Estimation of the Diameter Increment Function or Other Tree Relations ...

Forest Science, Vol. 33, No. 3, pp. 725-739. Copyright 1987by the Society of American Foresters

Estimation of the Diameter Increment Function or Other Tree Relations Using Angle-Count Samples JUHA LAPPI

ROBERT L. BAILEY

ABSTRACT. A formula is derived for the bias when an angle-countsampleis usedto estimate the mean of a tree variable that is correlated with the breast height diameter. This bias occurs,for instance,if averageincrementis estimatedwith incrementcores from an angle-countsample.Estimationof meanincrementfor a giveninitial diameteris studiedfurther by assumingthat incrementsare log-normallydistributed,in which case the samplingdistributionis a mixture of three log-normaldistributions.An estimate obtainedby weightingobservationsinverselyto the basalarea (i.e., with the estimated tree frequency)comparesfavorably in simulationswith a parametricestimatederived from the samplingdistribution of diameters. If incrementsare regressedon the initial diameters,then weightingproportionallyto the initial basal area and inverselyto the current basal area gives smallerbias and standarddeviation of parameterestimatesthan weightinginverselyto the currentbasalareaalone.FOR.SCI. 33(3):725-739. ADDITIONALKEY WORDS. Incrementcores,log-normaldistribution,mixture distribution, weighted estimate.

ANGLE-COUNTSAMPLINGprovides an efficient means to estimate stand basal area by just counting qualified trees. Additional measurementson sampletrees can provide data for unbiasedestimatesof other standtotals (Grosenbaugh1958).Angle-count(point) samplinghasbecomea major way to sampletrees from forest stands.It may thus be of interestto use anglecount samplesto estimatemodelsfor tree properties.Inferenceon average tree propertiesor on the dependencesof tree properties,however,requires special attention. In angle-countsampling, the samplingdistribution of diameter is produced by weighting the original distribution (Van Deusen 1986). See Patil and Rao (1977 and 1978)for a general statisticaldiscussionof weighteddistributions.The simplearithmeticmean of diametersor basalareasover the angle-countsampleis biasedfor the stand mean (Bitterlich 1984). If each sampletree is weightedinverselyto the basalarea, or equivalently,by the estimatedstem numberper hectare(C/B, where C is the basalarea factor and B is the basal area of the tree), an asymptoticallyunbiasedestimate follows. However, biased estimates occur in less evident situations when trees from angle-countsamplesare used in the same way as trees from random samples. The authors are Forest Biometrician, Finnish Forest Research Institute, Suonenjoki Re-

searchStation,SF-77600Suonenjoki,Finland,and Professorof ForestManagementand Biometrics,Schoolof ForestResources,The Universityof Georgia,Athens,GA 30602.This work was completedwhile the first author was a ResearchAssociateat The University of Georgia. Risto Ojansuu of the Finnish Forest Research Institute drew our attention to the bias and weightingproblemswhen using incrementcores from an angle-countsample.He also made useful commentsduring the progressof our study.Discussionswith Paul van Deusenof the USDA Forest Service and Thomas Lynch of Oklahoma State University helped to define certain investigationsin the study.ManuscriptreceivedJanuary28, 1987.

SEPTEMBER 1987/ 725

The following problem motivated this study. Assume that we have used an angle-gaugeto sample trees and we have extracted increment cores to measure the diameter increment for the last k years. From these measure-

ments we would like to examinerelationshipsbetween diameterincrement and characteristicsof the trees and stands, such as initial diameter and site index.

If the diameterincrementsare regressedon the initial diameters(or just averaged within initial diameter classes), the parameter estimates are biased. For any two trees having the same initial diameter, the tree with larger diameter incrementhas a greater samplingprobability.If we weight observationsinversely to the samplingprobability (i.e., current basal area), then for each initial classthe weightedaverageincrement(the weightedsum dividedby the sum of weightsin the class)is asymptoticallyunbiased.The current basal area is composedof the initial basal area and basal area increment, and only the incrementpart is the sourceof the bias. But if we regress the incrementson the initial diameterand weightinverselyto the basalarea, then also the trees with small initial diameterget (on average)larger weight than observationswith larger initial diameter.This implicit weightingwith respectto the initial diameterincreasesthe varianceof parameterestimates. S6derberg(1986)presentsa discussionof thesebias and weightingproblems in angle-countsampling. In this paper severalaspectsof the utilization of angle-countsampleswill be discussed.First, a formula is derived for the expected value of a tree variablein an angle-countsample.Second,the exact connectionbetweena log-normal distribution of diameter increment and its samplingdistribution is presented.Third, assuminga log-normaldistributionof increment, the use of the weighted estimate is compared with a parametric (distributionbased) estimateby simulations.An attempt is also made to correct the bias in estimatesof averageradial incrementsas given by Sheffieldet al. (1985). Fourth, the estimation of regressionparameters is studied using simulations. A weightingprocedure(used also in S6derberg1986)is tested, which locally (around each initial diameter)removesthe bias and yet does not give greaterweight to trees with smallinitial diameters. In this paper we treat the number of sampledtrees as a fixed variable, i.e., we conditionon the numberof sampledtrees. We do not considerhow the spatialdistributionof trees affectsthe actual samplesizesin angle-count sampling.Populationsize is assumedto be infinite. Furthermore, we assume that the random errors (residualerrors) do not correlatewith the spa-

tial arrangement of trees.We thusconsider angle-count sampling asa method to draw a fixed number of treesfrom infinite tree populationswith selection probabilities proportional to tree basal area. By these assumptions, the basalareafactor or the numberof samplepointsis irrelevant. See Holt et al. (1980) for related problems in finite population surveys with varying sampling probabilities. The problems are discussedin general terms, but examples are presentedfor diameter (radial) increment as a function of the initial diameter. Before discussingangle-countsamplingin particular, we first describein general terms the samplingdistributionwhen sampling probabilitiesdependon the sampledvariable. SAMPLING DISTRIBUTIONS PROBABILITY SAMPLING

IN

Supposewe are interestedin a randomvariabley havingprobabilitydensity p(y), andwe sampleY'Swith samplingprobabilitiesproportionalto w(y) (see 726/FOREST SCIENCE

Patil and Rao 1978for closely related derivations).The probability of having a specific value y in the sampleis then proportional both to p(y) and w(y), i.e., the probability density for sampledy's, Ps(Y)is:

Ps(Y)= [w(y)p(y)]/c,

(I)

where c is a constant.In this paper,a subscripts alwaysrefersto the sampling distribution. In order to provide a proper density function, the scaling constant c must be:

c = fw(y)p(y)dy = E[w(y)].

(2)

That is, c is the mean of w(y) in the population. Supposethe samplingweightis originallydefinedin termsof anothervariable z, which has probabilitydensityp(z). Let z and y havejoint density p(z,y). If the samplingis proportionalto v(z), then the implied weight w(y) can be expressed as:

w(y) -- fv(z)p(zly)dz, where P(zly)is the conditional densityof z giveny. That is, p(zly) = p(z,y)/p(y). Then w(y)p(y) in (I) and (2) can be written:

w(y)p(y)= f v(z)p(zly)p(y) dz = fv(z)p(z,y) dz. Also in this case the scalingconstantis the expected value of the weight: c = E[v(z)].

(3)

In summary,if observationsare sampledproportionallyto w(y) or v(z), then the samplingdistributionsof y are: ps(y) = w(y)p(y)/E[w(y)],

(4)

and

Ps(Y)= f v(z)p(z,y)dz/E[v(z)],

(5)

respectively. In angle-countsamplingthe selectionprobability is proportionalto basal area, and the variable of interest may be diameter (as in Van Deusen 1986), basal area, volume, or any other tree variable. In the following sectionwe study the mean of such a variable in angle-countsampling. MEAN

IN AN

ANGLE

COUNT

SAMPLE

Let us assumethat the following model holdstrue for each tree i in a tree population:

Yi = fa(Xi) + el,

wherey is any tree variable,x is a vectorof standandtree variables(treated asfixed), a is a parametervectorand ei is a randomerror with zero mean.If x is a groupindicator,thenfa(Xi)is just the meanof the group.Denotethe breastheightdiameterat the samplingtimeby D andthe varianceof e by 0-2. Assumethat for a givenvalue of x, y and D have marginalprobabilitydenSEPTEMBER 1987/727

sitiesp(y) and p(D), respectively,andjoint densityp(y,D). Supposewe take an angle-countsamplefrom the tree population,and the randomerror (e) is independentof the spatiallocationsof trees. Sincethe samplingprobabilities are proportionalto D2, accordingto (5) abovethe probabilitydensity, Ps(Y),for a sampledy is:

Ps(Y)= fD2p(y,D)dD/E(D2) The expectedvalue of a sampledy, Es(y),is therefore:

Es(y) = = = =

[f fD2yp(y,D)dD dy]/E(D2) E(D2y)/E(D2) [E(D2)E(y) + cov(D2,y)]/E(D2) E(y) + cov(D2,y)/E(D2)

(6)

Recall that all probabilitiesand expectationsare for the givenvalue of x. From Equation (6) we can see that the y's in an angle-countsample are unbiasedfor the populationmean of y givenx if and only if y and D • are uncorrelated(for givenx). The simplestand mostcommonsituationwhen y andD 2 are uncorrelated for a givenx is whenD is an elementof thex-vector (i.e., D is consideredto be fixed, and hence all of its covariancesare zero). Note that this is the casein remeasurementsampleswhere the initial sample is selectedusingangle-countsamplingand the final diametersare regressed on the initial diameter [which is then the D-variable in (6)]. Stage (1960) suggests that incrementcoresfrom angle-countsamplescan be usedto predict future basal area incrementsby assumingthat the ratio between previous basal area incrementand current basal area is equal to the ratio between the future increment and current basal area. He estimates this ratio

by taking the arithmeticaverageof tree ratiosin the sample.This is essentially equivalentfor regressingthe past incrementon the currentbasalarea, and the bias problem can be avoided. However, it may not be logically appealingto regresspast variableson the current variables. An evidentsituationwhen y and D 2 are correlated(and bias occurs)is when y is D or is a function of D. Assumenow that we want to derive a model for the diameter at the samplingtime, i.e., D is the y-variable. Denotingf = f(x), and usingthe definitionsof varianceand covariance: E(D 2) = [E(D)]2 + var(D) = f2 + cov(D2,y) = cov(D2,D) = E(D 3) - E(D2)f, where

E(D 3) = E[0c + ½)3]

= Ee3 +f3 + 3f(r2, and therefore

cov(D2,D) = Ee3 + 2f• 2 Thus, in this case Equation (6) becomes:

Es(D) = E(D) + (Ee3 + 2fo'2)/(o '2 q-f2).

728/FOREST

SCIENCE

(7)

Supposethat we are interestedin incrementsof diameter during some time interval precedingthe samplingtime. To apply the above results,D is assumedto remain as the y-variable. Let D• be the initial diameterand let i = D - D• be the increment.ThusD = D• + i, andD• is necessarilyamong the x-variables, which are treated as fixed (the increment may also be function of D•). Becausethe covariancewith a fixed variable is always zero, cov(D2,y)in (6) is:

cov(D2,D) = cov(D2, D 1 + i)

= Cov(D2, D l) + Cov(D2, /) = Cov(D2, i). Thus, the bias is the same if the dependentvariable is D or a previous increment of D. The bias is also the same for the inside bark increment, if the bark thicknessis assumedto be known at both time points. Note that if Equation (7) is used to evaluate the bias of increments,f must still be the

expectedvalue of the final diameter,i.e., f = D• + E(i). The amount of the bias is, however, dependent on the variance of the random error. If, for instance,the number of incrementrings used in increment measurementsincreases,the bias causedby the angle-countsampling increases. Because volume and diameter are correlated, the same kind of bias occurs if the dependent variable is volume or volume increment for fixed values of initial diameter and volume.

Equation (6), or Equation (7) if applicable,gives the expectedvalue of sampledy in terms of the populationexpectations.Thus it haslimited use in practice when the populationparametersare unknown. There are two possible approaches to estimating population averages using angle count samples.First, by assuminga specificdistributionalform for the diameter distribution, the parameters can be estimated from sample statistics. Second, the averagevalue can be estimatedby weightingobservationsinversely to the basal area. In the next sectionwe describehow the parameters of the samplingdiameter distributioncan be derived from the parameters of the diameter distribution, and vice versa, if the diameter (or increment) distributionis log-normal. The log-normalmodel will be used later to study the propertiesof the weightingmethodof estimation. ANGLE-COUNT

SAMPLING FROM A LOG-NORMAL DIAMETER DISTRIBUTION

The log-normaldistribution (i.e., the logarithm of the variable of interest is normally distributed) has some useful properties in growth studies (Fiewelling and Pienaar 1981). For instance, growth cannot be less than zero, which is the lower bound of the lognormaldistribution. Some diameter distributions(for a whole standand notjust for a giveninitial diameter)are also similar to a log-normaldistribution.If the followingresultsare appliedfor the diameter distribution of a stand, the "initial diameter" means the lower limit of diameters, and "diameter increment" is the difference between diameter and the lower limit.

For our purposes, the log-normal distribution has the following useful property. We denote the log-normaldensityfunction with parameters• and

tr2 by "ln(y;•,tr2)" (• is the meanand tr2 the varianceof the naturallogarithm of y). If y hasdensityln(y;•,tr2)andy is sampledwith samplingprobabilitiesproportionalto y, then the samplingdensity(4) is ln(y;• + tr2,tr2)

SEPTEMBER 1987/729

(Patil and Rao 1978). If we sampleagainfrom the log-normalsamplingdistribution ln(y;p• + cr2,•r 2) with probabilitiesproportionalto y, the overall samplingis proportionalto y2 and the densityis ln(y; (p• + cr2) + cr2,•r2). In other words, when w(y) = y the samplingdensityfunctionis

f•(y) = y ln(y;•,•r2)/E(Y) = ln(y;[[ + ff2,ff2),

(8)

and when w(y) = y2 f•(y) = y2ln(y;[r,½r2)/E(y2) = ln(y;p•+ 2cr2,{x2).

(9)

The moments E(y) and E(y2) can be obtained from the general moment equationof a log-linearvariable (e.g., Fiewellingand Pienaar 1981),

E(yt) = exp(tp•+ t2cr2/2). Thus

E(y) = exp(• + cr2/2),

(10)

E(y2) = exp(2p•+ 2or2)

(11)

and

The diameter distribution in angle-countsamplingcan now be derived. We assume that the diameter increment

i = D - A for an initial diameter A is

log-normallydistributedwith densityln(i;•,•r2). With samplingprobabilities proportionalto D 2, the probability density of the sampledincrements, f•(i), is

f•(/) = c (A + /)2ln(i;g,cr2) = c [A2 ln(i;[[,• 2) + 2Ai ln(i;•,• 2) + F ln(i;[[,•2)]. From (8) and (9) we see that

i ln(i;•x,cr 2) = E(/)ln(i;p• + 0'2,0'2), and

/2ln(i;p,cr2)= E(i2) ln(i;p•+ 2cr2,o'2). Thus:

fs(i) = c[A21n(i;tx,cr 2) + 2A E(i)ln(i;tx+ or2,0 '2) + E(F)ln(i;g + 2cr2,o'2)].

(12)

The samplingdistribution is a mixture of three log-normal distributions. In order to provide a proper density,the scalingconstantc in (12) has the value:

c = 1/[A2 + 2A E(i) + E(F)].

(13)

The momentsE(i) andE(F) neededin (12) and(13) are givenin (10) and (11). Equation (12) showsclearly how the samplingdistributionapproachesthe original distributionas the initial diameter (lower limit) increases. The mean and second moment of the samplingdistribution can be obtained from the mixture density(12) usingmomentEquations(10) and (11):

730/FOREST SCIENCE

Es(i) = c[A2exp(F• + cr2/2)+ 2A exp(2F•+ 2cr 2) + exp(3F•+ 9cr2/2)]. Es(F) = c[A2exp(2F• + 2cr 2) + 2A exp(3F•+ 9cr2/2) + exp(4tx+ 8cr2)].

(14) (15)

Equation (14) can also be derived directly from Equation (7). Conversely, if the mean and secondmoment of the samplingdistributionare known, then the populationparameterscan be solvediteratively from Equations (14) and (15). We usedboth the Newton-Raphsonmethod(see, e.g., Maindonald 1984) and a slightly modified version of the simplex procedureof

O'Neill (1985).Bothworkedwellif specialattentionwasgivento keepingcr2 alwayspositive.Toobtainstartingvaluesfor txandcr2, we first solveparameters ix and cr2 in termsof the first and secondmomentfrom the moment Equations (10) and (11):

tx = 21n[E(y)]- ln[E(y2)]/2,

(16)

o-2 = ln[E(y2)] - 21n[E(y)].

(17)

and

Then the startingvaluesfor txandcr2 are computedby usingthe first and secondmoment of the samplingdistributionof y to approximateE(y) and E(y2).If the meanand secondmomentof the samplingdistributionare not known but are estimatedwith the samplemoments,this methodprovides method-of-moments estimatesof the parameters.Note that the "moment estimates" of Van Deusen (1986) are not moment estimatesin the traditional sense;i.e., they are not obtainedby equatingthe first samplemomentsto the theoretical

moments.

Maximum

likelihood

estimation

would also be

quite straightforward(see Van Deusen 1986),but it requiresmore computation and is not considered here.

In order to illustratethe influenceof angle-countsamplingin a real set of increment data, we recalculated the increment results of Sheffield et al. (1985). They calculated average increments in initial diameter classesfor

three differenttime periods.For the first period they usedincrementcore measurements(five-year cores)from angle-countsamples,and they did not take into accountthe biasgivenin Equation(7). This biasaccountsfor part of the apparentgrowth reductionpresentedby Sheffieldet al. (1985). We assumedthat the sampledincrementsare lognormallydistributed. Using the meansand standarddeviationsgiven in Sheffieldet al. (1985)for 1oblollypine (Pinustaeda L.) in the GeorgiaPiedmontand Mountains(Table 48 p. 86), correctedmean incrementswere determined(Figure 1). The differencebetweencorrectedvaluesandpublishedvaluesis 16%for the initial diameter classof 3-5 in. and reducesto 3% for large initial diameters(over 19 in.).

If the incrementshave a beta distribution,the samplingdistributionis also a mixture of beta distributions(see Patil and Rao 1978 and the derivations above). Thus the connection between the diameter distribution and the samplingdistributioncan be utilized in the sameway as for the log-normal distribution. Bias estimates for the data used in Figure 1 were also computed by using the beta distribution(a heuristicrule was used to estimate the upperbound).Resultswere very closeto thosepresentedin Figure 1.

SEPTEMBER 1987/731

Radial

increment

(in)

0.16

0.14

/

0.12

\,

(Sheffield etal

0.10

0.08

/

V

•corrected forbias)

0.06

0.04,

0.02,

0.00 I

'

0

I

2

•

I

4

•

I

J

6

Initial

I

'

8

I

10

'

I

12

diameter

'

I

J I

14

16

'

I

18

'

I

20

(in)

FIGURE1. Publishedaverageannualradialincrementsfor loblollypine and estimatescorrectedfor biasplotted over initial diameter.

ESTIMATING

THE MEAN DIAMETER BY A WEIGHTED OR PARAMETRIC ESTIMATE

In the previoussectionwe described how the meandiameter(increment) canbe estimatedby assuming thatthe diameterdistribution is log-normal. An alternative method is based on the theory of angle-countsampling without distributionalassumptions.

In angle-count sampling withbasalareafactorC, 4C/•rD2 is anunbiased estimateof the numberof treeshavingdiameterD. As describedby Husch, Miller, andBeers(1982)andBitterlich(1984),the mostnaturalway to estimatetree averagesis to calculatethe averagein the sameway as if these estimatedtree numberswere the true tree numbers,or equivalentlyby cal-

culatinga weightedaveragewiththeweightsbeinginverselyproportional to 732/FOREST

SCIENCE

the selection probability (basal area). If these estimated tree numbers are used to estimate stand totals, the estimates are unbiased. However, the ratio of two unbiased estimates is a biased estimate for the ratio of the estimated

quantities.Thus, even if the standtotals and tree numberscan be estimated unbiasedly,their ratio is biasedfor the correspondingpopulationratio (tree average). The distributionof a ratio of two random variables(e.g., the estimated total divided by the estimatednumber of trees) generallycannot be easily written in explicit form, even if the distributionsof numeratorand denominator are known. Thus, we were not able to derive any generalresultsfor the natural estimate of an arithmetic mean of diameter or any variable correlatedwith it (giventhe valuesof x-variables).In this sectionwe againlimit our discussion to the case where the variable of interest is diameter or diameter increment.

In the case of only one tree, the weighted averageequalsthe observedy, and thus the biasis obtainedfrom Equation(7) for a generaldistributionand from (14) for the log-normaldistribution.This givesthe upper boundfor the bias. Simulationis usedin this sectionto givean ideaof how rapidlythe bias and standard deviation may decrease. Mean diameter increment is estimated by assumingthat the increment is log-normallydistributed. When using large values for mean and variance of the increment, the results will correspondto the estimationof the mean diameterof the diameterdistribution in a stand.

It is also of interestto comparebias and standarddeviationof an anglecount samplewith the standarddeviationof a plot samplewith the same number of sampledtrees. With populationstandarddeviation tr, the stan-

darddeviation in a plotsample withn sampled treesistr/X/•n. Our simulationsare basedon the followingassumptions.The initial diameter at time t• is A, and the incrementis assumedto be log-normallydistributed with a givenmean and variance.The parametersix and tr2 of the lognormal distribution are calculated from (16) and (17) using the assumed values for the mean and variance of the increment(or diameter). The angle-

count sampleis assumedto be basedon the final diameter(i.e., on A plus the randomlygeneratedincrement).The propersamplingdistributioncanbe obtainedby first selectingwith a uniformrandomnumberwhich of the three distributions in mixture (12) has generatedthe variable, and then using a normal variate generatorto generatethe variable from that distribution. Three caseswere investigated(Table 1). The first two couldrepresent,for instance,the five-year diameterincrementsof loblollypine. The third case representsa possiblegeneraldiameterdistribution. With only one observedtree, the standarddeviationof the samplingdistribution cannot be estimated, hence the parametric estimate cannot be computed. Also the weighted estimate is the observationas such, and the bias can be computedanalyticallyfrom (14). With only a few observations the parametricestimateis also slightlybiasedbut less so than the weighted estimate.The initial biasin both estimatesdecreasesrapidlyto zero as the samplesize increases. With a smallnumberof sampledtrees, the estimatesbasedon the anglecount samplehave greaterroot mean squareerror than the estimatesbased on the random plot sample. As the sample size increases,both sampling

methodsseemto provideaboutequallygoodestimates.Simulationswith larger samplesizesand more simulationrunsthan presentedin Table1 indicatedthat the standarderrorsmay not decreaseasymptoticallyat the same rate. However, no clear commonpattern was found for differentparameter SEPTEMBER 1987/ 733

TABLE 1. Bias and standard error (SE) in estimationof average diameter incre-

ment when incrementsare log-normallydistributedwith mean (m) and standard deviation(SD), and initial diameteris A. Samplesizeis n. In angle-countsampling mean incrementis estimatedparametrically(par., methodof moments)or with a weighted(weig.) estimate using 10,000 simulatedobservations.Standard errors in

plot sampleswith the samenumberof sampledtreesare alsogiven(plot);theseare computedanalytically. A

m

SD

4

1.4

.7

n

par. 1

2 5 10 20

10

1.4

0.7

I 2 5 10 20

2

5

1.8

1 2 5 10

20 100

weig.

plot

Bias

--

0.20

0

SE

--

0.84

Bias SE Bias SE Bias SE Bias

0.04 0.49 0.02 0.30 0.01 0.21 0.01

0.08 0.51 0.03 0.30 0.01 0.21 0.01

0.70 0 0.50 0 0.31 0 0.22 0

SE Bias SE Bias

0.15 --0.01

0.14 0.09 0.77 0.04

0.16 0 0.70 0

SE Bias SE

0.48 0.00 0.30

0.49 0.01 0.30

0.50 0 0.31

Bias SE Bias SE Bias SE Bias SE Bias SE

0.00 0.21 0.00 0.15 --0.24 1.49 0.11 0.88

0.00 0.21 0.00 0.15 0.99 2.21 0.49 1.46 0.19 0.89

0 0.22 0 0.16 0 1.80 0 1.27 0 0.81

Bias

0.07

0.09

0

SE

0.62

0.62

0.57

Bias

0.04

0.04

0

SE

0.45

0.44

0.40

Bias

0.01

0.01

0

SE

0.20

0.19

0.18

values. We didn't try to derive any analyticresultsfor the asymptoticbehavior of the estimates.

When estimatingmeanincrementor meandiameterwe would usuallylike to alsoget an estimatefor the standarderror (or root meansquareerror) of the estimatedmean. If the meanincrementis estimatedfrom an angle-count sample,there are two easy approximatemethodsto estimatethe standard error (SE). These methodscan be applied both for the parametric and weighted estimate, but assume, for simplicity, that we are using the weighted estimate. The first method is to use the usual standarddeviation (SD) of the sampleddiametersin the sameway we would use standarddeviation in a plot sample to estimate the standard error of mean [SE =

(sd)/g/-•n]. Thisestimate isconservative; it hasa slight positive biasevenif it is used to estimate the root mean squareerror. This can be seen in Table 1 by dividing the standarderror for samplesize one (= standarddeviation in 734/FOREST

SCIENCE

thesampling distribution) byX/•nandcomparing withthestandard errors(or root mean square errors) found in the simulations.

A secondpossibilityis to use the bootstrapmethod(see, e.g., Efron and Tibshirani 1986). This we can accomplishby taking with replacementrepeatedsamples(bootstrapsamples)from the originalsampleand computing the weighted(or parametric)estimatefor each.Then the bootstrapestimate of the standard error is the standard deviation of the estimates in these

bootstrap samples.In our case the bootstrapestimate of standarderror seemsto be biaseddownwardsslightlymore than the previousmethodwas biasedupwardsfor the moderatesamplesizesused.Thus, and alsobecause the bootstrapmethodrequiresmorecomputations, the previousmethodcan be recommended.

ESTIMATION

OF REGRESSION

PARAMETERS

Resultsin the previoussectionindicatethat the parametricapproachto estimatingmeandiameteror meanincrementmay not be better than simple weightingby the estimatedstemnumber(i.e., inverselyto the basalarea). If the variable of interestis not directly related to diameter,the parametric approachwould be moredifficultto apply.Thus, the weightingseemsto be the most practicalway to estimatethe mean of any variabley for a given value of x (e.g., for a given stand). Recall that if D is an element of x, no weighting is needed.

In a general situationwe are estimatingE(y) simultaneouslyfor different

valuesof x; that is, the regression functionE(ylx). The regression function can be estimatedunbiasedlyonly if observedy's are unbiasedfor each x (after a possiblebias correction). Assumethat the error variance is constant over x-values. From the linear model theory we know that when error variance is constant,each observationshouldhave equal weight. However, if we weight the observationsinverselyto the basalarea and D alsodepends on x, then we use an implicit weightingwith respectto x. This implied weightingwith respectto x may not be reasonablewhen estimatingthe regressionparameters, as illustrated in Figure 2 where diameter incrementis regressedon the initial diameter.The excessiveweight of the first two observations(those with smaller initial diameter) is the causefor a counterintuitive estimationof the slopein Figure 2. Note that the first two observations have greater bias than the last two observations,and variance is assumedto be equal for all observations. We may also assumethat, in our case,the weightsshouldbe equalover the range ofx variables ff the error variance is constantover the x-values. To prevent the bias, the weightsshouldbe inverselyproportionalto D 2for each

valueof x (i.e., of the formg(x)/D2).Thusa reasonable weightmightbe:

w(D,x) = E(D2lx)/D 2.

(18)

This weightis approximatelyonefor all x valuesand stillweightsobservations inversely to D 2 for each x. If the error variance varies over x, then weight(18) canjust be multipliedby an ordinaryvariancestabilizingweight. The problemin the suggested weight(18) is that E(D21 x) is not generally known;ify = D2 thenE(D21 x) isjust whatwe try to estimate.But in most caseswe can easily find a reasonableapproximationfor it. For instance, whenregressing diameterincrement'onthe initialdiameterD•, D• couldbe usedasa roughestimateof D. The resultingweightis thenD•2/D2. Notethat in the standardweightingproceduresin regression analysisthe weightingis SEPTEMBER 1987/735

Diameter

increment

(in)

1.6

1.4

1.2

1.0

0.8

0.6

Data points

0.4

Regression

line

0.2

0

1

2

3

Initial

4

5

6

diameter

(in)

7

8

FIGURE2. Estimatedregressionline whendiameterincrementis regressedon the initial diameter and observationsare weightedinversely on the basal area.

a function of the x-variables only, contrary to our case where the weights depend on the y-variable. In the followingexamplethe estimationof a linear incrementfunctionis studiedusingsimulateddata. Let us assumethat the followingmodelholds true for diameterincrementi and initial diameterD•:

i= aD• + b + e.

(19)

Random error e is assumedto be log-normallydistributed(or more precisely:b + e is log-normallydistributed,b is the mean and e is the deviation from the mean). Note that fin the bias formula (7) is now (1 + a)D• + b. Table 2 shows some simulation results when a = 0 (but the assumedmodel is (19), and a is estimated), b = 1.4, and standard deviation of the error is 0.7. Parametersare estimated using observationsas is, weightingobserva-

tionsinverselyto D2, andusingtheproposed weighting D12/D 2. Resultsfrom angle-countsamplingare also comparedwith standarddeviationsof param736/FOREST

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TABLE 2. Simulated estimation of regressionparameters a and b in the linear regression(19) when a = O, b = 1.4 and standard deviation of the log-normally distributederror is 0.7. Thesevaluescould be reasonable,e.g., for 5-year diameter incrementofloblolly pine (Sheffieldet al. 1985).Initial diameterswere2, 5, and 8 in. For each value of initial diameterequal numberof treeswere taken (n = 1, 2, 5, 20). The mean and standarddeviationof the estimatedparameterswere computedwith no weighting(unw.), with weight 1/IY and with weight D?/D•. Also the standard errors in a plot samplewith the same numberof trees are given (plot); theseare computedanalytically. Intercept

n 1

2 5 10

Slope

unw.

1/D•

Di2/D2

plot

unw.

Bias

0.37

0.38

0.16

0

SE

1.17

1.21

1.03

0.92

0.199

0.212

0.182

0.165

Bias SE Bias SE

0.36 0.81 0.37 0.51

0.15 0.74 0.06 0.44

0.08 0.67 0.02 0.40

0 0.65 0 0.41

-0.035 0.136 -0.035 0.088

-0.019 0.132 -0.008 0.078

-0.007 0.117 -0.002 0.070

0 0.117

Bias SE

0.38 0.34

0.03 0.28

0.03 0.26

0 0.29

-0.038 0.058

-0.004 0.051

-0.004 0.047

0 0.052

-0.036

1/D2 -0.051

Di2/D• -0.012

plot 0

0.074

eter estimatesfrom a random (plot) sample of the same size. In a random sample, the variancesof the parameter estimatesare the diagonalelements of the matrix cr2(X'X)- •. The sampledincrementstend to be biggerthan the incrementsin the population, but the difference is smaller for big initial diameters. Thus, without weighting, the estimate of intercept is biased upwards and the estimate of the slope is biased downwards. When the sample size increases,the bias remains the same but standarddeviation decreases.When there is only one observation for each initial diameter, the weighting cannot decreasethe bias for the estimatedaveragediameter correspondingto each initial diameter. If the weighting is inversely proportional to basal area, the observationwith the smallest initial diameter, and hence the largest bias, gets the biggest weight. The resultingparameter estimatesare even worse than without any weighting. With the proposed weighting the parameter estimatesare only slightly worse than the estimatesin a plot sample. When the number of observations for each initial diameter increases, the

bias in the weighted estimatesapproacheszero. The estimateswith weight

Di2/D2 seemto becomeeven slightlybetter than the estimatesin a plot sample.With weightlID 2 we needabout15to 20% moreobservations to get the sameprecisionas with weightD•Z/D2. The initial bias of parameter estimatesincreasesif the bias of observed incrementsin Equation (7) increases.If the standarddeviation of the error were greater, or parametersa or b were smaller than used in Table 2, then the bias term in (7) and hence the bias in the parameter estimateswould be increased.

DISCUSSION

The resultspresentedassumethat the random error of the diameterincrement does not correlate with the spatial arrangementof trees. The effect of such correlation on the estimationof the mean increment is probably quite difficult to analyze. We anticipate, however, that the effect of the spatial SEPTEMBER 1987/ 737

distribution of random errors on the samplingdistribution is minor compared to the direct weightingeffect of the angle-countsamplingdiscussedin this paper. The spatial distributionof trees determines,of course, how the sample size (number of sampled trees), which we assumed to be fixed, varies in practice. It may be of interest to compareour simulationresultsfor the weighted estimatewith the theoreticalresultsof M•tern (1969). He found that when usingangle-countsamplingin a "Poissonforest," fewer treesmustbe measured(or counted)than in plot samplingin order to estimatebasalarea with equal precision. This evidently holds true also for the estimation of total (i.e., per area average) diameter. Furthermore, when estimating stem numbers,we need to measuremore trees with angle-countsamplingthan in plot sampling.The weightedestimatefor the mean (per tree) diameteris the ratio of estimated

total diameter

to the estimated

stem number. We found

that with a small number of sampledtrees the mean diameter (or increment) cannotbe estimatedas preciselywith an angle-countsampleas with a plot sample.When the samplesizeincreases,the two samplingmethodsseemto become quite closeto each other. A theoreticalanalysiswould be neededto derive exact resultsfor the asymptoticbehavior. Maximum

likelihood estimates would be more efficient than the method-

of-momentsestimatesthat were used to estimate the parametersof the lognormal increment distribution.In practical applications,the benefits of a more efficient estimatorwould be offset, in comparisonwith the nonparametric weightingmethod,by the fact that incrementsneverfollow any parametric model exactly. If a distribution-basedmethod is used, then the most efficient estimation method should, of course, be used. Our analytic or simulationresults do not directly tell if angle-countsampling shouldor shouldnot be used to collect data for estimatingmodelsfor tree properties.First, the amount of bias is dependenton the samplesize. We consideredthe samplesize to be fixed in both angle-countand plot sampling. In practice the number of sampledtrees varies, and it varies differently in different samplingmethods.Angle-countsamplingis often usedas a methodto obtain more practicalsamplesizesthan can be obtainedby using fixed area plots. Second, in the simulatedestimation of regressionparametersfor the diameter increment, the initial diameterswere taken to be fixed. In practice the samplingmethod also determinesthe spread of initial diameters. With angle-countsamplingwe get more trees with largeinitial diameters,which is generally advantageousin estimation. If angle-countsamplingis used and the diameterat the samplingtime is not an independent variable in a model describingrelations between tree properties,then the bias shouldbe correctedeither by weightingor by using specificparametric(distribution-based)methods.The weightingmethod is safe to use becauseit is not basedon any distributionalassumptions.It is easy to apply when the variable of interest is basal area increment, volume increment, or any other tree variable. The weightingmethod for estimation of the mean or regressionparameters would probably work well also for other tree variables.

LITERATURE

CITED

BITrERLICH, W. 1984. The RelascopeIdea. CommonwealthAgric. Bur., Slough,England. EFRON, B., and R. TIBSHIRANI. 1986. Bootstrap methodsfor standarderrors, confidenceintervals, and other measuresof statisticalaccuracy.Stat. Sci. 1(1):54-77.

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ELFWELLING, J. W., and L. V. PIENAAR. 1981. Multiplicative regression with lognormal errors. For. Sci. 27(2):261-269.

GROSENBAUGH, L. R. 1958.Point-sampling and line-sampling:Probabilitytheory,geometric implications,synthesis.USDA For. Serv., South. For. Exp. Stn. Occ. Pap. No. 160. HOLT, D., SMITH, T. M. E, and WINTER,P. D. 1980.Regressionanalysisof data from complex surveys.J. Royal Stat. Soc. seriesA 143:474-487. HUSCH, B., MILLER, C. I., and BEERS,T. W. 1982. Forest mensuration. Ed. 3. New York.

MAINDONALD,J. H. 1984. Statisticalcomputation.Wiley, New York.

M.•TERN,B. 1969.WiegrossistdieRelaskop-Flfiche? Allgemeine Forstzeitung (Wien,Austria) 79(2):21-22.

O'NEILL, R. 1985. Function minimizationusing a simplex procedure.P. 79-87 in Applied statisticsalgorithms,P. Griffiths and I. D. Hill (eds.). Ellis Horwood, Chichester. PATIL, G. P., and RAO, C. R. 1977. Weighted distributions:A survey of their applications.P. 383-405 in Applicationsof statistics,P. R. Krishnaiah(ed.). North Holland, Amsterdam. PATIL,G. P., andRAO,C. R. 1978.Weighteddistributions and size-biased samplingwith applicationsto wildlifepopulationsandhumanfamilies.Biometrics34:179-189. SHEFFIELD,R. M., et al. 1985. Pine growth reductionsin the Southeast.USDA For. Serv. Resour.Bull. SE-83. 112p. STAGE,a. R. 1960. Computinggrowth from incrementcores with point sampling.J. For. 58(7):531-533.

S•DERBERG, U. 1986.Funktioner for skogliga produktionsprognoser. Tillvfixtochformh6jdfor enskildatrfid av inhemskatrfidslagi Sverige.(Functionsfor forecastingof timber yields. Incrementand form heightfor individualtreesof native speciesin Sweden.)SwedishUniv. Agric. Sci., For. Mensuration& Manage.Rep. No. 14. 251 p. VAN DEUSEN,P. C. 1986.Fitting assumeddistributionsto horizontalpoint samplediameters. For. Sci. 32(1):146-148.

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