Age Models when Stand Parameters and

ForestSc•ence• Vol 40, No 4, pp 715-731

Random-ParameterHeight/Age Models when Stand Parameters and

StandAge Are Correlated Jui• LAPPi JuH^MALINEN ABSTRACT.

Height/agecurvesof dominant treesin a standcanbe predictedusingmodelscontaining randomstandandtree parameters.ff the population meancurveis estimatedfrom the datawhere stand-specific parametersare correlatedwith standage (e.g., becauseold standsaregrowingonpoorsites),thiscorrelation maycausebiasedestimates.The same bias problemoccursin traditionalsite indexmethods.It is shownhow ordinaryleast squaresestimatesof tree specificparameterscanbe usedfor computing unbiased estimatesof the population meanparametersandfor estimatingthe relationship between standparametersandstandage. The application of the estimatedmodelseveralyears after the datacollection time is problematic becauseit shouldbe knownhowthe populationchangesover time. Fog. SCl. 40(4):715-731.

ADDITIONAL KEYWORDS.Site index,variancecomponents, bias,changing population, prediction.

HIS PAPER DEVELOPS FURTHER therandom-parameter height/age model of Lappi andBailey (1988). Thespecific problem addressed ishowthees-

timation and use of the model shouldbe modifiedif the stand-specific parametersof the model are correlatedwith the age of the stand. Using the association betweensite qualityand heightgrowthlying behindsite index concepts,that kindof correlationcanbe causedby a relationship betweensitequality andstandage. Sucha correlation,in the form "oldstandsare on poorsites,"is foundin severaldatasets sampledfrom existingforests(Monsemd1987, Walters et al. 1989).LappiandBailey(1988)useddatafromcontrolled experiments where standagewas a genuinefixedvariable,andthusit wasnot necessaryto consider suchcorrelationin their study. If a height/agemodelis fitted usingmethods(e.g., with OLS, OrdinaryLeast Squares)basedon the standardassumption that errors are uncorrelatedwith the independent variables,andthe standageandsitequalityare in factcorrelated,the estimatedmodelwill be biased.Walterset al. (1989) suggestinsmental variableestimation for obtaining unbiased estimateswhena height/age modelis fitted to temporaryplot data. With temporaryplots, i.e., with one point per stand, instrumentalvariablesmay be the best (only?)tool availablefor unbiasedestimation of the mean model.

Biging(1985) notedthat OLS estimationdoesnot properlytake into account between-treedifferences.Usingremeasurement data, he fitted a differentcurve for eachtree and then estimatedthe parametersof the meanheight/agecurve

NOVEMBER 1994/715

usingGLS (GeneralizedLeast Squares)methodsbasedon a randomcoefficient regressionmodelof Swamy(1970). Biging'sdiscussion is closelyrelatedto the mainpointof this paper. It is argued,however, that his solution,i.e., estimation with GLS, will stillproducebiasedestimatesof the parametersof the meancurve even if the biasis generallysmallerthan with eLS. GumpertzandPantula(1989)discussthe simplemethodwherethe parameter vector is estimatedseparatelyfor each individualwith eLS, and the mean of estimatesis usedas the estimateof the meanparametervector.They suggest that the estimatemay be a reaso•ablealternativeto GLS estim•ation in the case wherethe assumptions for GLS estimationare valid.It is arguedin thispaperthat the simpleestimateis unbiasedwhenstandparametersandstandage are correlated while the GLS estimateis generallynot unbiased. In the modelof Lappiand Bailey(1988), a hierarchical stand-parameter/tree parameter(variancecomponent) structureis assumed.Their methodfor estimatingvariancesandcovariances of the randomparametersis improvedin thispaper so that the estim•ates are unbiasedundermore generalconditions. The randomparametermodel studiedin this paper can be used to predict height/agecurvesof singletrees or standswhenanycombination of heightmeasurementsor remeasurements is availablefroma givenstand.An interestingand nontrivialquestionrelatedto the maintopicof the paperis howwe canutilizethe correlationbetweenstandparametersandstandage in predictions.These problems were discussed previouslyby Lappi(1990).

DATA The datausedin the studywere collectedfor computinggrowthandyieldtables for artificiallyregenerated(mainlysown)even-agedScotspinedominatedstands in Finland.The data have been describedin more detailby VuokilaandV'•liaho (1980). The 226 studystandswere subjectively selectedsothat they wouldcover differentsites and geographical areas and wouldmeet certainhomogeneityrequirements.Sampletree measurements includedcurrentheightandthe heights at 5, 10, 15, and20 yearsearlier, or as manymeasurements as couldbe reliably made. The total numberof sampletrees was 1808, but onlythe 1455 dominant andcodominant trees with 6161 heightmeasurements were includedin this analysis.The numberof trees per standwas between2 and8, the averagebeing6.4 and standarddeviation1.1. The (biological)age of standswas determinedas accuratelyas possiblefrommanagement recordsandannualrings.The difference between measuredtree age and stand age averaged0.2 yr with a standard deviation of 2.2 yr. It wasconcluded thatthe discrepancy betweenmeasuredtree agesandstandagewasmainlycausedby measurement errorsin countingannual rings, andthe obtainedstandage was usedfor all trees in the stand.The stand ages were between 26 and 112 yr (average52.5 yr), and the stand heights (averageheightof dominantandcodominant trees) were between6.8 and28.5 m (average15.6 m). The dataare analyzedas if measuredstandswere a randomsamplefrom the target population of stands,andtrees withineachstandwere a randomsample from trees in the stand.The purposeof the paperis to discussmethodological issuesin the analysisof height/agedata. Any discrepancy betweenthe sample

716/FOmSTSCmNCE

dataandthe intendedpopulation hopefully doesnotinvalidate the proposed methods. The obtained values of the estimates must be treated with caution.

OLS AND GLS IN A RANDOM COEFFICIENT MODEL The mainpointof this paperis easierto presentfor a simplerandomcoefficient regressionmodelwherethe hierarchical stand/treestructureis ignored.This way we canlink the discussion better to that of Biging(1985). Let us assumethat the observeddatafor individual(e.g., standor tree) i, i = 1..... n canbe described with model:

Yi = Xibi + eo

(1)

whereXi is the rti x p matrixof regressorvariablesfor individual i, bi is thep X 1 parametervectorfor individual i, andei is the rti x 1 error vectorindependent

ofX/andvar(e/)= frei.Eachbi Canthenbeestimated unbiasedly withtheOLS estimate

fii = (X/•i)-•XJY• (2) andthevariance of theestimate is o•/(X/.•Ii)-•.Assume thenfurtherthatbi is interpretedto be a randomvectorwith unknownmean[5 andwith knownvari-

ance-covariance matrixI;. Assume furtherthatrr• is knownfor eachi. How shouldwe estimate[57We canwrite the modelfor the dataas:

Yi = X/• + X,di + eo

(3)

where 15+ di = b/, anddi is a randomvectorwith meanzero andvadancecovariance matrixI;. ThusX,di + ei canbe interpreted to be the errortermof the model with mean zero and vadance-covariance matrix

Vi = X/I;X• + .•I

(4)

When we stack the model matrices and vectors for different individualsi, and take into account that the overall variance-covariance matrix for the error terms is

block-diagonal, we get the standardGLS estimateof [5:

•=

I• 1-1 n•V -1. X'ivi-lxi ZX• i Yi.

i=1

(5)

i=1

Swamy(1970) hasshownthat this canbe presentedas a "weightedaverage"of the OLS estimates(2):

• ----

Wi-1 i=1

Wi-16i,

(6)

i=1

where

Wi = I; + .•(X;Xi)-•

(7)

NOVEMBER 1994/717

isthetotalvariance offii (i.e.,it isthesumofestimation variance andintrinsic varianceof b). This is, in fact, the estimatewe get if we estimate[3with GLS

fromthemodel fii = I[3 + ui. TheOLSestimate of[3isobtained from(6)if I; isthezeromatrixandallo-/•'s areequal. Biging(1985) suggeststhat the estimate(5) is the correctrandomparameter solutionfor the mean parameter[3 that is in principledifferentfrom the OLS estimate.Figure 1 illustratesthe estimationmethodsfor a hypotheticalcase where the slopeparametersof differenttrees are the sameandthe youngtrees in the datahavelargerintercepts(comparethe figurewith Biging'sFig. 2). This kindof situationcanarisewhenstandsare dearcutwhentrees reacha givensize, so that fast-growingstandsdo not get old (in Figure 1 there are differencesonly in the initialgrowth).In thiscasealsoGLS estimate(5 or 6) will be biasedfor the meanparameters,even if the biasis smallerthanin OLS estimates. The reasonwhy bothGLS andOLS estimatesare biasedis that the intercepts of differenttrees and the x-variableare dependent(correlationin the data of Figure1 is - 0.87). Thusthe estimatesare computedundera misspecified model. Both OLS andGLS estimatesare unbiasedonlyif the expectedvalueof the error

Height

I

-Rr'.LS s

20

40

60

80

100

120

Age, years FIGURE 1. Parallellinesare height/ageregressionlinesfittedseparatelyfor eachtree with OLS using data pointshavingthe samex-valuesas the pointson the lines fi.e., y-valuesvary aroundthe

regression lines).The "OLS"lineis theregression linefittedfrompooledobservations or computed fromtree parameters using(13)by takingI; in the weightmatrix(7) to be the zeromatrix.The "GLS"lineis fitted using(6) whereweights(7) are computedassuming that there is no variationin

the slopes butthevariance of the treeintercepts is twicethewithin-tree variance oa assumed to be the samefor all trees. The relativepositionsof the linesestimatedwith differentmethodswould alsobe the samein the casewherethere are more observations for treeswith smallinterceptsthan for trees with largeintercepts.The "R" lineis the true "randomparameter"meancurve,i.e., the curvewherethe parametersare equalto the expectedvaluesfor a tree selectedrandomlyfromthe population.

718/FOhssrsc•cE

vector,in thiscaseX,di + eo doesnotdependonthe regressorvariables.In the

example E(diIXi)• 0 andthusE(X,di)• O.

Anunbiased estimate of [3iseasily obtained bynoting thatE[fiil•]= bi and E[bi] = [3wherethefirstexpectation is withrespectto thedistribution of ei and the second expectation is withrespectto thedistribution of bi. Thusthe overall

expectation ofl•i is[3foreachi. Each l•iisanunbiased est'_nnate of[3andthusalso l•, thearithmetic meanof l•i's.Astheweight of l•i in l• (i.e.,unity)doesnot dependon Xo anydependence betweenbi andX i doesnot causeanybiasto the estimate.

Theestimate (6)willbeequal tog ifallweight matrices Wiareequal. Inthe GLS estimation,the matrix• will makethe weightmatricesmoreequalthanare the weightsin OLS estimation.This explainswhy GLS estimateswill generally, as in the exampleof Figure1, be lessbiasedthanthe OLS estimateswhenthere is a dependence betweenbi andX i. If the necessaryassumptions for unbiasedGLS estimation(e.g., that error term is independent of the regressors)are valid,then OLS estimateswill alsobe unbiased.Thus there cannotbe anylogicalor anticipateddirectionto whichGLS estimatesshoulddeviatefrom the OLS estimates(e.g., that GLS height/age croweshouldbe steeperthanthe OLS curve).If conditions for the GLS estimation are in effect, the onlyanticipatedimprovementover OLS is that the varianceof

estimates willbesmaller. If there isnodependence between bi and X/,then gwill notbe as efficientan estimateof 13as the GLS estimate.See Hsiao(1986)for a gooddiscussion aboutthe specification and estimationproblemsin varyingparmeter

models.

MODEL MODEL SPECIFICATION

The datawere analyzedbothusinga linearizedversionof Richards'equation(as in Lappiand Bailey 1988) and Schumacher's model(as usedby Walterset al. 1989). Bothmodelsleadto similarconclusions. The application of Schumacher's modelis more straightforward,so only those results will be presented.We assumethat a reasonable modelfor the heightdevelopment of tree i in standk is:

ln(H•i(t))= a•i + [3•i t-i + e•i(t),

(8)

wherees,(t)istheerrortermwithconstant variance o-2andisuncorrelated with the error terms of other trees or the sametree at differentages.

It is assumed furtherthat the tree specific parameters asi and13sicanbe expressedas: ot• = a + ak + a•.,

(9)

•Si = b + bk + bsi,

(10)

wherea andb are fixedunknown population meansof the parameters,as andbn are randomstandparameterswith meanzero but possiblycorrelatedwith t, and asiandbs•are randomtree parameters, uncorrelated withthe standparameters

NOVEMBER1994/719

and with mean zero. Both standand tree parametersare assumedto be inde-

pendent of the errortermeki(t).The variances andcovariances var(ak),var(b•), cov(a•,b•), var(a•i), var(bk/),andcov(a• b•/) are unknownconstants. Note that the assumedmodelimpliesthat the standand tree heightsare correlatedover

time,andtreeheights ina givenstandarecorrelated. The treeparameters a•i and b•opresenting randomfluctuations of tree heightcurvesaroundthe standheight curve, are assumedto be independentof the tree ages in the data, but such

independence isnotassumed for thestandparameters akandb•. If thedatawere from unevenagedstands,it mightbe necessaryto considerpossiblecorrelation betweentree parametersandtree ages(more predsely,deviationsof tree ages fromthe averagestandage).Writingthe expressions (9) and(10) intoEquation (8), the modelis:

ln(H•i(t))= a + bt-• + a• + b•t-• + aki+ b•it-• + e•i(t). (11) ESTIMATIONOF VARIANCESAND COVARIANCES Unbiased estimates of the unknown variances and covariances can be obtained as

follows(seeAppendixfor a summaryof all calculations). First estimatethe model

(8) separately for eachtree witheLS. As the errorterme•i(t)is assumed to be independent of the treeagesin thedata,the obtained estimates &•oand[3kiare unbiased for a•i and•i whateverthe realizedvaluesof the randomparameters. Theresidual variance (r2isthenestimated unbiasedly bydividing thetotalsum of squaredresidualsby the totalnumberof degreesof freedom[(totalnumberof heightmeasurements)-2* (numberof trees)]. The obtainedvalue of • was 0.024.

Estimatesof the variancesand covariances of the randomparametersare

obtained fromthetreewise eLSestimates &•iand•i- Theestimates &kiand •i areaccording tothemodel (11): &•i = a + ak + a•i + uki, and

(12)

•ki= b + b•+ b•i+ V•i

(13)

whereu•i andvkiaretherandomestimation errorsof theparameters a•i and

respectively. Anunbiased estimate ofvar[u•.,/v•i] isgivenbyO2(Xj,/Xki)1where X•i isthemodel(design) matrixoftreei instandk ineLS regression forEquation (8).

Estimationof variancesof randomparametersis based on the same idea as usedin the "analysis ofvariance"methodfor estimating variancecomponents (see Searle1971). The estimationmethodis describedin termsof the a-parameters, i.e., for var(a•)andvar(a•i).Equation(12) looksthe sameas the standard two level variancecomponentmodel where trees are nested within standsexcept there is an additionalrandomcomponentu•. As we akeadyhave an unbiased estimatefor var(uk•)for eachk andi, we canproceedexactlyas in the standard analysis of variancemethodif we carrythesevariances alongin thecomputations.

Morespecifically (compare withp.474inSearle 1971),letT•, TA,andTodenote the followingquadraticstatistics:

r•- N' 720/FOP,KsrSCm•CE

(14)

K

rA=

(lS) ki

k=l

(16)

i=1

where n• is the numberof measuredtrees in standk, K is the numberof measuredstands,N is the total number of measuredtrees, and a dot in a sub-

scriptindicatessummationover the corresponding index.

Thentheexpected values of T•, TA,andToarecomputed undermodel(12). Threeequations areformed byequating theobserved valueof T•, TA,andTo, respectively,to the corresponding expectedvaluesexpressedin terms of the unknownvariancesandthe meanvalue.Whenthe set of equationsis solved,we get unbiasedestimatesfor the variances.In our casewe get:

and

T• - T• - (K- 1)v•r(a•) + P.. - •

vfiz(aD =

Nk

whereVkiis the estimated varianceof the estimation errorof parameterOtkO i.e.,

Vki= vfir(&ki -- %i), V.. = ZkZi Vki,andPk.= (1/nk)•i Vki.If allVki'sare zero, then the estimatesare the standardanalysisof varianceestimatesgivenin Searle (1971, p. 474). Lappi and Bailey (1988) estimatedthe between-standand within-standvariancesby first computingthe analysisof varianceestimatesandthen subtracted 17.. from the estimatedwithin-standvariance.When these estimatesare comparedto the aboveestimates,we seethat theirvfir(aki)hasbias

k

andtheir v•r(ak) hasbias

Wemayexpectthattheaverage estimation errorvariance •.. doesnotdeviate muchfrom the averageof standaverages 1

NOVEMBER 1994/ 721

so that the biasesare probablyquitesmall.The aboveestimatesare as easyto computeas the estimatesof LappiandBailey(1988). The variances of bkandbkicanbe computed in the sameway with Equations (17) and (18). Analogous estimatesof covariances cov(a•, b•) andcov(a•/,b•i) couldbe computedby replacingthe squaresin Equations(14)-(16) with the corresponding crossproducts,andsolvingthenthe covariances fromthe resulting estimatingequations.Accordingto Searleand Rounsaville(1974), the sameestimatesare obtained if the variancecomponents of dki q- •ki are firstsolvedwith the abovemethod,andthe covariances are computedusingequationcov(x,y) =

¾•.(var(x• + y) - var(x)- var(y)).Notethatwhenapplying theabove method for &ki + 1•/, the estimation varianceterm V•i in (17) and(18) willbe:

Vki= v•r(&ki-a•i)+ v•r(•i- [•i) + 2cov(a•i-ak••i-

[•i), (19)

wherethevariances andthecovariance canbeobtained from{•2(X;iXki ) -1. The obtained estimates were:

sd(a•) = 0.216, sd(b•) = 11.85, corr(ak,bk) = -0.68 sd(a•/) = 0.077, sd(b•/)= 3.58, corr(a•b•/) = -0.35. ESTIMATIONOF MEAN PARAMETERS

Let us thenconsiderthe estimationof the meanparametersa andb. If we could assumethatrandomparametersare independent of the observedagesin the data

set,thena, + bkt-• + a,i + b•it-1 + e•i(t)would bea random errortermwith zero meanand independentof the regressorvariables.Thus EGLS (Estimated GLS) would be a natural method where the variance-covariance matrix of the error terms could be obtained from the variances and covariances of the random

parameters.Note that covariancestructurehas now one additionallevel comparedto theabovediscussion aboutGLSestimation, sothatEquations (5) and(6) couldnot be useddirectly.Also OLS wouldbe unbiasedeven thoughit is a less effidentmethodthanEGLS. As we are preparedto have dependence between standageandstandparameters,EGLSor OLScannotbe used(withoutspecifying the exact form for the dependence). There are two possibleways to applythe proposedmethodof estimatingthe mean parameterwith the arithmeticmean of individualestimates,either to use the meanof the OLS tree parametersor the meanof standmeans.As the number of trees measuredin a standhad slightcorrelation0.18 with the standage, the

firstestimate would CatTy onthebias__problem•we aretrying toavoid. Thusthe

standmeansof tree estimates(i.e., &,., and[•k.)were first computed for each stand,andthe population meanparameterswere thenestimatedwith the means

ofthestand means. Theresulting estimates were• = 3.563and/•= - 44.05. OLSestimates ofa andbwereg = 3.449,and• = - 37.63.ThustheOLScurve is flatter thanthe unbiasedmeancurve(Figure2). This is in accordance with the

expectations aswellaswithresults ofBiging (198•5) andWalters etal.(1989).As expected,the GLS estimates• = 3.517, and b = -40.92 were betweenthe unbiased estimates and the OLS estimates.

Simple unbiased estimates of the (estimation) variances of • and/•canbe

obtained bydividing thesample variance ofthestand means •,. and •,. byK - 1, 722/FORESTSCIENCE

Height,rn

25

20

3.5

3_0

5

0

20

40

60

80

100

120

Age, years Fmu• 2. _The mean height/age curve estimated with0LS("OLS") andusing themeans ofthestand averages &k.and•k. oftheindividually fittedheight/age curves ("R").

i.e., in the sameway we wouldestimatethe varianceof the meanof i.i.d variables

(seeGumpertz andPantula 1989).Theobtained standard errorsof• and• were 0.014 and0.78, respectively.Whenthesefiguresare comparedto the differences betweenobtainedestimatesandthe OLS estimates,we note that the differences are clearlysignificant.A formaltest aboutthe misspecification of the model assumingindependence of standparametersand standage couldbe doneusing Hausman'smisspecification test discussed by Hsiao (1986). If the modelwouldcontainother globalparametersin additionto the expected valuesof the randomparameters,then estimationof the globalparametersand variancesandcovariances of the randomparameterscouldbe basedon globaland tree specificOLS estimatesobtainedfor a simultaneous model containingall trees.In the derivation of thevariancecomponents of therandomparameters,the covariancesof the estimationerrors of tree parametersshouldbe taken into account(see Lappi 1991). The abovemodelis easierto estimateas the model assumptions implythat the estimationerrorsof the tree specificparametersare independent.

DEPENDENCE BETWEEN STAND STAND PARAMETERS

AGE AND

It was describedabovehow we canget unbiasedestimatesof the meanparam-

etersa andb of thetree specific parameters otki= a + ak + a•i and[3re i = b+ b• + b•iin casethestandparameters akandb•maybe dependent ontherecorded agevaluesin the dataset. The interpretation of thisdependence is thatthe stand

NOVEMBER1994/ 723

parametersare dependenton the standage. We will now considerhow we can describethisdependence. It willbe discussed laterhowthiskindof dependence couldbe interpretedwhen the modelis applied. For the moment,we want to modelthe situationin the population at the data collectiontime, and the standage is definedas the age of the standat the data collectiontime (this will be later called"currentage"). The followingderivations are theoreticallyconsistentwith the aboveresultsonlyif standageis interpreted as a randomvariable(otherwisethe population meanof standparameterswould not be generallydefined,onlyfor a fixedstandage).

Thedependence between stand parameters and_stand ag_• can bestu. died using

standmeansof the treewiseOLS parameters,&k.,and [3k..According to the

assumed model •. = a + a• + u•and•k. = b + b•_+vkwhere u•andv• consist ofrandom estimation errorsofestimates •i and•i andofrandom tree parameters akiandb•i. Thusukandv• are randomvariableswithmeanzeroand independent of ak, b•, or anystandvariables.Thusthe dependence of (a + a•) and(b + b•) onanystandvariables canbe studiedunbiasedly withOLSregression

methods using •. •d •k.asdependent variables. Bysubtracting estimates ofa

andb from•. and[3•.,respectively, wecould getequations fora• andb•, butit

is morestraightforward to workdirectlywith(a + ak)and(b + b•). The following equations were obtained forA• = a + a• andB• = b + b• withrespectto the standage, denotedas T:

fik = 0.029T• - 0.000179•+ 2.566, R2 = 0.29

(20)

/• = -0.490T•- 18.30,R2 = 0.37

(21)

The standparametersare dearly dependentonthe standage.Note thatif there

would benodependence, wewould get"regression equations".•k = 3.563,and /)• = -44.05,where 3.563and-44.05aretheestimates obtained above for meanparametersa andb. As the curveswere fittedwithOLS withouttakinginto

account thatdifferent observations havedifferent errorvariances, thegivenR2 values(or otherstandardOLS statistics)do not haveanydear theoreticalinterpretation. The dependence betweenstandparametersandstandageis easierto interpret usingheightpredictions withinthe agerangeof the data.Predictedstandheight at age40 wasusedto describethe initialdevelopment, andpredictedstandheight at age90 wasusedto describethe asyrnptote. According to Figure3, oldstands havebothslowinitialdevelopment andlow asyrnptote.Youngstandsin the data seemto havefast initialdevelopment and relativelylow asyrnptote.Figure 4 showspredictedheight/agecurvesfor standswith differentcurrentagesandthe predictedstandheightwith respectto the current standage. The last curve demonstrates nicelythat it is a differentthingto predictthe accumulated growth in crosssectionaldataandto predictthe growthover time. Table 1 showsthe mean,standarddeviation,andRMSE whenlog-heightor the incrementof the log-heightis predictedfor all trees with differentmodels.The "mostfitted"modelwithpredictedstandparametersis unbiased andhassmallest RMSE bothin heightandheightincrementprediction.The OLS curveis unbiased for heightpredictions,but slightlyunderpredicts heightincrements.This is in accordance withthe factthatthe OLS curveis not steepenough.The meancurve provideson averageheightpredictions that are too small,andheightincrement

724/FOmS•SC•CE

Predicted log-heightat age40 3

2.75

2.5

2.25

2

1.75 1.5

40

60

80

100

Currentstandage,years

Predicted log-heightat age90 3.4

ß

"

b)

3.2

3

2

8

2.6 40

60

80

1

0

Currentstandage, years FIGURE 3. Predicted logarithmic heightln(H•(t))at aget = 40 (Figure3a)andat aget = 90 (Figure 3b)asa function ofthecurrentstandage.Predictions arebasedonthestandlevelequation ln(H•.(t))

= An + Bnt-'. Points correspond to individual stands; stand parameters AnandBnareestimated

with •rx nand •n respectively (i.e.Fig.3aisfor•tn.+ •n./40 and Fig.3bisfor•n.+ •n./90) ßThe solidcurvesareobtained bypredicting standparameters An andBnusingEquations (20)and(21). NOVEMBER 1994/ 725

Height,m

25[

j.•.•-- H(t), T=80

2o 15

10

.... H(t),T=110

t

. ...- H(t) T=4 H(T)

40

60

80

100

120

Age t, or Currentstandage T FIGURE 4. The thickline (H(T)) presentsthe predictedcurrentstandheightas a functionof the currentstandageT. The curvepresentsonlythe cross-sectional situationandcannotbe usedfor predictionsover time. Other curvespresentpredictedheight/agecurvesH(t) for standswith currentage T = 40 yr (thin solidline), T = 80 yr (brokenline), and T = 110 yr (dottedline). Equations (20) and(21) are usedto predictstandparameters bothin H(T) andH(t) curves.

predictionsthat are too large. Becausethe effect of parametersin heightand heightincrement prediction is differentfor differentages,themeanparameters do not providegoodpredictions if the standparametersandstandageare relatedas is the case in the data.

ffA n (=a + an)andBn (=b + bn)are predictedwith (20) and(21), thenan' TABLE

1.

Mean,standard deviation andRMSE(X/(mean) 2 + sd•) oftheprediction errors when log-heightor the increaseof the log-heightof all trees in the dataset are predictedusingthe OLS curve,the meancurve(i.e., usingthe estimatedpopulationmeanof the randomparameters)or the curve where standparametersare predictedusingEquations(20) and(21). Upperfiguresare for the pooleddataof all measurements;the lower figures are for stand averages. The figures are multipliedby 100. Predictionsare madewithoutheightmeasurements. in(/'/.i(O) - predicted Model

Mean

sd

RMSE

[lniH•i(t)) - ln(H•(t - 5))] - predicted Mean

sd

RMSE

0

22.5

22.5

0.9

5.1

5.2

OLS curve

0.5

19.7

19.7

0.9

3.2

3.3

Mean curve

4.9

23.3

23.8

- 1.4

6.1

6.2

5.4

20.3

21.0

- 1.3

4.1

4.3

-0.1 0.2

22.1 19.4

22.1 19.4

-0.1 - 0.0

4.8 2.8

4.8 2.8

Predicted stand Parameters

726/FOP,ESTSCmNCE

= A• - '•k andbk'= B• - •k arenewrandom stand parameters. Var(a•'), var(b•'),and•cov(a,',b•') canbe estimated by making the similaranalysis for ln(H•i(t))- A• - Bkt-• aswasdoneforln(H•i(t)). Thefollowing estimates were obtained:

sd(a•') = 0.18, sd(bk')= 9.14, andcorr(a•',b•') = -0.58. Whenthe standarddeviationsare comparedwith the originalstandarddeviations0.216 and11.85,we notethatthe variationof the slopewasreducedmore

thanthevariation oftheintercept. Thisisinaccordance withtheR2'sofEquations (20) and (21). When there are heightmeasurementsavailablefrom one or more trees from

oneor more agesin a new standof the samepopulation, the randomstandand tree parameterscanbe predictedapplyingthe formulaspresentedin Lappiand Bailey(1988).For the purposes of thispaperwe justnotethatthe predictedstand parameters are shrunken towardsthe population meanof the parameters.If we usethe dependence betweenstandageandstandparameters,thenthe meanof the a- andb-parameters is obtainedfromEquations(20) and(21), otherwisejust the estimatedpopulationmeansare used. In the former case there is more shrinkageas the "free" varianceof the standparametersis smaller.Figure5 showsthe predictedcurveswhenthe standheightis assumedto be knownexactly at age 40 or at age 80 (i.e., it is assumedthat the numberof sampletrees is infinite).It doesmake a differencewhetherthe dependence betweenstandage

25

Height, rn

2O

15

10

40

60

80

100

Age, years FIGURE5. Predictedstandheightcurvesfor a standwith height13 m at currentage 40 (curves crossing at 40) andfora standwithheight19m at age80 (curvescrossing at 80). Solidlinespresent predictions withoututilizingthe dependence betweenstandageandstandparameters, anddotted linespresentthe casewherethe standparametersare first predictedwith Equations(20) and(21) andthe decreasein variancesof standparametersis alsotakeninto account.

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andstandparametersis utilizedor not, if the standheightis knownat one age. If the standheight is known at two ages, then the predictedstandheight is determinedexactlyaccording to the model,and the assumptions aboutthe dependencebetweenstandage andstandparametersdo not have any effect.

DISCUSSION The aboveanalysisshowsthat the dependencebetween stand age and stand parameterscanbe analyzedwithinthe randomcoefficient modelframework.The key pointis thatparametersare first estimatedwith OLS separatelyfor eachtree. Thereafter the expectationof the randomparametersin a randomlyselected standcanbe estimatedunbiasedlywith the arithmeticmeanof the standmeansof the OLS estimatesevenif standparametersare dependenton the standage. The form of dependencebetweenstandparametersand standage can further be studiedby OLS curvefittingusingstandmeansof the tree OLS estimatesas the dependentvariable(see Equations20 and 21). It is essentialhere that the analysisis basedon unbiasedestimatesestimatedas if the randomparameters were fixed. For instance,when Henttonen (1990) studiedthe varianceand trend of year effectsof diametergrowthusingGLS estimatesof fixedyear effects,the obtained estimates were unbiased even if there had been trends in stand vari-

ables.When a reasonablefunctionalform (linearin parameters)is foundfor this dependence, more effidentestimatesfor the parameterscouldbe obtainedif this functionis written as the part of the overall model, and the parametersare estimatedwith GLS (see, e.g., the between-subject part of the modelof Vonesh andCarter1992).GLSestimateswouldnowbeunbiased because therandompart of the model,after adjustmentfor the dependence betweenstandparametersand standage,wouldbe independent of the fixedparametersof the model.Suchmore complicated estimation methodswould,however,misleadingly drawthe attention from the application problemsof the estimatedmodel. Statisticalmodels are derived assumingthat the data represent some ideal population.When applyingthe modelsin predictions,we assumethat we are dealingwith the samepopulation. However,forestschangecontinuously so that the population is reallythe sameif we are dealingwith datafrom the sametime point.How, for instance,couldthe abovemodelbe appliedto a standk having standage g whenp yearsare elapsedfrom the datacollectiontime? Let us first considerthe casewhere we wouldlike to utilizethe dependence between standparametersand stand age. For old stands,the reasonfor the dependencebetweenstandparametersand standage may be that fast growing standsare cut first. If we canassumethat the samecuttingpolicycontinues,then we mightuse the currentstandageg as the standage in the equationspredicting standparameters.This wouldmeanthatwe are assuming in predictions madeat different times for the same stand that a stand gets poorer over time. The apparentlogicalcontradiction can be solvedby interpretingthe predictionsas follows:on the average,standsthat get old are on the poor sites. If the dependencebetweenstandparametersand standage is causedby the land-useor silviculturalpracticesnot havingthe same effect any more (e.g., becausedifferentareascanbe nowaccessed equallywell), thenwe mightusethe ageof the standat the datacollection time as the standagein prediction equa-

728/Fo•xrsc•cE

tions.An additionalproblemis what cotfidwe thenassumeaboutstandsthat were youngerat the data collectiontime than the youngeststandsin the data?

If the dependence betweenstandparametersand standage is causedby the interactionof site quality and regenerationorder of stands, then it would be logicallyconsistentto use the mean parametersall the time. However, if the

dependency is causedby silvictfitural practicesor by changes in howdifferenttree spedesor geneticsourcesare allocated to differentlands,we maynotexpectthe mean parametersto remainthe same over time. For instance,basedon Figures3 and4, we cotfidmakethe followingspeculationsaboutthe forestsrepresentedby the datausedin this study.Currentold standsare reallygrowingonpoorsites,as cotfidbe anticipated. Standsregenerated 50-80 years ago are growingon best sites. As best standswere first regenerated,standsregeneratedmorerecentlyare growingonpoorersitesreaching lower heightat oldages(Figure3b). However,thesestandsobtainedmore intensivemanagement duringthe reforestation phase,so that the initialdevelopment has been fast (3a).

The validityof the obtainedrestfitsfor the relationsbetweenstandparameters andstandage is dependenton the validityof the assumptions. Whenthe heightincrementmeasurements were over a relativelyshorttime range then the two-parametermodel fitted separatelyfor each tree may not properlydescribethe heightdevelopmentover the entire life spanof trees. Especially sincetheremaybevariationin theearlydevelopment of the standsthat is not directlyrelated to the growth rate afterwards.In the model used, the effectsof the early developmentare confounded with the rate parameter.Similarly, there is no guaranteethat the asymptoticbehaviorassumedin the modelis reasonable in standswhere the heightgrowthis not yet saturating. Theseextrapolationproblemscan probablynot explainaway the dependencybetween standageandstandheight/age curves,but the formof the dependency maynot be correctlyspecifiedabove.It is alsopossiblethat the relationsfoundaboveare to someextent distortedbecauseof the subjectiveselectionof standsinto the data set.

McDill andAmateis(1992) suggestthat traditionalsite indexmethodsbasedon one site-specificparameterare not capableof reflectingdifferencesbetween heightcurves.They suggesta modelwith bothglobalandstandspecificparameters,wherethenumberof (fixed)standspecific parameters is dependent onthe number of remeasurements

in each stand in the data set. In such a random

parameterheightcurvemodelas the abovemodelor the modelof Lappiand Bailey(1988), a basicassumption is alsothat for standheightcurveswe need severalstand-specific parameters. In a randomparametermodel, the numberof standspecificparametersis dependenton the age rangeof the datausedto estimatethe expectedvaluesof the randomparametersandthe variancesandcovariances of the parameters.In applications of a randomparametermodelwe canpredictseveralstandspecific parametersevenif we havemeasuredthe heightof onetree at oneage. It is specifically this casewhere the estimationproblemsdiscussed in this paperare more apparent(see, e.g., Figure 5). S'nrtilar problemswouldbe inevitableif a traditionalsiteindexmodelwith onestandspecificparameterwouldbe estimated fromthe samedata.If there are suffidentmeasurements for severalagesin a givenstand,then the prior information utilizedin the randomparametermodel

NOVEMBER1994/729

getslittleweight,andhencethereis no bigdifferencebetweenpredictions based on randomor fixed parametermodels. If data for a given standwould allow estimationof a three (fixed)parameterheightcurve (so that it wouldbe possible to differentiatebetweenthe initialdevelopment and growthrate of the mature stand),then obviously suchmodelshouldbe preferredto the abovemodel. It may be quite straightforward to extendthe modelto naturallyregenerated standswith someage variationwithinstands.One shouldthen take into account thattree parametersare probablycorrelatedwithtree age.This correlationwould be a logicalfeatureof the standgrowthprocesscontraryto the artifactcharacter of the correlationbetweenstandparametersandstandageanalyzedin thispaper. An extensionof themodelto genuineunevenagedstandswouldbe moredifficult.

LITERATURE

CITED

BIGING,G.S. 1985. Improvedestimatesof site indexcurvesusinga varying-parameter model.For. Sci. 31(1):248-259.

GUMPERTZ, M., and S.G. PANTULA. 1989. A simpleapproachto inferencein randomcoefficient models.Am. Star. 43(4):203-210.

HENTTONEN, H. 1990. Kuusenrinnakorkeuslfipimitan kasvunvaihteluEtel•i-Suomessa. Summary: Variationin the diametergrowthof Norwaysprucein SouthernFinland.Universityof Helsinki, Dep. of For. Mensur.andManage.Res. NotesNo. 25, Universityof Helsinki.88 p.

HSIAO,C. 1986.Analysisof paneldata.Cambridge UniversityPress,Cambridge. 246 p. LAPPI,J. 1990. Statisticalmethodsfor changing andnonhomogeneous forests.P. 115-122 in Proc. XIX IUFRO World Congress,Division6, Montreal, Canada. LAP•'I,J. 1991. Calibration of heightandvolumeequations withrandomparameters.For. Sci.37(3): 781-801.

LAPel,J., andR.L, BAI,•EY. 1988.A heightprediction modelwithrandomstandandthreeparameters: An alternativeto traditionalsite indexmethods.For. Sci. 34(4):907-927.

McDILL, M.E., and R.L. AMAT•IS.1992. Measmingforest site qualityusingthe parametersof a dimensionally compatible heightgrowthfunction.For. Sci. 38(2):409-429. MONSERUD, R.A. 1987. Variationson a theme of site index. P. 419-427/n Proc. IUFRO Forest growthmodelling andprediction conf.Soc.Am. For. Publ.SAF-87.12.Soc.Am. For., Bethesda, MD.

SE•mLE,S.R. 1971. Linearmodels.Wiley, New York. 532 p.

SF•mLE, S.R., andT.R. ROUNSAWLLE. 1974.A noteonestimating covariance components. Am. Star. 28(2):67-68.

SWAMY,P.A.V.B. 1970. Efficientinferencein random coeffidentregressionmodel. Econometrics 3(2):311-323.

VON•SH,E.F., andR.K. C•rr•R. 1992. Mixed-effectsnonlinearregressionfor unbalanced repeated measures. Biometrics48(1):1-17.

VUOrdLA, Y., andH. V•,LIAHO. 1980. Viljeltyjenhavumetsik'diden kasvatusmallit. Summary:Growth andyieldmodelsfor coniferculturesin Finland.Comm.InstitutiForestalisFenniae99(2). 271 p. WALTERS, D.K., T.G. GREGOI•, and H.E. BURKI•mT.1989. Consistentestimationof site index curvesfitted to temporaryplot data. Biometrics45(1):23-33.

APPENDIX:

SUMMARY

OF CALCULATIONS

Estimationand predictionmethodsused in this study can be summarizedas follows.

1. Equations (8)-(10)(ln(H•i(t))= ot•i+ limit -1 + ½•i(t); ot•i= a + a• + a•i and[•i

730/FOIlESTSCmNCE

= b + bn + bni)specifythe modelwherean,anobnandbniare randomparameters; k is the stand index, and i is the tree index.

2. Tree-specific parameters aniand[•niare firstestimated withOLSseparately for each tree.

3. Theresidual variance •r2 = var(en•(t)) isestimated bydividing thetotalsumofsquared residualsby the total numberof degreesof freedom.

4. Var(ani)andvar(an)areestimated usingEquations (17)and(18), respectively, where

T•, TA,and TOare defined inEquations (14)-(16), and Vni isthees',t•ated variance olt•heestimation erroroftheOLSestimate ofani,i.e., Vni= 82(XniXni)-•, where Xniisthemodel matrixoftreei instand k and62istheestimate ofo• obtained inthe previousstep.Var(bn/),var(bn),var(ani+ bni),andvar(an+ bn)are estimatedusing the sameformulas,Vnineededfor var(ani+ bni)andvar(an+ bn)is givenin (19). Covariancescov(anobni) and coy(an,bn) are estimatedusingequationcoy(x, y)

= ¾2(var(x + y) - var(x)- v•ty)).

5. Compute stand means •n.,and[•n.oftheOLSestimates ofan•_and [•n-•_Mean parametersa andb areestimatedwiththemeansofthe standmeans&n, and[•n. Standard

errors ofthese estimates areobtained using sample variances of•n.,and'6•'•..

6. Thedependence ofstand parameters anandbnonthe"current" stand age_ T, defined asthestand ageatdatacollection time,isstudied byregressing •n., and[•n.onT and transformations of T. The obtained Equations (20) and(21) canbe usedto predictAn = a + anandBn = b + bnfromthe currentstandage.

7. Thevariances oftheresidual stand parameters an'= An- Anandbn'= Bk- •n andtheircovariance canbe estimatedapplying the abovemethodfor ln(Hni(t))- A

8. Whenthe modelis appliedto a new stand,the expected(initial)valuesof aniand

intheequation ln(Hni(t)) = ani+ [•nit-• aregivenbytheestimates ofa andb,orby the regression Equations (20)and(21)forAn = a + anandBn = b + bn, if we know how to interpretthe currentstandage T. If there are heightmeasurementsin the stand,then we canget nonzeroestimatesfor the standparametersan andbn(or an'

andbn'if dependency onthe standageis utilized)andfor the tree parameters an•and bn•of measured treesapplying BLUP (BestLinearUnbiased Predictor)similarlyas doneby Lappiand Bailey (1988). Copyright ¸ 1994by the Societyof American Foresters Manuscript receivedMay 2, 1993

AUTHORS

AND

ACKNOWLEDGMENTS

JuhaLappiis with the FinnishForest ResearchInstitute,Suonenjoki ResearchStation,FIN-77600 Suonenjoki, Finland,andJuhaMalinenis withtheUniversityofJoensuu, Facultyof Forestry,Box111, FIN-80101 Joensuu,Finland.

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