Designs for estimating variability structure and implications for detecting watershed restoration effectiveness David P. Larsen –Western Ecology Division,

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

--

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Gra

die

nt

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Gra

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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