2, 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 s2.
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.

27. STATISTICAL MODEL CONSIDER A FINITE POPULATION OF SITES
{S1 , S2 , … , SN }
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

28. STATISTICAL MODEL -- II

29. STATISTICAL MODEL -- III

30. STATISTICAL MODEL -- IV
IF p INDEXES PANELS, THEN
Sites are nested in panels: p ( i ) and
Years of visit are indicated by panel with npj = 0 or npj> 0 for panels visited in year j.
The vector of cell means (of visited cells) has a covariance matrix S :

31. 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.

32. TOWARD POWER
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

33. 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.

34. 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% %
0% %

35. TEMPORAL LAYOUT OF [(1-0), (1-n)]
YEAR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
[1-0] X
[1-n]

36. FIRST TEMPORAL DESIGN FAMILY
30 site visits per year
[1-0] 30 20 10
[1-n]
ALWAYS
REVISIT
NEVER

37. POWER TO DETECT TREND FIRST TEMPORAL DESIGN FAMILY NO YEAR EFFECT
Always
Revisit
Never
Revisit

38. POWER TO DETECT TREND FIRST TEMPORAL DESIGN FAMILY, MODEST (= SOME) YEAR EFFECT

39. POWER TO DETECT TREND FIRST TEMPORAL DESIGN FAMILY BIG (= LOTS) YEAR EFFECT

40. SERIALLY ALTERNATING TEMPORAL DESIGN [(1-3)4 ] SOMETIMES USED BY EMAP
YEAR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
FIA X
[(1-3)4 ]

41. SERIALLY ALTERNATING TEMPORAL DESIGN [(1-3)4 ] SOMETIMES USED BY EMAP
YEAR 1 2 3 4 5 6 7 8 9 10 11 …
FIA X
[(1-3)4 ]
Unconnected in an experimental design sense
Very weak design for estimating year effects, if present

42. SPLIT PANEL [(1-4)5 , --- ]
YEAR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
FIA X
[(1-4)5 ]
AGAIN, Unconnected in an experimental design sense
Matches better with FIA
Still a very weak design for estimating year effects, if present

43. SPLIT PANEL [(1-4)5 ,(2-3)5 ] This Temporal Design IS connected
YEAR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
FIA X
[(1-4)5 ]
[(2-3)5 ]
This Temporal Design IS connected
Has three panels which match up with FIA

44. SECOND TEMPORAL DESIGN FAMILY
30 site visits per year
[1-4] 30 20 10
[2-3] 5 15

45. POWER TO DETECT TREND SECOND TEMPORAL DESIGN FAMILY NO YEAR EFFECT

46. POWER TO DETECT TREND SECOND TEMPORAL DESIGN FAMILY SOME YEAR EFFECT

47. POWER TO DETECT TREND SECOND TEMPORAL DESIGN FAMILY LOTS OF YEAR EFFECT

48. COMPARISON OF POWER TO DETECT TREND DESIGN 1 & 2 = ROWS
YEAR EFFECT
NONE SOME LOTS

49. POWER TO DETECT TREND VARYING YEAR EFFECT AND TEMPORAL DESIGN

50. 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

51. STANDARD ERROR OF STATUS TEMPORAL DESIGN 2, NO YEAR EFFECT
TOTAL OF 75 SITES
TOTAL OF 150 SITES

52. 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

53. 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 EPA/620/R-02/004. US Environmental Protection Agency, Washington, DC.

55. FUNDING ACKNOWLEDGEMENT
The work reported here today was developed under the STAR Research Assistance Agreement CR 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.
This research is funded by
U.S.EPA – Science To Achieve
Results (STAR) Program
Cooperative
Agreement
# CR