Post on 28-Mar-2015
Designs for estimating variability structure and implications for
detecting watershed restoration effectiveness
• David P. Larsen– Western Ecology Division, NHEERL, USEPA– 200 SW 35th St. Corvallis, OR 97333
• N. Scott Urquhart– Department of Statistics– Colorado State University– Ft. Collins, CO 80523
Topics
• Linear trend detection– Applying the tools to restoration
monitoring• Organizing variability• Expanding the linear trend
detection model• Variance summary• Trend detection
A 2% per Year Linear Trend (each point is a regional mean value)
2% / yr Increase ( Slope = 0 ?)
For any patterned trend, there is an underlying linear component.
A 2% per Year Linear Trend (each point is a regional mean value)
Treatment
Can we detect a difference is slope between “treated” and “untreated” systems?
Reference
Linear trend detection
• Hypothesis test: Slope = 0?• Power: If a trend is present, what is
the likelihood of detecting it?• Hypothesis: Slope between treated
and reference = 0• Power: likelihood of detecting if
different?
Linear trend detection
• Power depends on:- magnitude of the trend (slope),- variability of our measurements,- number of sites,- the duration of the study (how long we can wait for the information).
Variance of a trend slope: How precisely can we
estimate it?
v ar( )( )
slopeX Xi
2
2
Organizing Variability
• Four major components:–Spatial
•Site-to-site–Temporal (year to year)
•Year•Site x Year Interaction
–Residual
““SITE” VARIANCESITE” VARIANCE::
Persistent Site-to-Site DifferencesPersistent Site-to-Site Differences
due todue to
Different Landscape/Historical ContextsDifferent Landscape/Historical Contexts
Different Levels of Human DisturbanceDifferent Levels of Human Disturbance
--
----
Gra
die
nt
----
----
>--
----
Gra
die
nt
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> ----------Stream Size -----------> .
Year variation• Concordant year-to-year variation
across all sites• Caused by regional
phenomena such as:– Wet/Dry years– Ocean conditions– Major volcanic
eruptions
Interaction variation
• Independent year-to-year variation among sites
• Driven by local factors
Residual variation
• The rest of it including:– Temporal or seasonal variation during
sampling window– Fine scale spatial variation– Crew-to-crew differences in applying
the protocol– Measurement error– …
Design framework
• Multiple sites with revisits within and among years
• Need a sample size of 30-50 to get reasonable estimate of variance, i.e., 30 – 50 sites; 30-50 revisits within year; at least 5 years with some sites visited annually, or at least in pairs of adjacent years.
AUGMENTED SERIALLY ALTERNATING
TIME PERIOD ( ex: YEARS)PANEL 1 2 3 4 5 6 7 8 9 10 11 12 13 ... 0 X X X X X X X X X X X X X 1 X X X X 2 X X X 3 X X X 4 X X X
TIME PERIOD ( ex: YEARS)PANEL 1 2 3 4 5 6 7 8 9 10 11 12 13 ... 1 X X X X X X X 2 X X X X X X 3 X X X X X X 4 X X X X X X
SERIALLY ALTERNATING WITH CONSECUTIVE YEAR REVISITS
Variance of a trend slope(New sites each year)
v ar( )( )
slopeN
N
N
X X
s
sy
ir
v
s
i
22
22
2
siteyear interactio
n
residual
Xi = Year ; Ns= Number of sites in region; Nv= Number of within-year revisits
(Urquhart and Kincaid. 1999. J. Ag., Biol., and Env. Statistics 4:404-414)
Variance of a trend slope(Revisiting the same sites each year)
22
2
2
var( )( )
ri
vy
s
i
NN
slopeX X
Xi = Year ; Ns= Number of sites in region; Nv= Number of within-year revisits
(See Urquhart & Kincaid, 1999)
year interaction
residual
Implications
• Effect of site = 0 if sites are revisited across years
• Year is not sensitive to “sample size”and its effect can become dominant
• Residual is affected by within year revisits• Interaction and residual are affected by
number of sites in survey, therefore other factors being equal, better to add sites to the survey rather than revisit sites
Some options(after adding sites doesn’t help)
• Extend survey interval• Focus on subpopulations to
manage variance• Monitor hypothesized covariates
controlling “year”
Adaptations for Effectiveness Monitoring
• Context– Comparing two
watersheds
Adaptations for Effectiveness Monitoring
• Context– Comparing
multiple watersheds
– Some treated ( )
– Some reference ( )
5 10 15 20 25
46
810
12
Year
Indi
cato
r
Power to detect a 2% per year “drift” from reference?
Variance of the difference in two trend slopes
(New sites each Year) 2
22
2
var( ) 2( )
ri
s v
s s
i
NN N
slopeX X
Xi = Year ; Ns= Number of sites in each region; Nv= Number of within-year revisits
site
interaction
residual
Variance of the difference in two trend slopes
(Revisiting the same sites Each Year) 2
2
2
var( ) 2( )
ri
v
s
i
NN
slopeX X
Xi = Year ; Ns= Number of sites in each region; Nv= Number of within-year revisits
interaction
residual
Duration (yrs) (Xi – X)2
9 60
10 82.5
11 110
12 143
13 182
14 228
15 280
Denominator’s effect
Variance Summary(Large wood)
Monitoring area
Site Year Interaction
Residual
North Coast
0.131 0.003 0.009 0.033
Mid-Coast 0.081 0 0.003 0.014
Mid-South 0.234 0.007 0.004 0.020
South Coast
0.166 --- 0.006 0.019
Umpqua 0.138 0.002 0 0.020
Design for power curves
• Annual visits, # sites varies
• Serially alternating design, with annual panel
• Variance components values were selected as low and high for Log10(LW+0.1)
• Alpha = 0.1
POWER CURVES FOR LOW VALUES OF VARIANCE COMPONENTS
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
n = 25
n=5
PO
WE
R
YEAR
POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
n = 25
n = 5
PO
WE
R
YEAR
POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS; AUGMENTED
ROTATING PANEL DESIGN
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
n = 6, 24
n = 2,8
The first number gives the number of sites in the "always revisit" panel; the second number gives the size of each of the rotating panels.
PO
WE
R
YEAR
Summary
• Characterization of spatial and temporal variation• Design framework for estimating components of
variation• A framework for evaluating linear trend• How variation affects trend detection• Modifying the framework for evaluating
restoration• An example using large wood as an indicator