49

a preliminary SUr.

Use of Logarithmic Regression in the Estimation of Plant Biomassl

G. L. BASKERVILLE

Canadian Forestry Service, P.O. Box 4000, Fredericton, New Brunswick.

Received August 24, 1971

photography or (t)~ of photography;ation of a base martsaic or large-scal~.tead;ing as it is not ea.lume. Alternatively.made less intensivo'oader types.planner should bearea and level orinventory is les~

.ones. This is truc: size varies little

sampled. For ex-~ present inventory1 been 20 times a~

would only hav"

BASKERVILLE, G. L. 1972. Use of logarithmic regression in the estimation of plant biomass. Can.J. Forest Res. 2, 49-53.

The basic assumptions of regression analysis are recalled with special reference to the use of alogarithmic transformation. The limitations imposed on inference-making by failure to complywith these assumptions are discussed and ways to avoid the limitations indicated. A systematic biasof the order of 10 to 20% which is inherent in most, if not all, prior uses of the logarithmic equationto estimate plant biomass is noted as is the correction for the bias.

BASKERVILLE, G. L. 1972. Use of logarithmic regression in the estimation of plant biomass. Can.

J. Forest Res. 2, 49-53.Les hypotheses de base de I'analyse par regression sont enoncees avec reference speciale a I'emploi

de la transformation logarithmique. Les restrictions imposees sur I'inference, a defaut de satisfaireles hypotheses, sont discutees et les moyens d'eviter les restrictions sont indiques. Un biais systematiquede 10 a 2070 qui est inherent dans la plupart, sinon tous, les emplois anterieurs de I'equation logarith-mique pour I'estimation de la biomasse des plantes, est note comme la correction pour le biais.

Occasionally regressions have been calculatedin terms of combinations of x-variables (usual-ly D2H) which give a linear relation in arith-metic ~nits. Avoidance of the logarithm maybe dangerous when it leads to violation ofnecessary assumptions of regression analysis.

This paper briefly reviews the assumptionsof regression and the reasons for using atransformation and calls attention to theappropriate way of converting estimates froma logarithmic equation back to arithmeticunits. These considerations are inherent inany use of a logarithm transformation andnot limited to calculations of plant biomass.However, it is shown that in the past, misinter-pretation of estimates from logarithmic equa-tions has resulted in underestimates of biomassin most, if not all, cases where the logarithmictransformation has been used. While theready access to computers today makesperpetuation of the error almost automatic,it is not difficult to seek and use methods thatare appropriate to each data set and willremove the error.

s the largest single'orest inventory. I taffected by require-:ail of information.mized by stringentexamination of the::s, and alternative~educe the cost or

Introduction

The most common procedure for estimatingin forest stands is through the use.and stand tables. A few stems

destructively sampled and the weight ofcomponent determined and related by

rcgression to some dimension of the standingI ree. A stand table which classifies stems perunit area by units of the dimension used in theregression is then expanded to an estimate ofbiomass by multiplying the number, of stemsIn each dimensi9n class by the weight ~estimat-ed from regression) for that class. This generalI\pproach has been common for man)' yearsItnd had been called allometry in Europe andJupan (Kira and Shidei 1967) and dimensionalI\nalysis in North America (Whitaker andWoodwell 1968).

The weight of a plant component usuallycan be plotted over some dimension (e.g.diameter, height, or a combination thereof)10 yield a straight line on double-log paper .Thus it has been expedient to calculate re-gressions as linear in the logarithms of thevariables and to transform back to arithmeticunits by determining the antilogarithm forIhe expansion of the stand table to biomass.

The ProblemIn the general case, we have two variables

Yand X such that, on double-log paper, theplot of Yon X yields a straight line. Therelationship suggested is that of the allometricequation "

IResearch sponsored by the U.S. Atomic EnergyCommission under contract with the Union CarbideCorporation.

Canadian Journal of Forestry, 2, 49 {1972)

50 CANADIAN JOURNAL OF Fa

[I] Y = 13Xa

We require an efficient and unbiased expres-sion of this relation which will permit (a)the estimation of f.- with limits of uncertaintygiven X .-and, (b) the comparison of theparameters 13 and a among independent datasets.

Solution for the parameters 13 and a canbe accomplished in arithmetic units by com-puter programs using an iterative least-squarestechnique which minimizes the sum of squares

N A[2] ~ (Y.- -Y.-)2

;=1

where N is the number of paired observations.Alternatively, equation [I] can be writtenin logarithmic form, either base e or base 10,

[3] LN(Y) = LN 13 + a LN (X)

which is linear. The parameters of this equa-tion can be estimated by solving as in ordinarylinear regression minimizing the sum of

squares

N { ~ }2

.-~1 LN( Y;) -LN( Y;)[4]

The sums of squares given in [2] and [4] arenot equivalent and the importance of choosingthe proper regression model when solvingfor f:J and a has recently been emphasized{Zar 1968).

The Right ModelThere are three assumptions fundamental

to a least-squares regression :I) It is assumed that for each X there is a

normally distributed population of y fromwhich the sample Y's used in the regressionare taken as a random sample. Failureto comply with this assumption will limitthe inferences that can be made regardingthe original population.

2) It is assumed that the true means, ~, ofall the sampled populations fall along agiven path, for example in the linear model~ = a + bX. Failure to comply withthis assumption will result in asystematicbias in estimated values of Y.

3) It is assumed that the variance, 0"2, is thesame for all the populations. That is, the

IREST RESEARCH. VOL. 2, 1972

populations of Y at every X are normallydistributed about their respective J.li withcommon variance, 0"2. Failure to complywith this assumption results in an "aver-aged" estimate of 0"2 and invalidates esti-mates of uncertainty and comparisons of13 and a among data sets.

Since we often wish to set limits of uncer-tainty and to compare sets of 13 and a to de-termine the feasibility of pooling data (forwhich purpose 0"2 must be uniform), it isdesirable that the uniformity of 0"2 be ensured,if necessary by transformation. The proceduresfor checking the uniformity of variance donot lend themselves to an approach with pass-or-fail tests of significance and judgment isan important factor (Draper and Smith 1966).A sequence of steps which the author hasfound useful is as follows :A-the variance of Y is calculated for eachX class and plotted over the X -class centerson arithmetic paper1) If the variance shows a definite trend, in

plant material commonly increasing withincreasing X, proceed as in step B. SeeDraper and Smith (1966) or other standardreferences for equivocal cases.

2) If the plot of variance of Y over X -classyields a horizontal band (often with wide,but random, scatter), this indicates that thevariance of Yi i5 independent of Xi andit is reasonable to assume a model ofthe form

[5] .Y.. = 13X..a + E..

where Ei is a random error. The appro-priate sum of squares to minimize is thatgiven by equation [2]. There are severaliterative least-squares methods availablefor such a solution, for example see Hull(1967) and Zar (1968).

B-lf the variance of Y is not uniform acrossthe domain of X, this indicates that the var-iance of Y is not independent of Xi. In thiscase a pos~i.ble model. would be

[6] yo = (PX.a ) E....

which, when transformed to logarithms yields

[7] LN (Yi) = LN(13) + a LN(Xi) + LN(Ei)

To test this possibility, the variance of LN(Y)is calculated .and plotted over X-class asbefore.

normallye J.li with

) comply

in "avl'/'.

ates e'ilj.

.risons or

of unccr.a to dc.

data (for

m), it I.

ensurcIJ,roceduro~"iance dovith pUN~.

19ment iNith 1966),lthor 111\'

for each

;s ccntcrM

, Itrend, i"

sing will!p B. '-.standllrd

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model

BASKERVILLE: LOGARITHMIC REGRESSIOr

I) If the variance shows a trend away fromthe horizontal, proceed as in step c.

2) If the plot of variance of LN(Y) overX-class is essentially horizontal with ran-dom deviations, then the model is indeedof the form of [7] and the appropriate sumof squares to minimize is that given byequation [4]. The solution procedure isto transform each Yand X variate to its log-arithm and compile as in linear regression.

C-If both the arithmetic and logarithmicvariances fail to show uniformity, it will beI\ccessary to weight each Y. observation,I.'ommonly by the inverse of the variance of

~l1ch f; (Draper and Smith 1966) and then.olve for the regression constants using theweighted logarithms. ,

For determining the correct model, I haveround a FORTRAN program which cal-~ulates the variance of Y and of LN(Y)by X -classes and plots the variance of Y overX and variance of LN( Y) over X useful( Bl\skerville 1970). Plots are also obtained ofY over X, LN( Y) over LN(X) and of thedeviations from regression in both arithmeticAnd logarithmic units. Such a display makesIt relatively easy to evaluate the validity()f the assumptions discussed above and todetermine the appropriate model.

The above is a minimal, but often sufficient,procedure for ensuring the correct choice ofmodel. The reader is referred to standardrcrerences for definitive treatments.

~

le

ze IS

"e

see

rm

t the .

'i. In thi.

lms ~

+ L .

of LN( r)

'-class II.

InterpretationI f the model is of the form [5] and if the

M(}lution for the parameters is by iterativetechniques that minimize the sum of squaresIn equation [2], then using the proper degrees'1( freedom: a) The sample variance (i.e.,the variance yielded by the (&;)2) is an unbiasedestimate of 0"2 and is the appropriate value touse in the comparison of regression para-

meters; b) the estimate Y; is an unbiasedestimate of J1 at X;; and c) The limits of un-certainty about y can be calculated in theusual way using 82.

If the model is of the form of [7] and if theKolution is by linear regression after transfor-mation to logarithms thus minimizing thelum of squares given by equation [4], then:a) The sample variance (i.e., in terms of (LN-

An Example

As an example of the difference betweenretransformation to the median and mean,Table 1 shows the estimate~ weight of foliageon balsam fir trees (Abies balsamea (L.)

~ IN THE ESTIMATION OF PLANT BIOMASS 51

(£j))2) is an unbiased estimate of 0"2 at LN(Xj);""""'

b) The estimate LN(Yj) is an unbiased esti-mate of J! at LN(Xj); and c) the limits ofuncertainty about LN(Y) are calculated inthe usual way using &2.

Conversion of LogarithmicEstimates to Arithmetic Units

When the logarithmic transformation isused, it is usually desirable, indeed necessary,to be able to express estimated values of Yin arithmetic (i.e., untransformed) units.However, the conversion of the unbiasedlogarithmic estimates of the mean and var-iance back to arithmetic units is not direct.This results from the fact that if the distribu-tion of LN( Y) at a given X is normal, thedistribution of Y cannot be normal but willcertainly be skewed. In fact, if the distributionis normal in logarithms, the solution of [3]for a given Xj and the determining of theantilogarithm of LN( Yj) yields the medianof the skewed arithmetic distribution ratherthan the mean (Brownlee 1967; Finney1941) ! The corrections for skewness are givenby Brownlee (on p. 62) as follows :

""""' Aif Ji = LN( Y) = 13 + ciLN(X)

and &2 = sample variance of the logarithmic

equation;Then

[8] y.;, e(~ + g2p>

[9] cr A2 .;, e (2~2 + 2~) -e(~2 + 2~).

where y is the estimated mean in arithmeticunits of the (skewed) Y distribution at Xand GA2 is the estimated variance (for theskewed Y distribution) in arithmetic units.Uncertainty limits can be retransformedfrom logarithms in a manner similar to yand these will be asymmetric about theregression line but the asymmetry will bein a direction appropriate to account for theskewness.

Weight (kg) of foliage determined from-DBH class

(inches} Median Mean Weighted mean

000259

142333476485

110140

123456789

1011121314

52 CANADIAN JOURNAL OF FOF

Mill.) from I to 14 in. (2.54 to 35.56 cm)in diameter at breast height: a) determinedfrom the median ti' that is the antilog of[13 + a LN(Xi)]; b) determined from themean ii as calculated by [8]; and c) deterrnin-ed from a weighted mean ti. The last is an

TABLE 1. Comparison of three solu~ions of the allo-metric equation LN(Y) = fJ + a[LN(X)]

adjustment for the fact that the slope of theallometric curve is continuously increasingover the domain of X and therefore the Y atthe X..class mid-point is always a slightunderestimate of the mean for all the possibleY's for the class. The regressions on whichthis table is based contained 102 observationsand by virtue of the scheme outlined aboverequired transformation to logarithms forcompilation. Further, examination of theplottings of the y variable and deviationsfrom the model over X showed that thedistribution of Y at a given X was normalin logarithm form and skewed in arithmeticform.

The differences in Table 1 are seen to beappreciable, particularly between the medianand mean estimates. For some cases it maybe reasonable to use the mediaft value, butin estimating biomass (and the chemicalinventories which depend upon it) it is clearthat the centroid of the class is the desiredvalue and this is given by the mean. Theliterature contains many estimates of plantbiomass based on logarithmic relationships,but I ~m ftot aware of any case (including

Conclusions

Proper use of regression techniques oftenmakes it necessary to transform data to theirlogarithms since failure to do so invalidateslimits of uncertainty and the comparison ofregression constants (for example to examinethe possibility of pooling data for stands or fora group of species). However, the transforma-tioa from the logarithmic form back to arith-metic units by simply determining the anti-logarithm has,.by failing to account for theskewness of the distribution in arithmetic

.03''.26.97.45.04.09.95.02..69.35.43.35.54.46

tEST RESEARCH. VOL. 2, 1972

my own data) in which the median was notinadvertently used in place of the mean inthe expansion to biomass per unit area. Theproblem was recognized by Madgwick (1970)although he did not pursue the matter .

It'ljs evident that the error introduced bythe use of the median Y where the mean Yis appropriate increases with the average sizein the X dimension. The effect could bedevastating when stands of different structureare being compared since a differential erroris introduced. For example, when the standtable for a young stand was expanded bymeans of appropriate logarithmic equationsto biomass per hectare determined by eachof the above three estimating procedures, itwas apparent that retransformation of re-gression estimates to median values as opposedto mean values introduced an error of theorder of 10-20% of the total biomass for atree component. This error will always be inthe nature of an underestimate.

I have examined some 40 regressions forvarious components of four broad-leaved andtwo coniferous tree species each having some70 to 100 observations. In every case, the var-iance was highly unstable in arithmetic unitsand the logarithmic transformation rectifiedthis problem. In every case, the plotted data(YIX, LN(Y)ILN(X), (Y- Y)IX) indicated thedistribution of Y to be normal in logarithmsand markedly skewed in arithmetic units.Thus, in every case it was necessary to applyequation [8] in the retransformation. Casualinspection of several similar data sets in theliterature indicates that while the use of alogarithmic transformation was valid, theretransformation was to the median whenit was intended to have been to the mean.

Nas not

rlean in

ea. The

c (1970)

Iced by

1lean y

age size

lu}d be

tructure

a} error

e stand

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)yeach

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BASKERVILLE: LOGARITHMIC REGRESSION

units, yielded the median rather than themean value of Yi for a given Xi. This hasresulted in a systematic underestimation of

biomass whenever the logarithmic transfor-mation has been used. Simply to avoid thelogarithm is not the solution since this will,in most cases, retain the unstable varianceand associated doubts about limits of un-

certainty.Inasmuch as there is ready access to com-

puters at virtually all centers of investigation,it is not a great burden to choose the regressionmodel and retransformation appropriate to

the data at hand. The bias is sufficiently largethat tests for its existence and, where neces-sary, its correction are well worth the effort.

The author acknowledges the assistance and advice ofDr. I. I. Beauchamp of the Oak Ridge National Labo.ratory, Biometrics Group, in the preparation of thematerial for this paper .

BASKERVILLE, G. L. 1965. Dry matter production inimmature balsam fir stands. Forest Sci. Mono. 9.42 pp.

.ons forved and19 somethe var-ic units"ectifieded datalted thea.rithms

units.) applyCasualin the

e of ald, thel whenmean.

softento theirilidatesison ofxamines or for

forma-) arith-e anti-lor thehmetic

IN THE ESTIMATION OF PLANT BIOMASS 53

-1970. Testing the uniformity of variance inarithmetic and logarithmic units of a Y-variablefor classes of an X-variable. Oak Ridge Nat. Lab.Publ. ORNL-IBP-70-1. 38 pp.

BROWNLEE, K. A. 1967. Statistical theory and method-ology in science and engineering. Second Edition.John Wiley and Sons, N.Y. 400 pp.

DRAPER, N. R., and SMrrH, H. 1966. Applied regressionanalysis. John Wil~y and Sons, N.Y. 405 pp.

FINNEY, D. I. 1941. On the distribution of a variate whoselogarithm is normally distributed. I. Royal Stat.Sci., Series B 7, 155-161.

HuLL, Norma C. 1967. STATPAK. U.S. At. EnergyComm. Doc. 1863, (see NONLIN).

KIRA, T., and SHIDEI, T. 1967. Primary production andturnover of organic matter in different forestsecosystems of the western Pacific. Iap. I. Ecol. 17,70-87.

MADGWICK, H. A. I. 1970. Biomass and productivitymodels of forest canopies. In Analysis of temper-ate forest ecosystems. Edited by D. E. Reichle,Springer Verlag, Heidelberg and New York, pp.47-54.

WHITAKER, R. H., and WOODWELL, G. M. 1968. Estimat-ing primary productivity in terrestrial ecosystems.Amer. Zool. 8, 19-30.

ZAR, I. H. 1968. Calculation and miscalculation of theallometric equation as a model in biological data.BioSci. 18, 1118-1120.