United Stales 

Department of 

Aoricullure 

and S emspec 

ume Ei on Equa 

Southeastern Forest Joe P. I\JlcGlure 

Experiment Station Jana Anderson 

Research Paper S E-264 Hans 9"". Sckreuder 
A Comparison of Regiona 

and Si te-Speci f ic 

Volume Estimation Equations 

Joe P. McClure, Project Leader 

Forest Inventory and Analysis 

Southeastern Forest Experiment Station 

Ashevillie, Msrth Carolina 

Jana Anderson, Statistician 

Business School, Colorado State University 

Fort Collins, Colorado 

Hans T. Schreuder, Project Leader 

Multiresource lnventory Techniques 

Rocky Mountain Forest and Range Experiment Station 

Fort Collins, Colorado 
ABSTRACT 

Re ressfon eguatfons for volume by region and site elass were examined 

for lobfolly pine. The regresslons for the Coastal Plain and Piedmont 

regions had sfgnfficantfy dlfferent slopes, The results shmed Important 

practicai dtfferences In percentags of conftdence fntervals contafnlng the 

true total voluns and in percentage of estimates wf thin a specif lc proportion of the true total in a stmlation study, Sampling from a modified 

Coastal Plain population with the same diameter dfstrlhtion as the Piedmont 

population and a modified Pledmnt population with the same dimeter distrlbution as the Coastal Plain population shaued that havf ng the proper 

d i mter dl str l but ion d id not improve predict ions either i n terms of conf ldsnce i nterva Is or I n gett l ng more est imtes w i th in a spec l f i ed percentage 

of true volume, 

The regressions for site class populations had significantly different 

s lopes, The simulation study shcwed that it mattered considerably which 

s i tes the smp l es were drawn from, 

S: Hensuration, loblolly pine, practical differences, statistical 

differences, weighted regression equations, 

There is a belief in forestry that 

trees of a given species requi re separate vo? ume tables for different sites 

and regions of the country. This, despite the fact that vol ume regression 

equations of the type V = a + b D2H, 

where V = volume and D" = diameter at 

breast height squared times height, tend 

to give very simi lar-looking estimated 

regression coefficients a and b for most 

data sets of a given species and across 

many spec i es , 

This Paper examines the gain associated with population-specific equations 

for merchantable cubic-foot volume inside 

bark of lob101 ly pine (Pinus -- taeda L.) 

by qregions and sites, Data for the 

study are from the Southeastern Forest 

Experiment Station's Forest Inventory 

and Analysis (FIA) unit. Because the 

sample sizes are very large, we expeGted 

the difference in equations to be statistically significant, We therefore 

also tried to assess the practical importance of the differences, 

Weighted simple linear regressions 

with assumed known weights k = 1,s were 

lylce!ure, Joe P,; Schreuder, %ns T,; Wl i son, 

Rodnw L, 1983, A emparison of several volume 

table equations for loblofly pins and white oak, 

Res, Pap, SE-240, Ashevl l le, NC: U,S, Department 

of Agr i a l t ure, Forest Ssrv f ce, Southeastern 

ForesB Experiment S1at ion, 8 pp, 

used in all cases. Weighted regression 

is similar to simple linear regression 

but assumes an increase in variabi 1 i ty 

i n the vari abl e of interest (vol ume) 

(measured by k) with an increase in the 

i ndependent variable (D2H). McCl ure and 

others1 estimated k to be 1.5 for large 

white oak (Quercus alba Lo) and lobloily 

pine data sets for themodel : 

where the mean value of e is 0 agd the 

variance of e is Va (e) = o* (D2H) , where 0' is the mean square residual. 

The literature is unclear on the 

need for separate equations for different regions or even di fferent species, 

Gevorki antz and 01 sen2 devel oped composite board-foot, cubic-foot, and 

cordwood volume tables combining conifer 

and hardwood species, The model used 

for cubic-fosl volume was the special 

case of equation (1): 

'evcark 1 an~z, S,R ,; Ol sen, L,P, 1955, Compos i 4-s 

volume tabfes for Plmber and their appl ication 

I n the Lake Saates, Tech, Bur l, 1104, Washing- 

*on, DC: U,S, Department of Agriwlture, 51 pp, 
with D in inches and H in feet, The authors concluded that these tables were 

adequate for large tracts generally and 

for small tracts for reconnaissance surveys or for timber of low value. They 

also pointed out that the definition of 

a good model is highly subjective. 

Van Deusen and others3 compared 

vol ume equations for 01 d-f i el d 1 oblol ly 

pine plantations for 702 trees from the 

Georgia Piedmont; 300 trees from A1 abama, Arkansas, and Mi s s i s s i ppi Gul f 

Coastal Plains; 378 trees from the Piedmont and Coastal Plain of Virginia, Delaware, Maryland, and North Carolina; and 

340 trees from the Tennessee, A1 abama, 

and Georgia high1 ands. They used the 

cmbined variable equation 

based on model (1) with k = 2. They 

el iminated the inf 1 uence of diameter 

distribution on the equation comparison 

by using the Mississippi diameter distribution as a model and modifying the 

other data sets to yield the same 

diameter distribution. They found 

statistical ly significant differences 

(at the 0.05 level) in outside-bark 

volume equations and attribute them to 

differences in measurement technique. 

The question of practical significance 

remai ned. 

Methods 

In our analyses, we addressed the 

subject of practical significance. If 

a popul ation-speci fic equation does not 

have to be used, inventory costs can be 

reduced because tree vol umes general ly 

do not have to be measured separately by 

region. Our criteria for determining 

the practical importance of di fferences 

in models wre: 

1. The relative proportions of 95- 

percent confidence intervals containing 

the total volume of a population (region 

or site class) for samples from that 

population (say Coastal Plain) or a different population (say Piedmont). 

2, Relative proportions of regression estimates that are within a set 

percentage (1, 2, 3, 4, and 5 percent) 

of the total volume for a population 

(region or site class) for samples from 

that population or a different but similar population, 

Data Sets 

The 1 obl 01 ly-t ree-vol ume data base 

described by McClure and others1 was used 

as a starting point in our study, Total 

volume in cubic feet, total height in 

feet, and diameter in inches were measured, These trees were selected to ensure a large sample of trees over the 

range of volumes and D2H values. Numbers of trees by size and site class, 

therefore, are not representative of the 

populations of interest, which are the 

actual loblolly pine populations in the 

Southeast. Since there were only 75 

trees in the mountain region and 55 in 

site class 5, the comparisons were limited to the Piedmont (P) and Coastal 

Plain (C) regions and productive sites 

(S, -,) and unproductive sites (S4+) .4 

Estimates of 2-inch diameter distrlbutions for these two regions and two 

site classes are available from the very 

1 arge permanent-pl ot data base maintained by the Southeastern Forest Experiment Station FIA unit. One-inch diameter class frequencies were obtained by 

interpolation. The tree-vol ume data 

base was then modified by random addition and subtraction of trees in the 

proper diameter classes to yield data 

sets with about the same diameter frequencies as the interpolated estimates, 

d~an Deusen, Pau I C,; Su l l f van, Al f red Dm; 

Matney, Thomas G, 1981, A predict ion system For 

cubfc foot volume of loblol ly pine appl lcable 

through much of its range. Southern Journal of 

Appl i ed Forestry 5(4) : 186-189, 

4~rodoctlon classes are based on cubic feet of 

y ield for ful ly stocked natural stands at cu lrnination of mean annual f ncrement, Sl tss 1 to 3 are 

those growing more than 85 cubic feet, site 4 

those w lth 50 to 85 cub lc feet, and s ite 5 those w l th 20 to 50 cub l c feet per acre per year, 
The resulting four large samples (two 

regions and two site classes) were 

treated as populations for purposes of 

this study, and two new data sets, P* and 

C*, were generated. P* consisted of 

Piedmont trees with the same diameter 

distribution as the Coastal Plain, and 

C* of Coastal Plain trees with the same 

diameter distribution as the Piedmont 

population. 

Results 

Regional and site class comparisons 

will be discussed separately. For each 

region or site class, the prediction 

equation 

or can be combined into one general 

equat i on 

with 3,601 trees and mean square error = 

0.00001. 

Because of the large sample sizes 

i nvol ved, the regression equations are 

significantly different. The 95-percent 

confidence interval constructed around 

the difference between the slopes of the 

two regional regressions is the interval 

(0.00002316, 0.00008105) which does not 

contain 0. The regression equations 

appear to be very similar, and the question of practical importance of differences remai ned . 

was used, where N is total number of The weighted Coastal Pl ai n regrestrees and is the mean for the popu- sion equation based on C* is 

lation (region or site) and a and b are 

regression coefficients estimated from VCr = -0.688 + 0.00211 DzH (5) 

the sample drawn. with 1,800 trees and mean square error = 

0.00001. The Piedmont equation based 

Regional Comparisons on P* is 

Coastal Plain (C) and Piedmont (P) 

populations consisted of 1,801 and 1,800 

trees , respect i vely . Skewness (bl ) and 

kurtosis (6) for volume are bl = 3.1 

and = 14.6 for C, bl = 5.5 and 4% = 

61.9 for P, bi = 3.7 and 4% = 22.6 for 

C*, and b, = 5.2 and 4% = 46.4 for P*. 

These values show that the populations 

are highly skewed; there are many trees 

with small and few with large volumes, 

There were two analyses: one for 

statistical significance and one for 

practical uti 3 i ty, The Pi rsl question 

was whether the weighted regression 

equatims with known ~eiyhts k = %,5 

for the two regions, 

with 1,801 trees and mean square error = 

0,00001 and 

with 1,800 trees and mean square error = 

0,00001 are sign1 fieantly different 

with 1,801 trees and mean square error = 

0.00001e 

The second question is : how we1 1 

do the regional regression equations 

predict total volume for each region? 

Because the test is limited to 

regression models, average DzH per tree 

and number of trees N in the region for 

which predictions were being made were 

assumed known and used in both equations. One thousand samples of sizes 

20, 40, 60, and 80 were drawn at random 

from each region. Total volume of the 

reyion was predicted for each sample by 

computing the regression eoe-l'f icients 

for each sa and combining this with 

known N and DZH for the region, The 

y soporti on of 95-percent samgl e conf f - 

dence limits containing the ac~al total 

volume for the reyion was computed along 

with the persentage sf volume estimates 

within 1, 2, 3, 4, and 5 percent of the 

actual total volume, 
The confidence interval results are 

shown in table 1. Samplings from either 

C or C* are better in predicting volume 

for C then sampling from P or P*. When 

P or P* is used rather than C, there is 

a loss from 15.4 to 16.9 percent for 

samples of size 20, and from 53.6 to 

56.0 percent for samples of size 80. 

Whether P or FM' is used seems to make 

1 ittle difference. Using C* rather than 

C actual ly improves confidence interval 

estimation when predicting volume for C. 

When predicting total volume for P, 

using C or C* results in slightly poorer 

confidence intervals than using P or P*. 

The loss ranges from 2.3 to 2.4 percent 

for samples of size 20 and from 5.7 to 

8.5 percent for samples of size 80. 

There seems to be a slight edge in using 

P* rather than P when predicting P 

vol ume. 

The percentages of time that the 

estimated volume is within 1, 2, 3, 4, 

and 5 percent of true volume for the 

predicted region are shown in table 2 

for n = 20 and n = 40 and in table 3 for 

n = 60 and n = 80. Using samples for C* 

rather than C to predict volume for C 

gives inconsistent results. For n = 20 

and n = 60, sampl i ng C* is somewhat less 

successful and for n = 40 and n = 80 

somewhat more successful than sampl i ng C. 

Sampl i ng from P* in predi cti ng volume 

for C is quite unsuccessful relative to 

sampling C or C*. Sampling from P* is 

only sl ightly better than sampl ing from 

P. Having a population with the same 

diameter distribution as the population 

being sampled results in only slightly 

improved estimates. 

When predicting volume of P, samp1 ing from P* rather than from P results 

5n higher percentages of times that the 

estimated volume is within 1, 2, 3, 4, 

or 5 percent of the true volume even 

though P* has a different diameter distribution than P. Sampling from C or C* 

is clearly worse than sampl ing from 

either P or P*. There is little difference in the results from C or C*, even 

through C* has the same diameter distribution as P. 

For site cl ass comparisons, there 

were 1,798 trees in sites 1-3 (SIe3) and 

1,401 trees in sites 4-5 (S4..5). Skewness (B,) and kurtosis () for volume 

are b, = 4.79 and 4% = - 40.47 for S,_, 

and b, = 2.91 and 4b2 = 14.83 for S4-5. 

Table 1.--Proportion of confidence intervals containing actual total volume for 

a region for samples of sizes 20, 40, 60, and 80 from each region and percentage 

loss in this proportion resulting from sampling in the other region, Each value 

is based on 1,000 iterations. 

Popul at i on 

sampl ed 

PROPORTION 

PEHCENTAGE LOSS 

Other region 16,9 31.8 45,O 53.6 2,3 4,5 2.7 8-5 

C* 0,O 0,O 0.0 0,O 2.4 2.9 0,7 5.7 

P * 15,4 30.4 45,2 56,O 0,3 0,O 0.0 0.0 

"If sampling from the other region yields a better result, the percentage loss 

is set at 0,O percent, 
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Table 3.--Percentage of regression estimates that are within 1, 2, 3, 4, and 5 percent of true total volume for 

a region and the relative success rates for samples of sizes 60 and 80 drawn from regions C, C*, P, and P* 

Region Percent Kegi on sampl ed (n=60) Success rate Regi on sampl ed (n=80) Success rate 

predicted deviation 

for-- a1 l owed C C* P P* C* R~ P* C C* P P* C* Ka P* 

Coastal 1 

Bldin 2 

3 

4 

a~atio of successful predictions in the unsampled region to successful predictions in the sampled region in 

percent, 
Equations S, -, were compared with 

equations S4_, in the same way as those 

for regions were compared. The better 

sites, S,_,, were combined because sample 

sires for the individual sites were 

i nadequate. The site equations are: 

with 1,798 trees and mean square error = 

0 . 00001 

with 1,401 trees and mean square error = 

0 .00001 

A general weighted regression for 

all trees is 

with 3,199 trees and mean square error = 

0.00001. 

The regression equations of S, .,., and S4+ are significantly different 

because the 95-percent confidence interval s constructed around the di fference 

of the slopes of the two regional regressions is the interval (-0.00005917, 

-0.00000172) which does not include 0. 

Because the test is limited to regression models, average D2H per tree 

and number of trees N in the site class 

for which predictions were being made 

were assumed known and used in all prediction equations. One thousand sampl es 

of sizes 20, 40, 60, and 80 were drawn 

at random from each site class. Total 

volume of each site class was predicted 

for each sample by computing the regression coefficients and combining this 

with N and for each site class. The 

percentages of 95-percent sample confidence limits containing the actual total 

volume for each site class were computed 

along with the percentages of volume 

estimates within 1, 2, 3, 4, and 5 percent of the actual total volume, 

Table 4 shows that there is a loss 

in sampling the wrong site class in 

terms of percentage of confidence intervals containing the true total volume. 

The loss increases with increase in sample size, going from 24.6 to 83.6 percent for n = 20 to n = 80 when sampl ing 

for S,-, and from 17.8 to 65.6 percent for 

n = 20 to n = 80 when sampling for S4-,. 

The percentages of times that the 

estimated volume is within 1, 2, 3, 4, 

and 5 percent of true volume for the 

predicted site are shown in table 5 for 

n = 20, n = 40, n = 60, and n = 80. It is 

clearly critical to use the regression 

from the correct site class to predict 

total volume. Predicting for Sl-, with 

the regressions from S,-, results in 

success rates of 53.4 percent or less 

for n = 20 to 37.8 percent or less for 

n = 80 with the 1, 2, 3, 4, and 5-percent 

deviations a1 lowed relative to the success rate for Sle3 regressions. Similarly, predicting for S4-, with regressions from S1 -, results has success 

rates of 75.4 percent or less for n = 20 

to 65.9 percent or less for n = 80 

relative to the success rate for S4-, 

regressions. 

Conclusions 

Although the results are subjective 

because the practical util ity criterion 

used differs between practitioners, firm 

conclusions can be drawn. A considerable price is paid by sampling from the 

other regions even when population size 

and average FrFi are known for a population. Although regression equations 

for regions seemed to differ very little, they had significantly different 

slopes that yielded differences of practical importance. Percentage of conf i - 

dence intervals containing the parameter 

of interest and percentage of estimates 

close to this parameter were considerably better when samples were drawn from 

the proper rqion. Waving the proper 

diameter distribution did not help when 

drawing samples from the other region. 

For site class popul ations, the 

regressions were significantly different and there was considerable difference in terms of percentage of conf idence intervals containing the parameter 

of interest and percentage of estimates 

close to this parameter, 
Table 4.--Percentage of confidence intervals containing actual total volume for 

a site for samples of sizes 20, 40, 60, and 80 from each site and percentage 

loss in this proportion resulting from sampling the other site class. Each 

value is based on 1,000 iterations. 

Site class Predictions 

sampled from-- Sites 1-3 Sites 4-5 

n=20 n=40 n=60 n=80 n=20 n=40 n=60 n=80 

PROPORTION 

Sites 1-3 

equation 0.907 0.892 0.902 0.885 0.735 0.564 0.421 0.306 

Sites 4-5 

equation 0.684 0.452 0.268 0.145 0.894 0.921 0.907 0.890 

PERCENTAGE LOSS 

Loss from 

sampl i ng 

other site 

class 24.6 49.3 70.3 83.6 17.8 38.8 53.6 65.6 
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I McC lure, Joe P,; Anderson, Jana; Schreuder, Hans T, I 

I I 

I A comparison of regional and site-specific volume estimation 

equations, Res, Pap, SE-264, Ashevi I le, NC: U,S, Department 

I 

I of Agr l cu l ture, Forest Servi ce, Southeastern Forest Exper iment I 

I Station; 1987, 9 pp, I 

I 

I Regression equations for volume by region and site class were exam- 1 

ined for loblol ly pine, The regressions for the Coastal Plain and I 

I Piedmont regions hadsignificantly dlfferentslopes. Theresults I 

1 showed important practical differences in percentage of confidence 

i nterva Is containing the true total volume and in percentage of I 

I estimates within a specific proportion of the true total in a simu- I I lation study. I 

I KEYWORDS: Mensuration, Ioblolly pine, practical differences, 

1 statistical differences, weighted regressionequations, 

I 

I 

11 I 

I 

I I 

I 1-.I .--.-m--P---..--.--.---.-(L---.s.-s---..- ----------CI -.-- - 

McClure, Joe P,; Anderson, Jana; Schreuder, Hens T, 

A comparison of regional and site-specific volume estimation 

equations, Res, Pap, SE-264, Ashevi l le, NC: U,S, Department 

I 

of Agr i cu l ture, Forest Servi ce, Sou theastern Forest Exper iment I 

Stat ion; 1987, 9 pp, I 

Regress ion equations for volume by region and s lie c lass were exam- I 

ined for loblolly pine. The regressions for the Coastal Plain and 

Piedmont regions had signif fcantly different slopes, The results 

showed important practical differences in percentage of confidence 

I 

i nterva Is containing the true total volume and In percentage of 1 

estimates within a specific proportion of the true total In a simulation study, I 

S: Mensuration, l oblol ly p ins, practical d i f ferences, 

statistical differences, weighted regression equations, 

I 

I