# 1SARMMM 9/8/05

DEVELOPMENTS DEVELOPMENTS IN IN

TREND DETECTION TREND DETECTION IN IN

AQUATIC SURVEYSAQUATIC SURVEYS

DEVELOPMENTS DEVELOPMENTS IN IN

TREND DETECTION TREND DETECTION IN IN

AQUATIC SURVEYSAQUATIC SURVEYS

N. Scott Urquhart

STARMAP Program Director

Department of Statistics

Colorado State University

# 2SARMMM 9/8/05

PATH FOR TODAYPATH FOR TODAYPATH FOR TODAYPATH FOR TODAY

Review a model I have used to project power of

EMAP-type revisit or temporal designs

Recent developments Impact of removing planned revisits – Grand Canyon

Power of differences in trend – Oregon plan & others

Generalize model to allow distribution of trends

Each part has different collaborators

# 3SARMMM 9/8/05

A SIMPLE MODEL for a A SIMPLE MODEL for a SURVEYED ECOLOGICAL RESPONSESURVEYED ECOLOGICAL RESPONSE

A SIMPLE MODEL for a A SIMPLE MODEL for a SURVEYED ECOLOGICAL RESPONSESURVEYED ECOLOGICAL RESPONSE

Consistent with annual or less frequent observation

Represent the response time series by an annual

departure

Represent space by a site effect only

Allow sites to be visited in panels

Regard trends across time as a contrast over the

panel by annual response means

SARMMM 9/8/05

wherei INDEXES PANELS 1, 2, ... , s(all sites in a panel have the same revisit pattern)

j INDEXES TIME PERIODS ( years in EMAP)k INDEXES SITES WITHIN A PANEL 1, 2, ... , ni

and (uncorrelated):

A STATISTICAL MODELA STATISTICAL MODEL

ijk ik j ijkY S T E

2 2 2~ ( , ) ~ (0, ) ~ (0, )ik S j T ijk ES T E

SARMMM 9/8/05

A STATISTICAL MODEL - continuedA STATISTICAL MODEL - continued

Consider the entire table of the panel by time-period

means, Without regard to, as yet, whether the design prescribes

gathering data in any particular cell

Ordered by panel within time period (column wise)

With this ordering, we get

11 21 1 12 2( , , , , , , , , )s s stY Y Y Y Y YY

22' '2'

' 'cov( , ) ii jj Eii Sij i j T

i i

Y Yn n

SARMMM 9/8/05

Now let X denote a regressor matrix containing a column

of 1’s and a column of the numbers of the time periods.

The second element of

contains an estimate of trend.

STATISTICAL MODEL - continuedSTATISTICAL MODEL - continued

1ˆ ( ' ) ' 1 1X X X Y

1 ' 2 1

If we let cov ( )= and cov( ) , then

cov( )

( ) ( )

S T

S i T s s E t in n

S Σ T Σ

Y

Σ D Σ 1 1 I D

SARMMM 9/8/05

STATISTICAL MODEL - continuedSTATISTICAL MODEL - continued

But this estimate of cannot be used because it is

based on values which, by design, will not be

gathered.

Reduce X, Y and to X*, Y*, and *, where

these

represent that subset of rows and columns

from

X, Y, and corresponding to where data will

be

gathered. Then

11 1

1 1

ˆ ( * ' * *) * ' *

ˆand cov( ) ( * ' * *)

X X X Y

X X

SARMMM 9/8/05

A STANDARDIZATIONA STANDARDIZATION

Note that

can be rewritten as

Consequently power, a measure of sensitivity, can be

examined relative to

22' '2'

' 'cov( , ) ii jj Eii Sij i j T

i i

Y Yn n

2 2' ' 2'

' ' 2 2cov( , ) ii jjii S T

ij i j Ei E E i

Y Yn n

2 2

2 2 and S T

E E

SARMMM 9/8/05

Trend: continuing, or monotonic, change. Practically,

monotonic trend can be detected by looking for

linear trend.

We will evaluate power in terms of ratios of variance

components and where this denominator

depends

on the ratios of variance components and the

revisit or temporal design.

TOWARD POWERTOWARD POWER

0 2ˆ

ˆ/ , so approximately, ~ ( , )E N

# 10SARMMM 9/8/05

POWER REFERENCEPOWER REFERENCEPOWER REFERENCEPOWER REFERENCE

Urquhart, N. S., S. G. Paulsen and D. P. Larsen.

(1998). Monitoring for policy-relevant regional

trends over time. Ecological Applications 8: 246 -

257.

# 11SARMMM 9/8/05

DESIGN and POWER DESIGN and POWER ofof

VEGETATION MONITORING STUDIESVEGETATION MONITORING STUDIESforfor

THE RIPARIAN ZONE NEAR THE THE RIPARIAN ZONE NEAR THE COLORADO RIVER COLORADO RIVER

in in

THE GRAND CANYONTHE GRAND CANYON

DESIGN and POWER DESIGN and POWER ofof

VEGETATION MONITORING STUDIESVEGETATION MONITORING STUDIESforfor

THE RIPARIAN ZONE NEAR THE THE RIPARIAN ZONE NEAR THE COLORADO RIVER COLORADO RIVER

in in

THE GRAND CANYONTHE GRAND CANYON

COOPERATORS

Mike Kersley, University of Northern Arizona, and

Steven P. Gloss, Program Manager-Biological Resources

Grand Canyon Monitoring & Research Center, USGS

# 12SARMMM 9/8/05

POWER TO DETECT TREND IN VEGETATION COVER,POWER TO DETECT TREND IN VEGETATION COVER,ZONE = 15, VARYING % TRENDZONE = 15, VARYING % TREND

POWER TO DETECT TREND IN VEGETATION COVER,POWER TO DETECT TREND IN VEGETATION COVER,ZONE = 15, VARYING % TRENDZONE = 15, VARYING % TREND

0

0.2

0.4

0.6

0.8

1

1 5 9 13 17 21

YEARS (-2000)

PO

WE

R

1%,

2%,

3%

5%

PER YEAR

# 13SARMMM 9/8/05

TODAY’S PATHTODAY’S PATHTODAY’S PATHTODAY’S PATH

Bit of historical background Distribution of sample sites along river Inquiry about your stat backgrounds Variation and its structure Power

Responses Zone

Responses to some questions asked during oral presentation

How the sample sites were selected How the power was calculated

Available Info – Probably not for today

# 14SARMMM 9/8/05

VIEW DOWN TRANSECT AT MILE 12.3VIEW DOWN TRANSECT AT MILE 12.3VIEW DOWN TRANSECT AT MILE 12.3VIEW DOWN TRANSECT AT MILE 12.3

# 15SARMMM 9/8/05

MARKING TRANSECT AT MILE 12.3MARKING TRANSECT AT MILE 12.3MARKING TRANSECT AT MILE 12.3MARKING TRANSECT AT MILE 12.3

# 16SARMMM 9/8/05

MIKE & MIKE & SCOTT AT SCOTT AT THE END!THE END!

MIKE & MIKE & SCOTT AT SCOTT AT THE END!THE END!

# 17SARMMM 9/8/05

CLIFF AT MILE CLIFF AT MILE 135.2135.2

(PARTIAL HEIGHT)(PARTIAL HEIGHT)

CLIFF AT MILE CLIFF AT MILE 135.2135.2

(PARTIAL HEIGHT)(PARTIAL HEIGHT)

# 18SARMMM 9/8/05

LOCATION OF SITES BY RIVER MILELOCATION OF SITES BY RIVER MILELOCATION OF SITES BY RIVER MILELOCATION OF SITES BY RIVER MILE

-15 25 65 105 145 185 225

RIVER MILE

Revisit Sites

2002 Sites

2001 Sites

# 19SARMMM 9/8/05

RESPONSE SIZE AND VARIATIONRESPONSE SIZE AND VARIATIONRESPONSE SIZE AND VARIATIONRESPONSE SIZE AND VARIATION

Data 2001 & 2002, including revisit sites Vegetation cover

Other responses, but not discussed here

Analysis model River Width (fixed)

Year (random) – proxy for roughness of immediate terrain

Station = river mile (random)

Residual = Year by Station interaction/remainder

# 20SARMMM 9/8/05

MEANMEAN and and STANDARD DEVIATIONSTANDARD DEVIATIONof VEGETATION COVER vs ZONE (RIVER FLOW LEVEL)of VEGETATION COVER vs ZONE (RIVER FLOW LEVEL)

MEANMEAN and and STANDARD DEVIATIONSTANDARD DEVIATIONof VEGETATION COVER vs ZONE (RIVER FLOW LEVEL)of VEGETATION COVER vs ZONE (RIVER FLOW LEVEL)

0

5

10

15

20

25

30

35

15 25 35 45 55 65

ZONE (RIVER FLOW LEVEL)

CO

VE

R

# 21SARMMM 9/8/05

STRUCTURE OF VARIANCESTRUCTURE OF VARIANCESTRUCTURE OF VARIANCESTRUCTURE OF VARIANCE

The common formulas for estimating (computing)

variance assume UNCORRELATED data.

Reality: This rarely is true. Examples -

Data from the same SITE, but different years are correlated

Data from the same YEAR, but different years are correlated

Total variance = var(site) + var(year) + var(residual)

Subsequent figures show this

# 22SARMMM 9/8/05

COMPONENTS of VARIANCE COMPONENTS of VARIANCE of VEGETATION COVERof VEGETATION COVERSITESITE, , YEARYEAR, and , and RESIDUALRESIDUAL

COMPONENTS of VARIANCE COMPONENTS of VARIANCE of VEGETATION COVERof VEGETATION COVERSITESITE, , YEARYEAR, and , and RESIDUALRESIDUAL

0

100

200

300

400

500

15 25 35 45 55 65

ZONE (RIVER FLOW LEVEL)

CO

VE

R

# 23SARMMM 9/8/05

SAMPLE SIZE ASSUMPTIONSSAMPLE SIZE ASSUMPTIONSFOR POWERFOR POWER

SAMPLE SIZE ASSUMPTIONSSAMPLE SIZE ASSUMPTIONSFOR POWERFOR POWER

25 revisit sites Revisited annually

30 sites to be visited on a three-year rotating cycle “Augmented Rotating Panel Design”

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 X 2 X X X X … 3 X X X X

# 24SARMMM 9/8/05

POWER TO DETECT TREND IN VEGETATION COVER,POWER TO DETECT TREND IN VEGETATION COVER,ZONE = 15, VARYING % TRENDZONE = 15, VARYING % TREND

POWER TO DETECT TREND IN VEGETATION COVER,POWER TO DETECT TREND IN VEGETATION COVER,ZONE = 15, VARYING % TRENDZONE = 15, VARYING % TREND

0

0.2

0.4

0.6

0.8

1

1 5 9 13 17 21

YEARS (-2000)

PO

WE

R

1%,

2%,

3%

5%

PER YEAR

# 25SARMMM 9/8/05

RESPONSE TO A QUESTIONRESPONSE TO A QUESTIONRESPONSE TO A QUESTIONRESPONSE TO A QUESTION

“What would be the effect of revisiting sites only in

alternating years after the first?” Response 1: My greatest concern would be retaining the

skills and knowledge of those doing the evaluations.

(Changing personnel would almost certainly change

response definitions in subtle, but unrecognized ways.)

Response 2: Power to detect trend would be delayed

somewhat. (Actually a bit more than I initially

thought!)

This is illustrated in the next two slides.

# 26SARMMM 9/8/05

ALTERNATE REVISIT PLAN and SAMPLE ALTERNATE REVISIT PLAN and SAMPLE SIZES ASSUMPTIONS FOR POWERSIZES ASSUMPTIONS FOR POWER

ALTERNATE REVISIT PLAN and SAMPLE ALTERNATE REVISIT PLAN and SAMPLE SIZES ASSUMPTIONS FOR POWERSIZES ASSUMPTIONS FOR POWER

25 revisit sites Revisited annually, for first three years (as planned),

then in alternating years

30 sites to be visited on a three-year rotating cycle A revisit plan with no specific name

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 … 1 X X X 2 X X X … 3 X X

# 27SARMMM 9/8/05

0

0.2

0.4

0.6

0.8

1

1 5 9 13 17 21

YEARS (-2000)

PO

WE

R

15

25 35

45

POWER TO DETECT TREND (2%PER YEAR) IN COVER by ZONE andPOWER TO DETECT TREND (2%PER YEAR) IN COVER by ZONE and

REVISIT PLANSREVISIT PLANS:: CURRENT = CURRENT = n n ; ; ALTERNATE =ALTERNATE = llPOWER TO DETECT TREND (2%PER YEAR) IN COVER by ZONE andPOWER TO DETECT TREND (2%PER YEAR) IN COVER by ZONE and

REVISIT PLANSREVISIT PLANS:: CURRENT = CURRENT = n n ; ; ALTERNATE =ALTERNATE = ll

# 28SARMMM 9/8/05

OBSERVATIONS RELATIVE TO POWER UNDER THE OBSERVATIONS RELATIVE TO POWER UNDER THE BIANNUAL REVISIT PLANBIANNUAL REVISIT PLAN

OBSERVATIONS RELATIVE TO POWER UNDER THE OBSERVATIONS RELATIVE TO POWER UNDER THE BIANNUAL REVISIT PLANBIANNUAL REVISIT PLAN

The loss of power for biannual revisits compared to the augmented serially

alternating design has some noteworthy characteristics: Power is the order of a quarter to a third for all years less than a decade.

The time required to get to a given level of power is extended by 3-5 years in the biannual revisit design.

The "years" on the x-axis represents the starting point for ANY comparison

Power accrues from accumulating data, elapsed time, and accumulating trend

Detection of moderate trends requires a commitment to the continuing acquisition consistent and comparable data.

These power evaluations DO NOT relate to comparing years 10 to 11, or any specific two years.

# 29SARMMM 9/8/05

MODEL ADAPTATIONMODEL ADAPTATIONMODEL ADAPTATIONMODEL ADAPTATION

Have a set of panels for the “untreated” sites,

another for the “treated” sites.

Change X*, but not Y*or *:

Then get standard error of 0 1 0 1

untreated

Treated

X 01 01 X

X0 10

β

# 30SARMMM 9/8/05

POWER TO DETECT DIFFERENCES POWER TO DETECT DIFFERENCES IN TRENDIN TREND

((BETWEEN “TREATED and “UNTREATED”BETWEEN “TREATED and “UNTREATED”))

POWER TO DETECT DIFFERENCES POWER TO DETECT DIFFERENCES IN TRENDIN TREND

((BETWEEN “TREATED and “UNTREATED”BETWEEN “TREATED and “UNTREATED”))

COOPERATORS

Phil Larsen, WED(Part of a presentation for the American Fisheries Society next week)

Oregon Plan Team

# 31SARMMM 9/8/05

SOURCE OF ESTIMATES OF SOURCE OF ESTIMATES OF VARIANCE COMPONENTSVARIANCE COMPONENTS

SOURCE OF ESTIMATES OF SOURCE OF ESTIMATES OF VARIANCE COMPONENTSVARIANCE COMPONENTS

Data source -----

Response is log of large woody debris

Log10(LWD+0.1)

Variance components values were selected as low and high

All plans assume annual revisit

Number of sites in each set = 5, 10, 15, 20, 25

# 32SARMMM 9/8/05

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

ALL POWER CURVES – SET #1 (8/16/05)ALL POWER CURVES – SET #1 (8/16/05)ALL POWER CURVES – SET #1 (8/16/05)ALL POWER CURVES – SET #1 (8/16/05)

# 33SARMMM 9/8/05

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

n = 25

n=5

POWER CURVES FOR DETECTING DIFFERENCES IN TREND

(LOW VALUES OF VARIANCE COMPONENTS)

# 34SARMMM 9/8/05

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

n = 25

n = 5

POWER CURVES FOR DETECTING DIFFERENCES IN TREND

(HIGH VALUES OF VARIANCE COMPONENTS)

# 35SARMMM 9/8/05

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

POWER CURVES FOR DETECTING DIFFERENCES IN TREND

(LOW vs HIGH VALUES OF VARIANCE COMPONENTS, n = 10 EACH)

# 36SARMMM 9/8/05

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

POWER CURVES n = 20POWER CURVES n = 20ALWAYS REVISIT SAME SITES ALWAYS REVISIT SAME SITES versusversus AUGMENTED SERIALLY ALTERNATINGAUGMENTED SERIALLY ALTERNATING

FOR HIGH VALUES OF VARIANCE COMPONENTSFOR HIGH VALUES OF VARIANCE COMPONENTS

# 37SARMMM 9/8/05

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.

POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS; AUGMENTED ROTATING POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS; AUGMENTED ROTATING PANEL DESIGNPANEL DESIGN

POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS; AUGMENTED ROTATING POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS; AUGMENTED ROTATING PANEL DESIGNPANEL DESIGN

# 38SARMMM 9/8/05

TOWARD POWER TO TOWARD POWER TO DETECT REGIONAL TRENDDETECT REGIONAL TREND

WHEN TREND VARIES BY SITEWHEN TREND VARIES BY SITE

TOWARD POWER TO TOWARD POWER TO DETECT REGIONAL TRENDDETECT REGIONAL TREND

WHEN TREND VARIES BY SITEWHEN TREND VARIES BY SITE

COOPERATORS

Phil Larsen, WED, EPA

Tim Gerrodette, National Marine Fisheries Service,

Southwest Science Center, La Jolla, CA

Dawn Van Leeuwen, New Mexico State Univ

Will use Oregon Plan data

# 39SARMMM 9/8/05

INITIAL SIMPLIFYING CONDITIONSINITIAL SIMPLIFYING CONDITIONSINITIAL SIMPLIFYING CONDITIONSINITIAL SIMPLIFYING CONDITIONS

Every site in a region is revisited every year, and

Some relevant response is evaluated.

# 40SARMMM 9/8/05

2

2

2

~ (0, ), 1,2, ,

~ (0, ), 1,2, ,

~ (0, ).

ij i j j ij

i S

j T

ij

Y S T t R

S i s

T j t

R

Consider this model:

where

all uncorrelated

For now, let be a constant, but shortly we will investigate

the eff ect of it be

2 2' '

' '

( )( ) ( )ˆ ( )

( ) ( )

j ij i j ijj j

i j i j j ijjj j

j j

t t Y Y t t Y

c S T t Rt t t t

ing a random variable.

Now consider the site estimate of trend:

# 41SARMMM 9/8/05

' ' ' ' ' ''

22

' ' 2'

2

ˆ

ˆ ˆcov( , ) cov ( ), ( )

cov , var ,( )

ˆ1ˆvar var

The have a nonzero covariance:

so,

i

i i j j j ij j j j i jj j

Tj j j j j j

j j j jj

ii

c T t R c T t R

c T c T c Tt t

s s

''

22

2

ˆ ˆ ˆvar( ) cov( , )

1

( )

i i ii i i

Tj

j

t t s

# 42SARMMM 9/8/05

2

2

2

2

~ ( , )

~ (0, ), 1,2, ,

~ (0, ), 1,2, ,

~ (0, ).

Consider this more general (hierarchical) model:

where

all uncorrelated

Now that is a random variable, its varia

ij i j i j ij

i

i S

j T

ij

Y S T t R

S i s

T j t

R

2 2' '

' '

( )( ) ( )ˆ ( )

( ) ( )

nce will enter

into the variances we have considered. Now consider

the site estimate of trend:

j ij i j ijj j

i j i j i j ijjj j

j j

t t Y Y t t Y

c S T t Rt t t t

# 43SARMMM 9/8/05

2 22 2 2 2 2

2

22

2

ˆ ( ) ( )

ˆvar( )

1ˆvar( )

i j i j i j ij j j i j ijj j

Ti j T j j

j j jj

Tj

j

c S T t R c T t R

c c tt t

t t s

With random, previous f ormulas change to:

, and

,

and so

2B

s

# 44SARMMM 9/8/05

WHERE NEXT WITH RANDOM SLOPES?WHERE NEXT WITH RANDOM SLOPES?WHERE NEXT WITH RANDOM SLOPES?WHERE NEXT WITH RANDOM SLOPES?

Model adapts to matrices and varied revisit plans

should be incorporated into power

calculations

Estimate magnitude of Montana Bull Trout

Various Oregon Plan responses

Develop web-based software This is where Tim Gerrodette enters

– This generalizes something he did earlier

2B

s

2B