Wageningen 2004 # 1
USES OF POWER
IN DESIGNING
LONG-TERM
ENVIRONMENTAL SURVEYS
N. Scott Urquhart
Department of Statistics
Colorado State University
Fort Collins, CO 80523-1877
Wageningen 2004 # 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
Wageningen 2004 # 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
Wageningen 2004 # 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
Wageningen 2004 # 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
Wageningen 2004 # 6
RESPONSES of INTEREST
(continued II)

Grand Canyon National Park
 Erosion Around Archeological Resources
 Near-river Terrestrial Environment (GCMRC)
Wageningen 2004 # 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*
Wageningen 2004 # 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
Distribution Function (cdf)
 Pose some Interesting Statistical Inference Problems Due
to
 Variable Probability Sampling – Almost Always Needed
 Spatially Continuous Resources – No List Can Exist
Wageningen 2004 # 9
EXAMPLE OF
STATUS,
SUMMARIZED
BY A cdf
Wageningen 2004 # 10
ESTIMATED CUMULATIVE DISTRIBUTION
FUNCTION OF SECCHI DEPTH, EMAP AND “DIP-IN”
Wageningen 2004 # 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
Wageningen 2004 # 12
IMPORTANT COMPONENTS OF VARIANCE
2
(

 POPULATION VARIANCE:
LAKE )
2
(

 YEAR VARIANCE:
YEAR )
2
(

 RESIDUAL VARIANCE:
RESIDUAL )
Wageningen 2004 # 13
IMPORTANT COMPONENTS OF VARIANCE
( - CONTINUED)
2
(

LAKE )
 POPULATION VARIANCE:

VARIATION AMONG VALUES OF AN INDICATOR
(RESPONSE) ACROSS ALL LAKES IN A
REGIONAL POPULATION OR SUBPOPULATION
Wageningen 2004 # 14
IMPORTANT COMPONENTS OF VARIANCE
( - CONTINUED II)
2
(

 YEAR VARIANCE:
YEAR )



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)
Wageningen 2004 # 15
IMPORTANT COMPONENTS OF VARIANCE
( - CONTINUED - III)
 RESIDUAL COMPONENT OF VARIANCE

HAS SEVERAL SUBCOMPONENTS

YEAR*LAKE INTERACTION
(
2
RESIDUAL
)
 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
Wageningen 2004 # 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
Wageningen 2004 # 17
COMPOSITION OF TOTAL VARIANCE
Acid Neutralizing Capacity
LAKE COMPONENT OF VARIANCE
Ln(Conductance)
Ln(Chloride)
pH(Closed system)
Secchi Depth
Ln(Total Nitrogen)
Ln(Total Phosphorus)
Ln(Chlorophyll A)
YEAR
Ln( # zooplankton taxa)
Ln( # rotifer taxa)
Maximum Temperature
0.00
RESIDUAL COMPONENT OF VARIANCE
0.20
0.40
0.60
0.80
1.00
PROPORTION OF VARIANCE
Wageningen 2004 # 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
Wageningen 2004 # 19
COMPOSITION OF TOTAL VARIANCE
GRAND CANYON -- NEAR RIVER VEGETATION
Richness - 60K cfs
Richness - 45K cfs
SITE COMPONENT OF
VARIANCELAKE COMPONENT
YEAR
RESIDUAL COMPONENT OF VARIANCE
Richness - 35K cfs
Richness - 25K cfs
Richness - 15K cfs
Veg - 60K cfs
Veg - 45K cfs
Veg - 35K cfs
Veg - 25K cfs
0.00
0.20
0.40
0.60
0.80
PROPORTION OF VARIANCE
1.00
Wageningen 2004 # 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.
Wageningen 2004 # 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
Wageningen 2004 # 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
Wageningen 2004 # 23
DESIGN TRADE-OFFS: TREND vs STATUS
( - CONTINUED)
 VARIANCE OF A MEAN:
var (mean) 
2
m

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
--Wageningen 2004 # 24
DESIGN TRADE-OFFS: TREND vs STATUS
( - CONTINUED - II)
 Consequently
 Becomes
var (mean) 
2
m
1
2
var (regional mean slope) 
m  ( ti  t ) 2
 Note that the regional averaging of slopes
has the same effect as continuing to monitor
at one site for a much longer time period.
Wageningen 2004 # 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.
Wageningen 2004 # 26
STATISTICAL MODEL
 CONSIDER A FINITE POPULATION OF
SITES

{S1 , S2 , … , SN }
 and A TIME SERIES OF RESPONSE VALUES
AT EACH SITE:
{Y1 (t ), Y2 (t ),, YN (t )} and their average: Y (t )
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
Wageningen 2004 # 27
STATISTICAL MODEL -- II
AGAIN CONSIDER THE UNDERLYING TIME SERIES
DURING AN INDEX WINDOW
{Y1 (t ), Y2 (t ), , YN (t )}
and their averages: Yi (), Y (t ), and Y ().
 2SITE = var{Yi ()},
2
 YEAR
 var{Y (t )}
 2RESIDUAL  var{Yi (t )  Yi ()  Y (t )  Y ()}
Wageningen 2004 # 28
STATISTICAL MODEL -- III
{Yi (t )}  {Yij }
i indexes sites
R
where S
Tj indexes " years"
Yij  Y  (Yi   Y )  (Y j  Y )  (Yij  Yi   Y j  Y )
 Y  Si  Tj  Eij
2
and Si ~ (0,  2SITE ), Tj ~ (0,  YEAR
), and Eij ~ (0,  2RESIDUAL ),
with these random variables otherwise uncorrelated.
Wageningen 2004 # 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 npj = 0
or npj> 0 for panels visited in year j.
 The
vector of cell means (of visited cells) has a
covariance matrix S :
ch
cov Ypj  S (
2
SITE
,
2
YEAR
,
2
RESIDUAL
, n pj )
Wageningen 2004 # 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
1
1
1

  (X'S X ) X'S Y ,
1
1

cov(  )  ( X ' S X )
and
contain an estimate of the regional trend
and its variance.
Wageningen 2004 # 31
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
2
 2SITE /  2RESIDUAL and  YEAR
/  2RESIDUAL
Power depends on the ratios of variance
components, the panel plan, and on
   0/  RESIDUAL ; approximately, ˆ ~ N (, 2ˆ )
Wageningen 2004 # 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.
Wageningen 2004 # 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
MATH RECOME
100%
50%
 Pure panel =[1-0] =“Always Revisit”
0%
50%
 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
Wageningen 2004 # 34
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
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
[1-n]
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Wageningen 2004 # 35
FIRST TEMPORAL DESIGN FAMILY
 30 site visits per year
[1-0]
30
20
10
0
[1-n]
0
10
20
30
ALWAYS
REVISIT
NEVER
REVISIT
Wageningen 2004 # 36
POWER TO DETECT TREND
FIRST TEMPORAL DESIGN FAMILY
NO YEAR EFFECT
1
30:0
20:10
10:20
0:30
0.8
POWER
Always
Revisit
0.6
0.4
0.2
0
0
5
10
15
Never
Revisit
20
YEARS
Wageningen 2004 # 37
POWER TO DETECT TREND
FIRST TEMPORAL DESIGN FAMILY,
MODEST (= SOME) YEAR EFFECT
1
30:0
20:10
10:20
0:30
POWER
0.8
0.6
0.4
0.2
0
0
5
10
15
20
YEARS
Wageningen 2004 # 38
POWER TO DETECT TREND
FIRST TEMPORAL DESIGN FAMILY
BIG (= LOTS) YEAR EFFECT
1
30:0
20:10
10:20
0:30
POWER
0.8
0.6
0.4
0.2
0
0
5
10
15
20
YEARS
Wageningen 2004 # 39
SERIALLY ALTERNATING TEMPORAL
DESIGN [(1-3)4 ] SOMETIMES USED BY
EMAP
YEAR
1
FIA
X
[(1-3)4 ]
X
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
21
X
X
X
X
X
X
X
Wageningen 2004 # 40
SERIALLY ALTERNATING TEMPORAL
DESIGN [(1-3)4 ] SOMETIMES USED BY
EMAP
YEAR
1
FIA
X
[(1-3)4 ]
X
2
3
4
5
6
7
8
9
10 11
X
X
X
X
X
X
…
X
X
X
…
X
X
 Unconnected in an experimental design
sense

…
Very weak design for estimating year effects,
if present
Wageningen 2004 # 41
…
…
SPLIT PANEL [(1-4)5 , --- ]
YEAR
1
FIA
X
[(1-4)5 ]
X
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
21
X
X
X
 AGAIN, Unconnected in an experimental
design sense
Matches better with FIA
 Still a very weak design for estimating year
effects, if present

Wageningen 2004 # 42
SPLIT PANEL [(1-4)5 ,(2-3)5 ]
YEAR
1
FIA
X
[(1-4)5 ]
X
2
3
4
5
6
7
8
9
X
12
13
14
15
16
17
18
19
20
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
21
X
X
X
X
11
X
X
[(2-3)5 ]
10
X
X
X
X
X
X
X
X
X
X
This Temporal Design IS connected
Has three panels which match up with FIA
Wageningen 2004 # 43
X
SECOND TEMPORAL DESIGN FAMILY
 30 site visits per year
[1-4]
30
20
10
0
[2-3]
0
5
10
15
Wageningen 2004 # 44
POWER TO DETECT TREND
SECOND TEMPORAL DESIGN FAMILY
NO YEAR EFFECT
1
30:0
20:5
10:10
0:15
POWER
0.8
0.6
0.4
0.2
0
0
5
10
YEARS
15
20
Wageningen 2004 # 45
POWER TO DETECT TREND
SECOND TEMPORAL DESIGN FAMILY
SOME YEAR EFFECT
1
30:0
20:5
10:10
0:15
POWER
0.8
0.6
0.4
0.2
0
0
5
10
YEARS
15
20
Wageningen 2004 # 46
POWER TO DETECT TREND
SECOND TEMPORAL DESIGN FAMILY
LOTS OF YEAR EFFECT
1
30:0
20:5
10:10
0:15
POWER
0.8
0.6
0.4
0.2
0
0
5
10
YEARS
15
20
Wageningen 2004 # 47
COMPARISON OF POWER TO DETECT TREND
DESIGN 1 & 2 = ROWS
YEAR EFFECT
1
NONE
SOME
0.6
1
0.6
0.8
0.4
0.6
POWER
POWER
0.8
POWER
LOTS
0.8
1
0.4
0.4
0.2
0.2
0.2
0
0
0
0
5
10
15
5
10
20
15
20
0
0
YEARS
5
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.2
POWER
1
0.4
0.2
0
5
10
YEARS
15
20
15
20
15
20
0.4
0.2
0
0
10
YEARS
POWER
POWER
YEARS
0
0
5
10
YEARS
15
20
0
5
10
YEARS
Wageningen 2004 # 48
POWER TO DETECT TREND
VARYING YEAR EFFECT AND TEMPORAL
DESIGN
1
TEMPORAL DESIGN 2
0.8
NONE
POWER
TEMPORAL DESIGN 1
0.6
SOME
0.4
LOTS
0.2
0
0
5
10
15
20
YEARS
Wageningen 2004 # 49
STANDARD ERROR OF STATUS
TEMPORAL DESIGN 1, NO YEAR EFFECT
0.5
SE STATUS
0.4
TOTAL OF 30
SITES
0.3
30:0
20:10
10:20
0:30
0.2
110 SITES
VISITED BY
YEAR 5
0.1
0
0
5
10
410 SITES
VISITED BY
YEAR
20
15
YEARS
Wageningen 2004 # 50
20
STANDARD ERROR OF STATUS
TEMPORAL DESIGN 2, NO YEAR EFFECT
0.5
SE STATUS
0.4
30:0
20:5
10:10
0:15
TOTAL OF
75 SITES
0.3
0.2
0.1
TOTAL OF
150 SITES
0
0
5
10
15
20
YEARS
Wageningen 2004 # 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
Wageningen 2004 # 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.
Wageningen 2004 # 53
Wageningen 2004 # 54
FUNDING ACKNOWLEDGEMENT
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.
This research is funded by
U.S.EPA – Science To Achieve
Results (STAR) Program
Cooperative
# CR - 829095
Agreement
Wageningen 2004 # 55