TWG 2/27/03 

DESIGN and POWER 

of

VEGETATION MONITORING 

for

THE RIPARIAN ZONE NEA

COLORADO RIVER 

in 

THE GRAND CANYON

DESIGN and POWER DESIGN and POWER 

of

VEGETATION MONITORING 

VEGETATION MONITORING 

for

THE RIPARIAN ZONE NEA

THE RIPARIAN ZONE NEA

COLORADO RIVER COLORADO RIVER 

in 

THE GRAND CANYON THE GRAND CANYON

N. Scott Urquhart

STARMAP Program Direc

Department of Statistics

Colorado State University
TWG 2/27/03 

MY PLAN versus REALIT

continued

MY PLAN versus REALIT

MY PLAN versus REALIT

continued continued

 I agreed to give a management-orient

technical talk I gave at GCMRC in De

Î Interrupting my vacation!

 That talk was about 85% done when 

night I received a message from Dennencouraging me to address a VERY dquestions.

 NOW: Briefly respond to several of D

questions, then give most of the talk I 
TWG 2/27/03 

QUESTIONS DENNIS ASK

(Abbreviated versions)

QUESTIONS DENNIS ASK

QUESTIONS DENNIS ASK

(Abbreviated versions) (Abbreviated versions)

(1) Compare roles of scientists and managers in adaptiv

programs? 

(2) Prioritizing monitoring and research: compromises r

declining budget & conflicting views 

(3) Evaluating utility of gathered information what are t

to managers of information gathered, analyzed, 

scientists?

(4) Does some information have higher value than other (5) Tradeoffs between sampling designs that allow extra

Grand Canyon and those that do not? When is eappropriate?

(6) Risk assessment: Potential effects of reduced samplin

consequent lower levels of detection of resource necessitated by funding or logistical limitations?

(7) When should we worry about Type II errors more th

How do they differ?
TWG 2/27/03 

MONITORING IS NOT RESEA

impacts on answers to questions 1 – 

MONITORING IS NOT RESEA

MONITORING IS NOT RESEA

impacts on answers to questions 1 impacts on answers to questions 1 – 

 Research concerns how & why things 

Î May need to be temporally intensive

 Monitoring concerns “What has happ

Î Major differences: measures & tempora

z EX: Mike Kearsley’s vegetation index versus d

 Adaptive management may require s

Î But managers need to look critically at th

research.

z Specifically: Its linkage to manageable actions
TWG 2/27/03 

QUESTION 5: What are the tradeoffs bet

designs that allow extrapolation to the entire G

those that do not? When is each most ap

QUESTION 5: QUESTION 5: What are the tradeoffs bet

What are the tradeoffs bet

designs that allow extrapolation designs that allow extrapolation to the entire G

to the entire G

those that do not? When is each most app

those that do not? When is each most ap

 When you need to make a statement a

area, sample it.

Î Biological investigators know far less abo

resources reside than they think they dothings turn up where they “aren’t suppo

 Model development: Targeted site se

appropriate – even necessary

Î Pick gradient of sites which will su

estimation of model components.
TWG 2/27/03 

POWER TO DETECT TREND IN VEGETA

ZONE = 15, VARYING % TREN

POWER TO DETECT TREND IN VEGETA

POWER TO DETECT TREND IN VEGETA

ZONE = 15, VARYING % TREN

ZONE = 15, VARYING % TREN

0

0.20.40.60.8

1

1 5 9 13 

YEARS (-2000)

POWER
TWG 2/27/03 

TODAY’S PATH TODAY’S PATH

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

Î ResponsesÎ Zone

 Responses to some questions asked during oral prese How the sample sites were selected How the power was calculated

A

Prob
TWG 2/27/03 

REPORT

from

THE PEER REVIEW PANE

on

THE TERRESTRIAL COM

of the 

BIOLOGICAL RESOURCES PR

of

THE GRAND CANYON MONIT

and 

RESEARCH CENTER

REPORT

from

THE PEER REVIEW PANE

THE PEER REVIEW PANE

on

THE TERRESTRIAL COM

THE TERRESTRIAL COM

of the 

BIOLOGICAL RESOURCES PRO

BIOLOGICAL RESOURCES PR

of

THE GRAND CANYON MONIT

THE GRAND CANYON MONIT

and 

RESEARCH CENTER RESEARCH CENTER
TWG 2/27/03 

THE “BEGINNING” THE “BEGINNING” THE “BEGINNING” – MARC

– MARC

MARC
TWG 2/27/03 

THE PANEL THE PANEL THE PANEL
TWG 2/27/03 

MONITORING COMPONENTS

PLANT AND ANIMAL INVEN

LONG-TERM MONITORIN

MONITORING COMPONENTS MONITORING COMPONENTS

PLANT AND ANIMAL INVEN

PLANT AND ANIMAL INVEN

LONG-TERM MONITORIN

TERM MONITORIN

 DEFINE THE DOMAIN WHICH BO

COVER

Î PANEL STRONGLY RECOMMENDS TH

MAIN-STEM CORRIDOR + RELEVANT 

 CONDUCT A PROBABILITY SAMPLE 

DOMAIN. PROBABILITY OF POINVARIED IN MANY PRACTICAL W

 CONDUCT INVENTORY AT THOSE P

Î PERHAPS SPREAD OUT OVER FOUR Y

z PERHAPS FOUR TIMES WITHIN EACH YEAz COVERING THE ENTIRE CORRIDOR EACH 
TWG 2/27/03 

RESULT OF REVIEW PAN

SUGGESTIONS

RESULT OF REVIEW PAN

RESULT OF REVIEW PAN

SUGGESTIONS SUGGESTIONS

 GCMRC advertised for someone to to 

vegetation monitoring studies alonsuggested by the panel.

 Mike Kearsley (NAU) bid on that RF

Î Bid included UNM + HYC

 Mike asked me to help determine and 

transects running up from the rive60 k cfs level.

 Transects laid out June/July, 2001.
TWG 2/27/03 

A QUESTION A QUESTION A QUESTION

 QUESTION: “What information did 

Review Panel have access to?

 RESPONSE: 

Î The Panel received about 15 documents, 

z Background information on the process for coordinating and 

Adaptive Management Working Group’s information needsmanagement objectives (MOs) and information needs (INs). 

z Melis, T., M. Liszewski, B. Gold, L. Stevens, F.M. Gonzale

W. Vernieu, and B. Ralston. (undated). Draft prospectus fomonitoring protocols for the Colorado River ecosystem.

z Webb, R.H., D.L. Wegner, E.D. Andrews, R.A. Valdez, and 

Downstream effects of Glen Canyon Dam on the Colorado Rreview. In “The controlled flood in Grand Canyon,” R.H. WGeophysical Monograph 110, American Geophysical Union
TWG 2/27/03 

VIEW DOWN TRANSECT AT MILE 

VIEW DOWN TRANSECT AT MILE 

VIEW DOWN TRANSECT AT MIL
TWG 2/27/03 

ELEVATION CONTROL POINT F

TRANSECT AT MILE 12.3

ELEVATION CONTROL POINT F

LEVATION CONTROL POINT F

TRANSECT AT MILE 12.3 TRANSECT AT MILE 12.3
TWG 2/27/03 

CLIFF AT MILE 

135.2

(PARTIAL HEIGHT)

CLIFF AT MILE CLIFF AT MILE 

135.2

(PARTIAL HEIGHT) (PARTIAL HEIGHT)
TWG 2/27/03 

LOCATION OF SITES BY RIVE

LOCATION OF SITES BY RIVE

LOCATION OF SITES BY RIVE

-15 25 65 105 145 

RIVER MILE

Revisit Sites 

2002 Sites2001 Sites
TWG 2/27/03 

“QUICK QUIZ” QUICK QUIZ” QUICK QUIZ”

 How many of you have taken a statist

Î HANDS UP!

 Within the last five years? Took more stat than was required? Remember hearing the word variance  Remember how to compute a varianc
TWG 2/27/03 

WHAT WAS VARIANCE

WHAT WAS VARIANCE

WHAT WAS VARIANCE

 It was something you could compute 

characterized how spread out a se

Î A small variance meant data was rather Î A large variance mean the data was spreÎ Next slide illustrates this idea

 Detail: variance = (standard deviatio

Î σ2 - unknown value Î s2 - its estimate 
TWG 2/27/03 

MEANS AND STANDARD DEVI

MEANS AND STANDARD DEVI

MEANS AND STANDARD DEVI

 How do means and standard deviatio

data?

Î The range:

z mean – standard deviation to mean + standard 

contains 60% to 70% of data values (67% for ndata)

 even for very non-normal data

z Multiply the standard deviation by above 2, and 

increase to 90 to 99% ( 95% for normal data)
TWG 2/27/03 

MEANS AND STANDARD DEVI

MEANS AND STANDARD DEVI

MEANS AND STANDARD DEVI

 Lets use vegetation index to illustrate 

Î At 25 kcfs, it has 

z a mean of about 30 ( μ = 30)z a standard deviation of about 11 ( σ = 11)z Illustration is very nonnormal (for purposes of 

Î Second illustration has same mean, smal

standard deviation:

z μ = 30 and σ = 5.5
TWG 2/27/03 

0.0000.0050.0100.0150.0200.0250.030

0 10 20 30 40 

VEGETATION INDEX (PSE

RELATIVE FREQUENCY
TWG 2/27/03 

0.0000.0050.0100.0150.0200.0250.030

0 10 20 30 40 

VEGETATION INDEX (PSE

RELATIVE FREQUENCY

Range μ - 2σ = 8 to μ - 2σ = 52

includes 99+% of frequency here

Range μ - σ = 19 to μ - σ = 41 

includes ~60+% of frequency 

here

ILLUSTRATION μ = 30 & σ = 
TWG 2/27/03 

0.0000.0050.0100.0150.0200.0250.030

0 10 20 30 

VEGETATION INDEX (PSEU

RELATIVE FREQUENCY

ILLUSTRATION μ = 30 & σ = 5

19 to 4125.5 to 35
TWG 2/27/03 

0.0000.0050.0100.0150.0200.0250.030

0 10 20 30 40 

VEGETATION INDEX (PSE

RELATIVE FREQUENCY

ILLUSTRATION - COMPARIS

μ = 30 & σ = 11 with μ = 30 & σ 
TWG 2/27/03 

RESPONSE SIZE AND VARI

RESPONSE SIZE AND VARI

RESPONSE SIZE AND VARI

 Data 2001 & 2002, including revisit si

z Vegetation coverz Richness of vegetation speciesz Diversity index (H’)

 Analysis model

z Width (fixed)z Year (random)z Station = river mile (random)z Residual = Year by Station interaction/remaind
TWG 2/27/03 

A QUESTION A QUESTION A QUESTION

 Data 2001 & 2002, including revisit sites

z Vegetation coverz Richness of vegetation speciesz Diversity index (H’)

 QUESTION: “How were all of the question

TWG represented in these three variabl

Î ANSWER: They weren’t. These were a

analysis. Many other variables were ev

z Insectsz Birdsz Reptiles
TWG 2/27/03 

MEAN and STANDARD DEVIAT

of VEGETATION COVER vs ZONE (RIVER 

MEAN and STANDARD DEVIAT

STANDARD DEVIAT

of VEGETATION COVER of VEGETATION COVER vs ZONE (RIVER 

ZONE (RIVER 

05

101520253035

15 25 35 45 

ZONE (RIVER FLOW LEV

COVER
TWG 2/27/03 

MEAN and STANDARD DEVIAT

of RICHNESS vs ZONE (RIVER FLOW 

MEAN and STANDARD DEVIAT

STANDARD DEVIAT

of RICHNESS of RICHNESS vs ZONE (RIVER FLOW 

ZONE (RIVER FLOW 

0123456

15 25 35 45 

ZONE (RIVER FLOW LEVE

RICHNESS
TWG 2/27/03 

MEAN and STANDARD DEVIAT

of DIVERSITY vs ZONE (RIVER FLOW 

MEAN and STANDARD DEVIAT

STANDARD DEVIAT

of DIVERSITY of DIVERSITY vs ZONE (RIVER FLOW 

ZONE (RIVER FLOW 

0

0.10.20.30.4

15 25 35 45 

ZONE (RIVER FLOW LEV

DIVERSITY
TWG 2/27/03 

STRUCTURE OF VARIAN

STRUCTURE OF VARIAN

STRUCTURE OF VARIAN

 The common formulas for estimating 

variance assume UNCORRELAT

 Reality: This rarely is true. 

Î Examples -

z Data from the same SITE, but different years z Data from the same YEAR, but different years 

Î Total variance = var(site) + var(year) + v

 Subsequent figures show this
TWG 2/27/03 

COMPONENTS of VARIANCE of VEGETA

SITE, YEAR, and RESIDUAL

COMPONENTS of VARIANCE COMPONENTS of VARIANCE of VEGETA

of VEGETA

SITE, YEAR, and RESIDUAL RESIDUAL

0

100200300400500

15 25 35 45 

ZONE (RIVER FLOW LEV

COVER
TWG 2/27/03 

COMPONENTS of VARIANCE of RIC

SITE, YEAR, and RESIDUAL

COMPONENTS of VARIANCE COMPONENTS of VARIANCE of RIC

of RIC

SITE, YEAR, and RESIDUAL RESIDUAL

05

1015

15 25 35 45 

ZONE (RIVER FLOW LEV

RICHNESS
TWG 2/27/03 

COMPONENTS of VARIANCE of DIV

SITE, YEAR, and RESIDUAL

COMPONENTS of VARIANCE COMPONENTS of VARIANCE of DIV

of DIV

SITE, YEAR, and RESIDUAL RESIDUAL

0

0.05

0.1

15 25 35 45 

ZONE (RIVER FLOW LEV

DIVERSITY
TWG 2/27/03 

POWER IS? POWER IS? POWER IS? 

 Variation causes uncertainty in maki

Î Statistical tests usually are described as 

z Significant (there is a “difference”), orz Not significant (there is no difference)

Î POWER describes the likelihood of findi

when an effect really is there.

z POWER = Prob(correct decision)

Î Depends on many things

z Amount of relevant data (“n”)z The size of the effect of interestz Amount of variance & its structure

– to a STA

to a STA
TWG 2/27/03 

POWER FOR TREND DETE

POWER FOR TREND DETE

POWER FOR TREND DETE

 Trend = generally continuing change 

direction 

Î Increasing, orÎ DecreasingÎ Even if it trend curves, it always will disp

 Revisits to previously visited sites

Î Important to remove the site effect from 

trend

Î Some sites need to be revisited annually 

effect of years from what it would otherw
TWG 2/27/03 

SAMPLE SIZE ASSUMPT

FOR POWER

SAMPLE SIZE ASSUMPTI

SAMPLE SIZE ASSUMPTI

FOR POWER FOR POWER

 25 revisit sites

Î Revisited annually

 30 sites to be visited on a three-year r

Î “Augmented Rotating Panel Design”

 TIME PERIOD ( ex: YEA

PANEL 1 2 3 4 5 6 7 8 9 10  0 X X X X X X X X X X  1 X X X X  2 X X X  3 X X X 
TWG 2/27/03 

POWER TO DETECT TREND IN VEGETA

ZONE = 15, VARYING % TREN

POWER TO DETECT TREND IN VEGETA

POWER TO DETECT TREND IN VEGETA

ZONE = 15, VARYING % TREN

ZONE = 15, VARYING % TREN

0

0.20.40.60.8

1

1 5 9 13 

YEARS (-2000)

POWER
TWG 2/27/03 

POWER TO DETECT TREND (2%PE

IN COVER by ZONE

POWER TO DETECT TREND (2%PE

POWER TO DETECT TREND (2%PE

IN COVER by ZONE IN COVER by ZONE

0

0.20.40.60.8

1

1 5 9 13 

YEARS (-2000)

POWER

15

25

35 45
TWG 2/27/03 

POWER TO DETECT TREND (2% PE

IN RICHNESS, by ZONE

POWER TO DETECT TREND (2% PE

POWER TO DETECT TREND (2% PE

IN RICHNESS, by ZONE IN RICHNESS, by ZONE

0

0.20.40.60.8

1

1 5 9 13 

YEARS (-2000)

POWER

15

25

45 
TWG 2/27/03 

POWER TO DETECT TREND (2% PE

IN DIVERSITY INDEX, by ZON

POWER TO DETECT TREND (2% PE

POWER TO DETECT TREND (2% PE

IN DIVERSITY INDEX, by ZON

IN DIVERSITY INDEX, by ZON

0

0.20.40.60.8

1

1 5 9 13 

YEARS (-2000)

POWER

25

35 45
TWG 2/27/03 

RESPONSE TO A QUEST

RESPONSE TO A QUEST

RESPONSE TO A QUEST

 “What would be the effect of revisitin

alternating years after the first?”

Î Response 1: My greatest concern would 

skills and knowledge of those doing the (Changing personnel would almost certresponse definitions in subtle, but unrec

Î Response 2: Power to detect trend woul

somewhat. (Actually a bit more than I i

Î This is illustrated in the next two slides.
TWG 2/27/03 

ALTERNATE REVISIT PLAN an

SIZES ASSUMPTIONS FOR 

ALTERNATE REVISIT PLAN and ALTERNATE REVISIT PLAN and 

SIZES ASSUMPTIONS FOR P

SIZES ASSUMPTIONS FOR 

 25 revisit sites

Î Revisited annually, for first three years (

then in alternating years

 30 sites to be visited on a three-year r

Î A revisit plan with no specific name

 TIME PERIOD ( ex: YEA

PANEL 1 2 3 4 5 6 7 8 9 10  0 X X X X X X  1 X X  2 X X  3 X X 
TWG 2/27/03 

0

0.20.40.60.8

1

1 5 9 13 

YEARS (-2000)

POWER

15

25 35

POWER TO DETECT TREND (2%PER YEAR) IN COV

REVISIT PLANS: CURRENT = n ; ALT

POWER TO DETECT TREND (2%PER YEAR) IN COV

POWER TO DETECT TREND (2%PER YEAR) IN COV

REVISIT PLANS REVISIT PLANS: CURRENT = CURRENT = n ; ALT

ALT
TWG 2/27/03 

OBSERVATIONS RELATIVE TO POWER 

BIANNUAL REVISIT PLAN

OBSERVATIONS RELATIVE TO POWER 

OBSERVATIONS RELATIVE TO POWER 

BIANNUAL REVISIT PLAN BIANNUAL REVISIT PLAN

 The loss of power for biannual revisits compared to t

alternating design has some noteworthy charact

Î Power is the order of a quarter to a third for all years lÎ The time required to get to a given level of power is ext

in the biannual revisit design.

 The "years" on the x-axis represents the starting poin

ANY comparison 

Î Power accrues from accumulating data, elapsed time, aÎ Detection of moderate trends requires a commitment to 

acquisition consistent and comparable data.

Î These power evaluations DO NOT relate to comparing 

or any specific two years.

z Neither design “fills up a tank with power” so you ca

comparisons regardless of how often you measure v
TWG 2/27/03 

ANOTHER QUESTION ANOTHER QUESTION ANOTHER QUESTION

 Can “Whole Canyon” estimates be ob

these results and sampling plan?

 RESPONSE: YES – with some quali

Î For some, but not all, of the responses ev

z Indices like diversity don’t combine across site

Î For the whole Canyon below the 60 kcfs 

z and by geologic reachz More accurate estimates would require quite a 

 Need areas associated with various flow elevations.
TWG 2/27/03 

SPATIALLY BALANCE

RANDOMIZATION ALONG 

SPATIALLY BALANCE

SPATIALLY BALANCE

RANDOMIZATION ALONG 

RANDOMIZATION ALONG 

 Illustrate, rather than explain in gene Consider 16 sites, from which we wan

balanced sample of 7.
TWG 2/27/03 

SAMPLE 

REPRESENTATION

1 0 0 0 0

2 0 0 0 13 0 0 1 0 4 0 0 1 15 0 1 0 06 0 1 0 17 0 1 1 08 0 1 1 19 1 0 0 0 

10 1 0 0 111 1 0 1 012 1 0 1 113 1 1 0 0 14 1 1 0 115 1 1 1 016 1 1 1 1

REAL SI(UNKNOSAMPLE 

REP

1 0 0 

2 0 0 3 0 0 4 0 0 5 0 1 6 0 1 7 0 1 8 0 1 9 1 0 

10 1 0 11 1 0 12 1 0 13 1 1 14 1 1 15 1 1 16 1 1 
TWG 2/27/03 

1 0 0 0 02 0 0 0 13 0 0 1 0 4 0 0 1 15 0 1 0 06 0 1 0 17 0 1 1 08 0 1 1 19 1 0 0 0 

10 1 0 0 111 1 0 1 012 1 0 1 113 1 1 0 0 14 1 1 0 115 1 1 1 016 1 1 1 1

RANDOMIZATIONS RANDOMIZATIONS RANDOMIZATIONS - HIERAR

- HIERAR

HIERAR

9 1 0 0 0

10 1 0 0 111 1 0 1 012 1 0 1 113 1 1 0 014 1 1 0 115 1 1 1 016 1 1 1 1

1 0 0 0 02 0 0 0 13 0 0 1 04 0 0 1 15 0 1 0 06 0 1 0 17 0 1 1 08 0 1 1 1

TAIL

9 1 0 0 0

10 1 0 0 111 1 0 1 012 1 0 1 113 1 1 0 014 1 1 0 115 1 1 1 016 1 1 1 1

5 0 1 0 06 0 1 0 17 0 1 1 08 0 1 1 11 0 0 0 02 0 0 0 13 0 0 1 04 0 0 1 1

H

T
TWG 2/27/03 

RANDOMIZATIONS RANDOMIZATIONS RANDOMIZATIONS - con

- con

con

11 1 0 1 012 1 0 1 1

9 1 0 0 0

10 1 0 0 115 1 1 1 016 1 1 1 113 1 1 0 014 1 1 0 1

5 0 1 0 06 0 1 0 17 0 1 1 08 0 1 1 13 0 0 1 04 0 0 1 11 0 0 0 02 0 0 0 1

12 1 0 1 111 1 0 1 0

9 1 0 0 0

10 1 0 0 116 1 1 1 115 1 1 1 014 1 1 0 113 1 1 0 0

5 0 1 0 06 0 1 0 18 0 1 1 17 0 1 1 04 0 0 1 13 0 0 1 01 0 0 0 02 0 0 0 1

T

H

TT

H

TT

H

12 1 0 1 1 11 1 0 1 0

9 1 0 0 0

10 1 0 0 116 1 1 1 115 1 1 1 014 1 1 0 113 1 1 0 0

5 0 1 0 06 0 1 0 18 0 1 1 17 0 1 1 04 0 0 1 13 0 0 1 01 0 0 0 02 0 0 0 1
TWG 2/27/03 

SELECTING THE SAMP

SELECTING THE SAMP

SELECTING THE SAMP

12 1 0 1 1 11 1 0 1 0

9 1 0 0 0

10 1 0 0 116 1 1 1 115 1 1 1 014 1 1 0 113 1 1 0 0

5 0 1 0 06 0 1 0 18 0 1 1 17 0 1 1 04 0 0 1 13 0 0 1 01 0 0 0 02 0 0 0 1

7/16 = 0.4375

14/16 = 0.875021/16 = 1.312528/16 = 1.750035/16 = 2.187542/16 = 2.625049/16 = 3.062556/16 = 3.500063/16 = 3.937570/16 = 4.375077/16 = 4.812584/16 = 5.250091/16 = 5.687598/16 = 6.1250

105/16 = 6.5625112/16 = 7.0000

0.31.32.33.34.35.36.3

12 1 0 1 11 1 0 1 

9 1 0 0 

10 1 0 0 16 1 1 1 15 1 1 1 14 1 1 0 13 1 1 0 

5 0 1 0 6 0 1 0 8 0 1 1 7 0 1 1 4 0 0 1 3 0 0 1 1 0 0 0 2 0 0 0 
TWG 2/27/03 

ADDING MORE POINT

ADDING MORE POINT

ADDING MORE POINT

 Points will not be usable for a variety 

no vegetation needs to be measurefaces.

Î A process called reverse hierarchical ord

to expand the spatially balanced list witspatially balanced sample. Additional pselected from that list in their order of a

Î I have no simple illustration of that proc

available. Sorry. See the Stevens & Ols
TWG 2/27/03 

EXTENSIONS FOR 

SPATIALLY BALANCED SAM

EXTENSIONS FOR EXTENSIONS FOR 

SPATIALLY BALANCED SAM

SPATIALLY BALANCED SAM

 Cut a stream up into lots of little piec

length

Î Do the spatially balanced sampling with 

z Randomly select a specific point in each samplz We actually used the 702 segments between flo

 they had unequal length so these lengths enter into 

Canyon” summaries

z Pieces can have different weights in the sampl

 This just stretches/shrinks segment lengths on the s

remains the number of points
TWG 2/27/03 

EXTENSIONS FOR 

SPATIALLY BALANCED SAMP

EXTENSIONS FOR EXTENSIONS FOR 

SPATIALLY BALANCED SAMP

SPATIALLY BALANCED SAMP

 This extends to two-dimensional samp

Î Effectively represent the coordinates of e

in a decimal (base 4)

Î Map the coordinates of the squares onto 

by interspersing the digits in their decim

Î Then Proceed as before

 Check at epa.gov/wed/arm for softw
TWG 2/27/03 

REFERENCE FOR THE SPA

BALANCED SAMPLING METH

REFERENCE FOR THE SPA

REFERENCE FOR THE SPA

BALANCED SAMPLING METH

BALANCED SAMPLING METH



http://www.orst.edu/dept/statistics/epas/spatial_balance_imperfect_frame.pd
TWG 2/27/03

where

i INDEXES SITE SETS 1, 2, ... , s

(all sites in a site set have the same revj INDEXES TIME PERIODS ( yeark INDEXES SITES WITHIN A SITE 

and (uncorrelated):

Sik ~ (μ, ) Tj ~ (0, ) Eijk 

A STATISTICAL MODE

A STATISTICAL MODE

Y S T

ijk = ik j 

+ +

σ 

2 σ T S2
TWG 2/27/03

A STATISTICAL MODEL A STATISTICAL MODEL - co

co

 CONSIDER THE ENTIRE TABLE OF THE 

TIME-PERIOD MEANS,

Î WITHOUT REGARD TO, AS YET, WHETHER 

PRESCRIBES 

GATHERING DATA IN ANY PARTICULAR C

Î ORDERED BY SITE-SET WITHIN TIME PERIO

Y = ( , ,,, , 

YY YY Y 

11 21 1 12 ⋅⋅ ⋅⋅ / /

s 

cov( , ) / 

YY n 

ij ⋅ i j' ' ii ' S i T 

⋅ = ++ δσ σ 

2 2 

With this ordering, we get
TWG 2/27/03

 NOW LET X DENOTE A REGRESSOR M

CONTAINING A COLUMN OF 1’S AND A THE NUMBERS OF THE TIME PERIODS. ELEMENT OFCONTAINS AN ESTIMATE OF TREND.

STATISTICAL MODEL STATISTICAL MODEL - con

con

If we let cov ( , , , ) 

1 2 T 

t T T Λ T = Σ = 

 cov(Y

I D 11 

)

() ' 

== ⊗ +⊗ + 

− 

Φ

σ 

St i T ss 

n 

Σ 21 

β) = −− − (' ) ' X XX Y 1 1 Φ Φ 1
TWG 2/27/03

STATISTICAL MODEL STATISTICAL MODEL - con

con

 BUT THIS ESTIMATE OF β CANNOT B

BECAUSE IT IS BASED ON VALUES DESIGN, WILL NOT BE GATHERED

 REDUCE X, Y AND Φ ΤΟ X*, Y*, AND Φ

THESE REPRESENT THAT SUBSET COLUMNS FROM X, Y, AND Φ COR

TO WHERE DATA WILL BE GATHE

and 

( *' * *) *' *

cov() ( *' * 

β

β

=

=

− 

− 

− X XX 

X X

Φ Φ

Φ

1 

1 

1
TWG 2/27/03

 NOTE THAT

CAN BE REWRITTEN AS

 CONSEQUENTLY POWER, A MEASURE 

CAN BE EXAMINED RELATIVE TO

A STANDARDIZATION A STANDARDIZATION

2 2 

'' 

' cov ( , ) / 

ij ii T ij S 

i 

YY n 

δσ σ δ

⋅ ⋅ = ++

cov ( , ) { ( / )/ 

' ' '

Y Y n

ij i j ii S E i 

⋅

⋅

= +

+

δ σ σ 

δ

2 2 

σ S σ E σ T σ 

22 2

/ / and 
TWG 2/27/03

 TREND: CONTINUING, OR MONOTONI

PRACTICALLY, MONOTONIC TRENDETECTED BY LOOKING FOR LINE

 SENSITIVITY (in the title) CAN BE EXPRE

POWER.

 WE WILL EVALUATE POWER IN TERM

VARIANCE COMPONENTS AND

WHERE THIS DENOMINATOR DEPEND

RATIOS OF VARIANCE COMPONENSAMPLING DESIGN.

TOWARD POWER TOWARD POWER

0 

ˆ

/ , so approximately, ~ 

λβσ β 

= E 
TWG 2/27/03 

POWER REFERENCE POWER REFERENCE POWER REFERENCE

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

(1998). Monitoring for policy-relevtrends over time. Ecological Applica

257.
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TOP OF TRANSECT AT MILE 12.3 TOP OF TRANSECT AT MILE 12.3 TOP OF TRANSECT AT MILE 12.3
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MARKING TRANSECT AT MILE 1

MARKING TRANSECT AT MILE 1

MARKING TRANSECT AT MILE 1
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CLIFF AT MILE 128.0 CLIFF AT MILE 128.0 CLIFF AT MILE 128.
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CLIFF AT MILE 

135.2

(FULL HEIGHT)

CLIFF AT MILE CLIFF AT MILE 

135.2

(FULL HEIGHT) (FULL HEIGHT)
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CLIFF AT 

MILE 223.5

(FULL SCALE)

CLIFF AT CLIFF AT 

MILE 223.5 MILE 223.5

(FULL SCALE) (FULL SCALE)
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CLIFF AT 

MILE 223.5

(CROPPED)

CLIFF AT CLIFF AT 

MILE 223.5 MILE 223.5

(CROPPED) (CROPPED)
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MIKE & 

SCOTT AT 

THE END!

MIKE & 

SCOTT AT SCOTT AT 

THE END! THE END!