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CompositeEstimation for
Forest Inventories Using Multiple
Value Groups and Strata
Michael S. Williams, MultiresourceInventory Techniques,
Rocky Mountain Forest and Range Experiment Station,
USDA Forest Service, 240 W. Prospect, Fort Collins, CO
80526-2098, and Ken A. Cormier, WashingtonOffice,
Timber ManagementService Center, USDA Forest Service,
Fort Collins, CO 80524.
ABSTRACT. Due to economicandpoliticalpressuresplacedon timberproduction,informationfrom timber
inventoriesneedsto be brokendowninto informationonparticular valuegroups,whichare groupingsbased
on individualtreecharacteristics.
Thusestimatesof totalsand variancesneedto be computed
for individual
valuegroups,all valuegroupscombined,and any collectionof valuegroups.A methodfor reportingthese
estimatesusinga variance-covariance
matrix is given.This isfollowed by two examplesof the estimationof
totalsandvariancesfor valuegroupsappliedto a populationcomprisingtwostrata.Poisson(3P) andsample-
treesamplingare usedin thefirst andsecondstrata,respectively.
Theseexamples
are includedto highlight
importantestimationconsiderations.
A workedexamplebasedon an artificial populationis also given.The
populationfor theworkedexampleis dividedintotwo strata,with threevaluegroupsin eachstratum.3 P and
sample-treesamplingare usedin thefirst and secondstrata.South.J. Appl.For. 20(2):103-109.
Formany
timber
inventories,
materials
being
sold
are
broken 2. In each stratum,use samplingmethodsthat will give
downintovaluegroups,whicharegroupings
basedonquality
factorssuchasspecies,
treesize,treegrade,orsomecombination
of thesefactors.Fortimberusers,somevaluegroupsaremuch
moreimportantthanothers.For example,a mixedhardwoodsoftwoodstandcouldcontainthreevaluegroupsthatconsist
of
highvaluehardwoods,
softwoodsawtimber,
andpulpwood.A
bidderwhohaslittleuseforpulpwoodwouldbeinterested
in an
estanate
ofthetotalvolumefortheothertwovaluegroups.
Some
measureof the precisionof the valuegroupestimates
is also
nmportant.
Thismeasureof precisionletstheuserknowwhether
the truetotal volumeis likely to differ substantially
from the
estanateof totalvolumein the valuegroupof interest.In the
previous
example,thesamebiddermaywantto lowerhisorher
bid if theestimate
for thehardwood
valuegroupis likelyto be
inaccurate.
To improvethe efficiencyof valuegroupestimatesand
reportmoreaccurateinformationaboutavailabletimber,the
USDA ForestServiceis implementingthefollowingchanges
to timber inventories.
1. Increasestratificationto improvethe precisionof estimates.
preciseestimatesof total volume for the minimum cost
and make the bestuseof available samplingresources.
3. Report a degreeof precisionon individual value groups
and any combinationof value groups.
To achievethesethreegoals,estimatesof total volume
(indiciated
byI7)andadegree
ofprecision
[indicated
by
var (Y)] need to be calculated in each stratumfor indi-
vidual valuegroups,all value groupscombined,and any
combinationof value groups.These estimatesthen must
be combined
•across
all strata.The methodgivenin this
paper
allows
Y andvar(I•) tobecalculated
forallpossible
value group combinationswith the fewest number of
calculations.
StatisticalBackground
Let M = the numberof value groupsandL = the number
of strata.An obstaclein reportingvariances
for anycollection
of valuegroupsis thatthenumberof possiblecombinations
of valuegroups,whichis
SJAF20(2)1996 103
1. The data structures used in the current NFS national cruise
computerprogramsdo not allow bootstrappingor jackknifing. Incorporatingthesewouldrequirea majorreinsionof the programs.
becomesvery largeevenwhenthenumberof valuegroupsis
relativelysmall.For example,if apopulationcontains7 value
groups,127 different varianceestimateswould be needed.
Ratherthanreportingvariancesfor all possiblecombinations
of valuegroups,we proposethefollowing.Let C = {j:j • 1,
ß.. M} be any setof valuegroupindices.Then thevariance
for any collectionof valuegroupsis givenby
2. Covarianceestimatorsarerarelygivenin theliterature;so,
beforeimplementation,
theywouldhaveto be derivedfor
all of the currentsamplingdesignsas well as any new
designs.
An Example
Requirednotation:
[,.j•C
J
j•C
j'•j
L
=
the number of strata.
g
= the numberof samplegroupsfor the sample-tree
method.
(Mood et al. 1974,p. 178)
M
= the numberof value groups.
where
i
= tree subscript.
j, j'
= value groupsubscript,
j, j' = 1... M.
k
= samplegroupsubscript,k = 1... g.
l
= stratasubscript,l = 1... L.
Xl
= totalof the samplingcovariatein stratuml.
•.
= estimated
total
ofthevariable
ofinterest
instratum
l
is the sum for value groupj acrossall L strata.Thus the
simplestmethodfor reportingvariancesfor anycollectionof
valuegroupsis to reporta variance-covariance
matrixandto
let the usercomputevarianceestimatesonly for the value
groupsof interestusing(1). For example,ifM = 3 we report
ff..j = estimated
total
ofthevariable
ofinterest
forvalue
groupj summedacrossall strata.
var(f.0
cov(:,f.1) var(f.:) cov(f.2,f.,)
(2)
•tj = estimated
total
ofthevariable
ofinterest
forvalue
groupj in stratuml.
cov(f.,f.0 cov(f.,f.o var(f.)
The disadvantage
of using(1) is thatcovariancesmaynot
be readily availablefor somesamplingdesigns(point-3P
samplingfor example).However,thesecovariances
canbe
obtainedusinga numberof methods,the simplestbeing
^
ß
'
^
Yli
= variableof interestfor tree i in stratuml.
Xli
= covariatefor tree i in stratuml.
Eli
= theprobabilityof selectionfor tree i in stratuml
f/c
= samplingfrequencyfor samplegroupk in sampletree sampling.
^
2
(3)
where
var
(I•(j&
j,)) =var
(•.j,+•.j).Coml•utation
ofthe
variancefor a singlevaluegroup,i.e. var(Y4)'is generally
a
straightforwardprocessas discussedlater. The only addi-
tionalinformalion
required
tocalculate
thecovariances
in (3)
n(e)! = expectedsamplesizein stratumI.
nl
= samplesizein stratuml.
ntj
= actualsamplesizein stratuml for value groupj
nt(tc) = actualsamplesizein stratuml in samplegroupk
Nl
= number of units in stratum l.
above
isvar(Y(
.&.,) This
requires
atmost
L2
• additional
.ss)'
computations.
In practice,
the•tual number
of•computati9ns
should
besmaller
since
var
(Y,¾
....) .= var
....
tt,v.s?
^\(Y,.'•
q! +var
^ \(Y,.,
q lI
whenever
mesamples
useato calculate
YtjandYq,,are
independent
in stratuml.
Alternative options for generating covariances are
bootstrapping
andjackknifingas well as directcalculation.
Certainpracticalconsiderations
makethesealternatives
less
attractive.
Some of the items that were considered
choosingthe methodwere:
104
SJAF20(2)1996
before
We usetwo strata (L = 2) and an unspecifiednumber
of value groups. In the first stratum 3P sampling is used
(Schreuder et al. 1993) and in the second stratum a
systematic sampling method, where every fth tree is
selected, is used. This method is referred to as sampletree sampling (USDA Forest Service Timber Cruising
Handbook 1993). While this method is not commonly
used in all regions of the United States, it is ideal for
illustrating sampleindependenceconsideration,which
wall be discussed
later.
Thentheestimatorfor thepopulationtotalfor valuegroups
j andj' is givenby
The desiredoutputfor theexampleis thevariance-covarlancematrixgivenin (2). In the followingsubsections,
the
E Yli
Yl(i&f)
=n(e)•
/11 i=1 •li
theory^forestimating
the variancefor value•group
j (var(r/j))andforanytwovalue
groupsj
andj'(var(r/(j&j,)))
(8)
m each stratuml is given (additionalinformationon the
estamationof these variances can be found in Cochran 1977,
p. 35-38, 140-144).We alsodiscuss
the independence
of
samples
in thetwo samplingmethods
andcontrast
them.
Variance estimators for the sum of the estimated totals for
anytwo valuegroups
j andj' aregivenby
^
Estimation for 3P Sampling
Poisson
sampling
wasusedin thefirststratum(l = 1). An
efficientestimatorof thepopulation
totalin thefirst stratum
lSthe adjustedPoissonsamplingestimator:
nl n(e)lY•j
var(Yl(j&j,))=i__El/.
•--•,/1
/ .I-(•(j&D)2/(nl
Note
^
that
when
^
3P
sampling
^
is
used,
Var(Yl(j&
j')) • var(Ylj ) q-var(Ylj') because
thevalue
grou•p
estimators
arenotindependent.
Thecorrelation
betweenYlj
^
andYlj'willbenegative
because
anincrease
inonevalue
•1.
=n(e)l
• Yl/
/•1 i=1•li
(4)
where
theprobability
ofselection
isKli= n(e)lXli/X1.1
To
estimatevaluegrouptotals,a newindicatorvariableneedsto
be defined. Let
0,otherwise
)
ßIYli,
iftree
ibelongs
to
value
group
j,)
(5)
Yli=
groupestimate
hasa tendency
to decrease
theestimate
inthe
othervalue group.
Estimationfor Sample-TreeSampling
Sample-tree
sampling
is usedin thesecond
stratum.
For
thissystematic
sampling
method,individualvaluegroupsor
collectionsof valuegroupsare assignedto oneof g groups,
referredto as samplegroups,andthe treesin that sample
groupareassigned
a sampling
frequencyrio
k = 1.... g.Thus
if treei belongs
tosamplegroupk,theprobability
of selection
is givenby •2i = 1/fk.The estimator
of the stratumtotalis
givenby
The estimateof the total for value groupj is
n2
•lj- n(e)l
• Y•i
/•/1 i=1Kli
(10)
(6)
An approximate
estimatorof thevariancefor the stratum
total•1.is
i=1•2i
Estimatorsfor thetotalof valuegroupj andvaluegroups
j andj' combinedare
2
Var
(•1.):
•/'n(e)
1-•yli
•1.
/ /(nl(nl
-1))
(7)
i=1[. gli
and
(Grosenbaugh
1974).
Substituting
Y•iforYliand•j for•1.in(7)gives
the
^ : • •2i
Y•
Y2(j&j')
^
i=l
estimated
variance
forvaluegroup
j, i.e., var(Ylj).
To estimatethepopulation
totalfor anytwovaluegroups
j andj', definethenew indicatorvariable
where
0,otherwise
,, lY•i,
iftree
ibelongs
to
value
group
jør
j'I
0,otherwise
ßlY2i
,iftree
ibelongs
to
value
group
jJ•
Yli=
1 isthesame
astheexpansion
factor.
•li
(12)
Y2i:
(13)
and
SJAF20(2)1996 105
^
0,otherwise
,, (Y2,,
,ftreet
belongs
to
value
groupj
orj')
Y2i=
^
because
Y2;andY2j'arenot•ndependent.
Combining Strata Estimates
The varianceestimatorof the total for all valuegroups
combinedin the stratumis givenby
Estimates in each of the L strata are combined to form the
variance-covariance
matrix. Sincestrataare sampledindependently,thediagonalelementsof thevariance-covariance
matrix are computedby summingacrossthe strata,i.e.
var(I•2
)=l1n21•(n(e)2y2i
I•2.)2/(n2(n2-1))(15)
L
Varianceestimatorsfor valuegrouptotalsaremorecomplicatedduetotheindependence
betweensamplegroupsand
dependence
withinsamplegroups.Foranysamplegroupthat
containsonly onevaluegroup,j, the outcomeof the sample
is independent
fromtheoutcomeof thesamplescollectedin
the other samplegroups.Thus, if the samplegroup only
containsone value group,the samplecan be viewed as an
(21)
I=1
To calculatethe off-diagonalelementsof the variancecovariancematrix,Equation(3) isusedwith thevariancefor
theestimated
totalsof anytwovaluegroupsjandj', givenby
L
independent
sample
of sizen2j= n2(k).
In thissituation
the
samplegroupandvaluegroupsamplesizesareequal.Hence
the varianceestimatorwhenvaluegroupj is the only value
groupin a samplegroupis givenby
I=1
A Worked Example
var(Y2J)
• m2)i=
1[••il
(16)
To illustratetheproposed
method,onesamplewasdrawn
fromeachstratum
inanartificialpopulation,
andthevaluegroup
estimators for total volume and their variances were calculated.
If samplegroupk containsmorethanonevaluegroup,the
varianceestimatorfor valuegroupj is givenby
where
The artificialpopulation
contains
datafromN = 1600loblolly
pinetrees(PinustaedaL.). Thedatacomprise
diametersquared
timesheight(xi) andtotal tree volume(Yi) measurements
for
eachtree.For testingpurposes,
oneof the threevaluegroup
codeswasrandomlyassigned
to eachtreeandthepopulation
wasdividedintotwostratawithN• = 800= N2.The800largest
treeswereassigned
to thefirststratum
where3P sampling
was
used.Theexpected
samplesizefor 3P sampling
wasn(e)l= 10.
The 800 smallesttreeswere assigned
to the secondstratum
becausetheyweremorehomogeneous,
andthe sample-tree
method is most effective in this situation. This stratum was
y*2i
=(Y2i,
iftree
ibelongs
to
value
group
k•
0, otherwise
(18)
Two casesneed to be consideredwhen estimatingthe
dividedintotwo samplegroups.The firstsamplegroupcontainedonlyvaluegroup1treesandhada sampling
frequency
of
fl = 40. Thesecond
samplegroupcontained
valuegroups2 and
3 andhada sampling
frequency
off2 = 50.
totalsforanytwovaluegroups.
•Ifvalue•roups
j andj'arein
different
sample
groups,
thenY2jandY2j'areindependent 3P SamplingEstimation
random variables and
var
(I72(j&/)):
var
(I72/)
+var
(I72j,)
[191
Variancesestimatorsfor the estimatedtotalsof any two
valuegroupsjandj' thatarebothin thesamesamplegroup
are givenby
^
mated totals are
I9•.
•--•i=1
y..2
2i-(Li=l
Y2i
=(l_n2(k))*n2(k)
f.n2(k)..)2
(2O)
var(Y2(j&J")
( N2
) (n2½,(n2(/O
-1))
106
Thetreesselected
by3Psampling
inthefirststratum
arelisted
in Table 1. In addition,theindicatorvariablesgivenin (5) are
alsolisted.Theexpected
andactualsamplesizesaren(eh = 10
andn! = 13.The estimates
of thetotalvolumefor eachstratum
and all pairsof strataare calculatedusing(4), (6), and (8),
respectively.
The varianceestimatesfor all possiblestratum
estimates
arecalculated
usingEquations
(7) and(9). The est•-
SJAF20(2)1996
= 9930.0
^
•11
= 2316.2
Table
1.Sample
values
selacted
from
stratum
1ofthe
test
population
using
3Psampling.
Y•t,J=1isthevalue
ofY•=
invalue
group
I
Treenumber Valuegroup
1
2
3
4
5
6
7
8
9
10
11
12
13
xli
2
1
Yli
2756.25
2866.88
3739.77
3794.56
6450.28
6629.24
8328.32
8760.96
10813.44
11727.81
12608.32
12736.08
13320.19
3
3
1
3
3
2
2
1
3
3
2
•r•i
4.28
4.86
Y•i,J= 1
0.005511
0.005732
0.007477
0.007587
0.012896
0.013254
0.016651
0.017516
0.021620
0.023448
0.025209
0.025464
0,026632
6.61
9.00
15.11
13.84
17.41
12.28
22.77
23.25
28.19
29.33
24.95
0
4.86
0
0
15.11
0
0
0
0
23.25
0
0
0
Y•i,J=2
Y•i,J=3
4.28
0
0
0
6.61
9.00
0
13.84
17.41
0
0
0
28.19
29.33
0
0
0
0
0
0
12.28
22.77
0
0
0
24.95
I212 = 2667.5
var(Y•o&3))
= 1997571.0
•3
var(Yv2&3))
= 1573412.8
^
= 4946.2
•1(1&2)
--4983.7
Note
that
forthevar(I•lj)
andvar(I;'•(j&j,))estimates,
the
sumgoesfrom i = 1.... nl. Thusthesummationtermfor the
varianceestimateof valuegroup1, i.e.
•l(l&3)
--7262.5
I•1(2&3)
= 7613.8
(.n(e)yfi
•li rlj
^
with variance estimates
,) =0.The
becomes
(I•lj)
2whenever
(Yli
same
istrue
for(9)
var(•.)
= 181184.1
vat(•1)
= 1499209.4
vat(•2)
= 1359952.0
whenever[Yli = O.
Sample-Tree Estimation
Thetreesselected
in thesecondstratumusingsample-tree
samplingare listedin Table 2. The value groupindicator
variablesgivenin (13) andthe samplegroupindicatorvariablesgivenby (18) arealsolisted.The samplingfrequency
for valuegroup1 wasfl = 40. Hence,everyfortiethtreewith
var(I213) = 2376776.5
var(•(l&2))
= 1846138.1
Teble2.Samplevaluesselectedfromstratum2ofthetestpopulationusing•mple-treesampling.
' , J' = 1 isthevalueofY2i
Y2i
' invelue
group1. y *2i,k = ! isthevelueof y •<2iin samplegroup1.
Tree
number
Value
group
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Y2•
3
1
2
2
1
2
1
2
3
1
2
1
2
3
3.87
3.92
2.19
1.40
2.00
3.52
2.75
1.00
1.15
2.48
1.15
1.27
3.75
5.52
1
3
1.37
1.60
•r2i
0.0200
0.0250
0.0200
0.0200
0.0250
0.0200
0.0250
0.0200
0.0200
0.0250
0.0200
0.0250
0.0200
0.0200
0.0250
0.0200
y•,j=l
0
3.92
0
0
2.00
0
2.75
0
0
2.48
0
1.27
0
0
1.37
0
y•,j=2
0
0
2.19
1.40
0
3.52
0
1.00
0
0
1.15
0
3.75
0
0
0
y•i,j=3
3.87
0
0
0
0
0
0
0
1.15
0
0
0
0
5.52
0
1.60
y•/,k=l
0
3.92
0
0
2.00
0
2.75
0
0
2.48
0
1.27
0
0
1.37
0
y•i,k=2
3.87
0
2.19
1.40
0
3.52
0
1.00
1.15
0
1.15
0
3.75
5.52
0
1.60
SJAF20(2)1996 107
value groupcode 1 was selected.Th•s resultedm a sample
I•.•= • +I22•
groupandvaluegroupsample
of sizen2(1)
= n21= 6. Trees
with valuegroupcodes2 or 3 wereassigned
to samplegroup
2 andselectedwith samplingfrequency
f2 = 50. Thusevery
fiftieth tree encountered
that was eithervalue group2 or 3
wasselected.
Thisresulted
in a sample
groupsizeofn2(2)=
10 with valuegroupsamplesizesof n22= 6 andn23= 4 for
value groups2 and 3 respectively.Estimatesof the total
volume,totalvolumein eachvaluegroup,andfor eachpair
of valuegroups,givenby (10), (11) and(12) respectively,
are
Any othercombinationof valuegroupscanbe calculated
in a similarmanner.All thepossiblecombinations
of value
groupvolumeestimatesfor the two samplesare
•
=
11894.0
I•.l
=
2945.9
1964.0
•..2
=
3264.1
629.7
•.3
=
5708.4
596.5
I•.o&2)
=
6185.8
762.2
I•.o&3)
=
8421.1
1202.1
•..(2&3)
=
8871.3
^
Y•0&3)= 1158.6
^
Y•(2&3)= 1257.5
The diagonalelementsof thevariance-covariance
matrix
arealsogenerated
by addingtheacrossthestratausing(21)
For example,the variancefor valuegroup1 is
The varianceestimatefor all value groupscombined
(var
(I•2.))
iscalculated
using
(15).Equation
(16)isus•ed
to
calculatethe variancein the first value group (var(Y20),
becausethefirst samplegrouponly containstreesfromthe
first value group.To calculatethe varianceestimatesfor
The covariances,which are the off-diagonalelementsof
(2), require the variancesfor every pair of value group
value
groups
2and3 (var(Y2j),J = 2,3),(17)isused.
Samples estimatesand are generatedby addingacrossthe strataas
drawn
indiff•erent
valuegroups^can
beassumed
independent. before.Thus,thevarianceestimatefor valuegroups1 and2
Thus,var(Y20&2))
andvar(Y2½&3
)) arecalculated
using is givenby
(19).Thevariance
estimate,
var(Y2(2&3)),
iscalculated
using
(20), becauseboth valuesgroups2 and 3 are assignedto
samplegroup2 andarethereforenot independent
samples.
The variance estimates are
var(•.)
= 81449.6
var(Yn)
= 12108.5
^
^
^
var(/o&2))= var([o&2))+ var
To concludethisexample,using(3) to generatethecovariancefor valuegroups1 and2 combinedgives
^^
cov(y1,Y2)var(Y12)
= 56109.1
var(•3)
= 104319.7
var(Y•0&2))= 68217.7
^
var(Y•o&3))= 116428.3
The variance-covariance
matrixfor the two samplesis
1511317.9
-506511.6
-939207.4
-506511.6
1416061.2
-1129016.9
-939207.4 -1129016.9 2481096.3
var(Y•(2&3))
= 65710.9
Conclusions
Combining the Strata Estimates
To estimatethetotalsin anyvaluegroup,theestimates
are
addedacrossthe strata.For example,the total acrossboth
stratafor valuegroup1 is
108
SJAF20(2)1996
The methoddiscussedfor estimatingthe varianceof an
arbitrarycollectionof value groupsis a combinationof
well-documented
statistical
results that minimizes
the
number of calculations and the amount of information that
must be reported Th•s method •s also be the easiest
methodto incorporateinto the currentForest Service and
•ndustrytimber inventories.
Literature
Cited
Coc•,n•, W.G. 1977.Sampling
techniques.
Ed.3. Wiley,NewYork.428p.
GROSENBAUGH,
L R 1974 STX 3~3-73 Tree content and value estima-
tion usingvarioussampledesigns,dendrologymethods,and V-S-L
conversioncoefficients.USDA For. Serv.Res.Pap. SE-117.112 p.
MoonA.M., GRAYroLL,
F.A., Am>BOES,
D.C. 1974.Introduction
tothetheory
of statistics.
Ed. 3. McGraw Hill, New York. 564 p.
USDA FOP,
ESTSERWCE.
1993.Timbercruisinghandbook.
FSH 2409.12.
SCUP,
EUVER,
H.T., T.G. GREC,
OmE,ANnG. B. WOO•),1993.Samplingmethods
for multiresource
forestinventory.Wiley, New York. 446 p.
SJAF20(2)1996 109