DEVELOPMENTS
IN
TREND DETECTION
IN
AQUATIC SURVEYS
N. Scott Urquhart
STARMAP Program Director
Department of Statistics
Colorado State University
SARMMM 9/8/05
# 1
PATH 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
SARMMM 9/8/05
# 2
A SIMPLE MODEL for a
SURVEYED 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
# 3
A STATISTICAL MODEL
Yijk  Sik  T j  Eijk
where
i 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):
Sik ~ ( ,  S2 )
SARMMM 9/8/05
T j ~ (0,  T2 )
Eijk ~ (0,  E2 )
A 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)
Y  (Y11 , Y21 , , Ys1 , Y12 , , Ys 2 , , Yst )
With this ordering, we get
cov(Yij , Yi ' j ' ) 
SARMMM 9/8/05
 ii '
ni
2
S
  T2 
 ii ' jj ' E2
ni
STATISTICAL MODEL - continued
If we let cov (S)=ΣS and cov( T )  ΣT , then
cov(Y)  
 ΣS  D (ni )  ΣT  1 1   I  D (ni )
1

'
s s
2
E t
1
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
1
1
1
ˆ
  (X '  X) X '  Y
contains an estimate of trend.
SARMMM 9/8/05
STATISTICAL 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
1
1
1
ˆ
  ( X * '  * X*) X * '  * Y
and cov( ˆ )  ( X * '  *1 X*) 1
SARMMM 9/8/05
A STANDARDIZATION

Note that
cov(Yij , Yi ' j ' ) 
can be rewritten as
cov(Yij , Yi ' j '

 ii ' S2
ni
  T2 
 ii ' jj ' E2
ni
  ii '   S2    T2   ii ' jj '  2
)   2  2 
 E
ni 
 ni   E    E 
Consequently power, a measure of sensitivity, can be
examined relative to
  S2 
  T2 
 2  and  2 
E 
 E 
SARMMM 9/8/05
TOWARD POWER

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.
   0 /  E , so approximately, ˆ ~ N ( ,  2ˆ )
SARMMM 9/8/05
POWER 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.
SARMMM 9/8/05
# 10
DESIGN and POWER
of
VEGETATION MONITORING STUDIES
for
THE RIPARIAN ZONE NEAR THE
COLORADO RIVER
in
THE GRAND CANYON
COOPERATORS
Mike Kersley, University of Northern Arizona, and
Steven P. Gloss, Program Manager-Biological Resources
Grand Canyon Monitoring & Research Center, USGS
SARMMM 9/8/05
# 11
POWER TO DETECT TREND IN VEGETATION COVER,
ZONE = 15, VARYING % TREND
1
POWER
0.8
1%,
0.6
2%,
3%
0.4
5%
PER YEAR
0.2
0
1
5
9
13
17
21
YEARS (-2000)
SARMMM 9/8/05
# 12
TODAY’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
SARMMM 9/8/05
Available Info –
Probably not for today
# 13
VIEW DOWN TRANSECT AT MILE 12.3
SARMMM 9/8/05
# 14
MARKING TRANSECT AT MILE 12.3
SARMMM 9/8/05
# 15
MIKE &
SCOTT AT
THE END!
SARMMM 9/8/05
# 16
CLIFF AT MILE
135.2
(PARTIAL HEIGHT)
SARMMM 9/8/05
# 17
LOCATION OF SITES BY RIVER MILE
Revisit Sites
2002 Sites
2001 Sites
-15
25
65
105
145
185
225
RIVER MILE
SARMMM 9/8/05
# 18
RESPONSE 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
SARMMM 9/8/05
# 19
MEAN and STANDARD DEVIATION
of VEGETATION COVER vs ZONE (RIVER FLOW LEVEL)
35
30
COVER
25
20
15
10
5
0
15
25
35
45
55
65
ZONE (RIVER FLOW LEVEL)
SARMMM 9/8/05
# 20
STRUCTURE 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
SARMMM 9/8/05
# 21
COMPONENTS of VARIANCE of VEGETATION COVER
SITE, YEAR, and RESIDUAL
500
COVER
400
300
200
100
0
15
25
35
45
55
65
ZONE (RIVER FLOW LEVEL)
SARMMM 9/8/05
# 22
SAMPLE SIZE ASSUMPTIONS
FOR POWER

25 revisit sites


Revisited annually
30 sites to be visited on a three-year rotating cycle

“Augmented Rotating Panel Design”
PANEL
0
1
2
3
SARMMM 9/8/05
TIME PERIOD ( ex: YEARS)
1 2 3 4 5 6 7 8 9 10 11 12 13 ...
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
# 23
POWER TO DETECT TREND IN VEGETATION COVER,
ZONE = 15, VARYING % TREND
1
POWER
0.8
1%,
0.6
2%,
3%
0.4
5%
PER YEAR
0.2
0
1
5
9
13
17
21
YEARS (-2000)
SARMMM 9/8/05
# 24
RESPONSE 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.
SARMMM 9/8/05
# 25
ALTERNATE REVISIT PLAN and SAMPLE
SIZES 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
PANEL
0
1
2
3
SARMMM 9/8/05
TIME PERIOD ( ex: YEARS)
1 2 3 4 5 6 7 8 9 10 11 12 13 ...
X X X
X
X
X
X
X …
X
X
X
X
X
X …
X
X
# 26
POWER TO DETECT TREND (2%PER YEAR) IN COVER by ZONE and
REVISIT PLANS: CURRENT = n ; ALTERNATE = l
1
POWER
0.8
15
0.6
25
0.4
35
45
0.2
0
1
5
9
13
17
21
YEARS (-2000)
SARMMM 9/8/05
# 27
OBSERVATIONS RELATIVE TO POWER UNDER THE
BIANNUAL 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.
SARMMM 9/8/05
# 28
MODEL ADAPTATION

Have a set of panels for the “untreated” sites,
another for the “treated” sites.

Change X*, but not Y*or *:
 1 X untreated
1 X   
0 0
0 0


1 X Treated 
Then get standard error of  0 1 0 1 β
SARMMM 9/8/05
# 29
POWER TO DETECT DIFFERENCES
IN TREND
(BETWEEN “TREATED and “UNTREATED”)
COOPERATORS
Phil Larsen, WED
(Part of a presentation for the American Fisheries Society next week)
Oregon Plan Team
SARMMM 9/8/05
# 30
SOURCE OF ESTIMATES OF
VARIANCE 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
SARMMM 9/8/05
# 31
ALL POWER CURVES – SET #1 (8/16/05)
1
0.8
0.6
0.4
0.2
0
0
SARMMM 9/8/05
5
10
15
20
# 32
POWER CURVES FOR DETECTING DIFFERENCES
IN TREND
(LOW VALUES OF VARIANCE COMPONENTS)
1
0.8
n = 25
0.6
n=5
0.4
0.2
0
0
SARMMM 9/8/05
5
10
15
20
# 33
POWER CURVES FOR DETECTING DIFFERENCES
IN TREND
(HIGH VALUES OF VARIANCE COMPONENTS)
1
0.8
n = 25
0.6
n=5
0.4
0.2
0
0
SARMMM 9/8/05
5
10
15
20
# 34
POWER CURVES FOR DETECTING DIFFERENCES
IN TREND
(LOW vs HIGH VALUES OF VARIANCE COMPONENTS, n = 10 EACH)
1
0.8
0.6
0.4
0.2
0
0
SARMMM 9/8/05
5
10
15
20
# 35
POWER CURVES n = 20
ALWAYS REVISIT SAME SITES versus AUGMENTED SERIALLY ALTERNATING
FOR HIGH VALUES OF VARIANCE COMPONENTS
1
0.8
0.6
0.4
0.2
0
0
SARMMM 9/8/05
5
10
15
20
# 36
POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS; AUGMENTED ROTATING
PANEL DESIGN
1
0.8
n = 6, 24
0.6
n = 2,8
0.4
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.
0.2
0
0
SARMMM 9/8/05
5
10
15
20
# 37
TOWARD POWER TO
DETECT REGIONAL TREND
WHEN 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
SARMMM 9/8/05
# 38
INITIAL SIMPLIFYING CONDITIONS


Every site in a region is revisited every year, and
Some relevant response is evaluated.
SARMMM 9/8/05
# 39
Consider this model:
Yij    Si  T j   t j  Rij
where
Si ~ (0, S2 ), i  1,2, , s 

2
T j ~ (0, T ), j  1,2, , t  all uncorrelated

2
Rij ~ (0, ).

For now, let  be a constant, but shortly we will investigate
the effect of it being a random variable.
Now consider the site estimate of trend:
ˆi
 (t  t )(Y Y

 (t  t )
j
ij
i
j
2
j'
j'
SARMMM 9/8/05
)
 (t  t )Y

 (t  t )
j
ij
j
2
j'
j'
  c j (  Si  T j   t j  Rij )
j
# 40
The ˆi have a nonzero covariance:


ˆ
ˆ
cov(  i ,  i ' )  cov   c j (T j   t j  Rij ),  c j ' (T j '   t j '  Ri ' j ' ) 
j'
 j



 cov   c jT j ,  c j 'T j '    c 2j var T j  
j'
 j
 j
 T2
 (t
j
t)
2
,
j
so,
  ˆi
var ˆ  var  i
 s


 

   1   var( ˆ )  cov( ˆ , ˆ ) 

i
i
i' 
  s 2   
i
i i '




 2
1

 
2

 T 
2 

  (t j  t )   s
 j

SARMMM 9/8/05
# 41
Consider this more general (hierarchical) model:
Yij    Si  T j  i t j  Rij
where
i ~ (  , 2 )
Si ~ (0, S2 ),
T j ~ (0, T2 ),
Rij ~ (0, 2 ).
Now that 


i  1,2, , s 
 all uncorrelated
j  1,2, , t 


is a random variable, its variance will enter
into the variances we have considered. Now consider
the site estimate of trend:
ˆ

i
 (t  t )(Y Y

 (t  t )
j
ij
i
j
2
j'
j'
SARMMM 9/8/05
)
 (t  t )Y

  c (  S
 (t  t )
j
ij
j
j
2
j'
j'
i
 T j  i t j  Rij )
j
# 42
With  random, previous formulas change to:
ˆ   c (  S  T   t  R )   c (T   t  R ), and

i
j
i
j
i j
ij
j
j
i j
ij
j
j
2
2







2
T
ˆ    c 2   2   2     c t   2 
var 


i
j
T
j j

,
2
 (t j  t )
 j 
 j

 
j
and so

 2
2


1




2
B
ˆ 
var 



T 
 (t  t ) 2   s
 s
 j

 j

 
SARMMM 9/8/05
# 43
WHERE NEXT WITH RANDOM SLOPES?


Model adapts to matrices and varied revisit plans
 B2
s


should be incorporated into power
calculations
Estimate magnitude of
 B2

Montana Bull Trout

Various Oregon Plan responses
Develop web-based software

This is where Tim Gerrodette enters
– This generalizes something he did earlier
SARMMM 9/8/05
# 44