Content uploaded by Curtis L. Vanderschaaf

Author content

All content in this area was uploaded by Curtis L. Vanderschaaf on Oct 18, 2019

Content may be subject to copyright.

Journal : FORSCI

Article Doi : 10.1093/forsci/fxz058

Article Title : Predictive Ability of Mixed-Eects Height–Diameter Models Fit Using One Species but Calibrated for Another Species

First Author : CurtisL. VanderSchaaf

Corr. Author : CurtisL. VanderSchaaf

INSTRUCTIONS

We encourage you to use Adobe’s editing tools (please see the next page for instructions). If this is not possible, please list responses clearly in an e-mail, using line num-

bers. Please do not send corrections as track changed Word documents.

Changes should be corrections of typographical errors only. Changes that contradict journal style will not be made.

ese proofs are for checking purposes only. ey should not be considered as nal publication format. e proof must not be used for any other purpose. In particular

we request that you: do not post them on your personal/institutional web site, and do not print and distribute multiple copies (please use the attached oprint order

form). Neither excerpts nor all of the article should be included in other publications written or edited by yourself until the nal version has been published and the full

citation details are available. You will be sent these when the article is published.

1. Permissions: Permission to reproduce any third party material in your paper should have been obtained prior to acceptance. If your paper contains gures or

text that require permission to reproduce, please conrm that you have obtained all relevant permissions and that the correct permission text has been used

as required by the copyright holders. Please contact jnls.author.support@oup.com if you have any questions regarding permissions.

2. Author groups: Please check that all names have been spelled correctly and appear in the correct order. Please also check that all initials are present. Please

check that the author surnames (family name) have been correctly identied by a pink background. If this is incorrect, please identify the full surname of the

relevant authors. Occasionally, the distinction between surnames and forenames can be ambiguous, and this is to ensure that the authors’ full surnames and

forenames are tagged correctly, for accurate indexing online. Please also check all author aliations.

3. Figures: Figures have been placed as close as possible to their rst citation. Please check that they are complete and that the correct gure legend is present.

Figures in the proof are low resolution versions that will be replaced with high resolution versions when the journal is printed.

4. Missing elements: Please check that the text is complete and that all gures, tables and their legends are included.

5. URLs: Please check that all web addresses cited in the text, footnotes and reference list are up-to-date, and please provide a ‘last accessed’ date for each URL.

6. Funding: Funding statement, detailing any funding received. Remember that any funding used while completing this work should be highlighted in a sepa-

rate Funding section. Please ensure that you use the full ocial name of the funding body, and if your paper has received funding from any institution, such

as NIH, please inform us of the grant number to go into the funding section. We use the institution names to tag NIH-funded articles so they are deposited

at PMC. If we already have this information, we will have tagged it and it will appear as coloured text in the funding paragraph. Please check the information

is correct.

All commenting tools are displayed in the toolbar.

To edit your document, use the highlighter, sticky notes, and the

variety of insert/replace text options.

DO NOT OVERWRITE TEXT, USE COMMENTING TOOLS ONLY.

AUTHOR QUERY FORM

Journal : FORSCI

Article Doi : 10.1093/forsci/fxz058

Article Title : Predictive Ability of Mixed-Eects Height–Diameter Models Fit Using One Species but Calibrated for Another

Species

First Author : CurtisL. VanderSchaaf

Corr. Author : CurtisL. VanderSchaaf

AUTHOR QUERIES - TO BE ANSWERED BY THE CORRESPONDING AUTHOR

Please ensure that all queries are answered, as otherwise publication will be delayed and we will be unable to complete the next stage of

the production process.

Please note that proofs will not be sent back for further editing.

e following queries have arisen during the typesetting of your manuscript. Please click on each query number and respond by

indicating the change required within the text of the article. If no change is needed please add a note saying “No change.”

AQ1 General comment: Alarge number of changes have been made to improve the English. Please check carefully throughout.

AQ2 Please give “HTs” and “HTCD” in full

AQ3 Please give “HTs” and “HTCD” in full

AQ4 What does the “T” in equation 3 denote?

AQ5 General comment: For all displayed and in-line equations, please ensure that variables and parameters have been correctly

formatted throughout for consistency–– roman, bold or italic as appropriate, according to standard mathematical style (a number

of changes have been made).

AQ6 “however it is suggested, for this particular topic of determining if the ability of calibrating mixed-effects models across a variety of

species improves model- fitting ability, is a very important research topic for this particular subject of study” does not make sense.

Please check and reword.

AQ7 Please provide publisher details for the reference ‘Bechtold, 2005’, ‘Greenhill, 1881’.

AQ8 Please provide accessed date for the reference ‘O’Connell, 2017’.

AQ9 The resolution of figures 1, 2, and 3 are lower in quality than usually printed. Please provide high resolution images for better

processing.

Forest Science • XXXX 2019 1

For. Sci. XX(XX):1–11

doi: 10.1093/forsci/fxz058

APPLIED RESEARCH

biometrics

Predictive Ability of Mixed-Effects Height–Diameter

Models Fit Using One Species but Calibrated for

Another Species

CurtisL. VanderSchaaf

Mixed-effects individual tree height–diameter models are presented for important pines in the Western Gulf, USA. Equations are presented for plantations of loblolly (Pinus

taeda L.), longleaf (Pinus palustris P.Mill.), shortleaf (Pinus echinata Mill.), and slash (Pinus elliottii Engelm.) pine. To produce localized individual tree height estimates, these

models can be calibrated after obtaining height–diameter measurements from a plot/stand of interest. These equations can help answer an interesting question of whether

a model ﬁt for one species can be calibrated to produce reasonable height estimates of another species. In situations where mixed-effects models have not been developed

for a particular species, perhaps an equation from another species can be used. This question was addressed by calibrating these models using independent data of loblolly,

longleaf, and slash pine plantations located in South Carolina. For each calibration species, in addition to the models developed described above, previously published models,

but of the same model form, ﬁt using other species from across the USA were examined.

Results show that models of a variety of species can be calibrated to provide reasonable predictions for a particular species. Predictions using this particular model form indicate

that model calibration is more important than species-speciﬁc height–diameter relations.

Keywords: Pinus, pine, ﬁr, growth and yield

H–D models are an integral component of forest inventories

used to reduce sampling times. Mixed-eects H–D models

have been developed for many species (e.g., Lappi 1991,

Lynch et al. 2005, Trincado et al. 2007, VanderSchaaf 2014,

Mehtätalo et al. 2015). e greatest advantage when using linear

mixed-eects models often is the ability to calibrate the model using

data independent of those used in model tting. Plot/stand-specic

H–D relations can be produced for trees from plots/stands not in the

model-tting dataset if H and D observations have been collected

from trees in that plot/stand. Hence, calibration for a specic stand

(or plot) in a sense “borrows strength across units,” supplementing the

information from the stand/plot by information from all the other

stands/plots used in model tting (Fitzmaurice etal. 2004). Unlike

ordinary least squares, which ignores individual plot trends when t-

ting an H–D curve of trees from several plots when estimating H only

as a function of D, a calibrated mixed-eects model better captures

the individual behavior of the H–D curve within a particular stand

(or plot). Mixed-eects models t using plot-level data can also be

calibrated at the plot level during operational inventories, providing

more localized H–D relations within a stand. To increase the e-

ciency of forest inventories in loblolly (Pinus taeda L.), longleaf

(Pinus palustris P. Mill.), shortleaf (Pinus echinata Mill.), and slash

(Pinus elliottii Engelm.) pine plantations in the Western Gulf, USA

region, individual tree mixed-eects models were developed.

Mixed-eects H–D models have been developed for loblolly

pine plantations across the southeastern United States (Trincado

etal. 2007) and for East Texas plantations (Coble and Lee 2011).

Additionally, Coble and Lee (2011) presented an equation for slash

pine plantations in East Texas. However, all models were devel-

oped exclusively using data from industrial lands. e equations

presented here use data from a range of ownerships including

nonindustrial private landowners (NIPF), state, federal, etc. lands,

and are developed from plots located throughout the Western Gulf,

USA region (Figure 1). ere has not been a mixed-eects H–D

model developed for longleaf pine and shortleaf pine plantations in

the Western Gulf. Mixed-eects H–D models have been developed

for naturally regenerated shortleaf pine stands for western Arkansas

and eastern Oklahoma (Budhathoki etal. 2008).

AQ1

Manuscript received April 4, 2018; accepted July 31, 2019; published online June 17, 2019.

Afﬁliations: Curtis L. VanderSchaaf (vandersc@latech.edu), Assistant Professor, 101 Reese Dr, School of Agricultural Sciences and Forestry, Louisiana Tech

University, Ruston, LA 71272.

Acknowledgments: Useful comments were received from Associate Editor Dr Scott Roberts, an Applied Research Editor, and three reviewers.

Copyright © 2019 Society of American Foresters

Extract2=HeadB=Extract=HeadB

Extract2=HeadA=Extract=HeadA

Extract3=HeadA=Extract1=HeadA

Extract3=HeadB=Extract1=HeadB

BList1=SubBList1=BList1=SubBList

BList1=SubBList3=BList1=SubBList2

SubBList1=SubSubBList3=SubBList1=SubSubBList2

SubSubBList3=SubBList=SubSubBList=SubBList

SubSubBList2=SubBList=SubSubBList=SubBList

SubBList2=BList=SubBList=BList

Keywords=Keywords=Keywords_First=Keywords

HeadA=HeadB=HeadA=HeadB/HeadA

HeadB=HeadC=HeadB=HeadC/HeadB

HeadC=HeadD=HeadC=HeadD/HeadC

Extract3=HeadA=Extract1=HeadA

EM_Ack_Text=EM_Ack_Text=EM_Ack_Text_WithOutRule= EM_Ack_Text

REV_HeadA=REV_HeadB=REV_HeadA=REV_HeadB/HeadA

REV_HeadB=REV_HeadC=REV_HeadB=REV_HeadC/HeadB

REV_HeadC=REV_HeadD=REV_HeadC=REV_HeadD/HeadC

REV_Extract3=REV_HeadA=REV_Extract1=REV_HeadA

BOR_HeadA=BOR_HeadB=BOR_HeadA=BOR_HeadB/HeadA

BOR_HeadB=BOR_HeadC=BOR_HeadB=BOR_HeadC/HeadB

BOR_HeadC=BOR_HeadD=BOR_HeadC=BOR_HeadD/HeadC

BOR_Extract3=BOR_HeadA=BOR_Extract1=BOR_HeadA

EDI_HeadA=EDI_HeadB=EDI_HeadA=EDI_HeadB/HeadA

EDI_HeadB=EDI_HeadC=EDI_HeadB=EDI_HeadC/HeadB

EDI_HeadC=EDI_HeadD=EDI_HeadC=EDI_HeadD/HeadC

EDI_Extract3=EDI_HeadA=EDI_Extract1=EDI_HeadA

CORI_HeadA=CORI_HeadB=CORI_HeadA=CORI_HeadB/HeadA

CORI_HeadB=CORI_HeadC=CORI_HeadB=CORI_HeadC/HeadB

CORI_HeadC=CORI_HeadD=CORI_HeadC=CORI_HeadD/HeadC

CORI_Extract3=CORI_HeadA=CORI_Extract1=CORI_HeadA

ERR_HeadA=ERR_HeadB=ERR_HeadA=ERR_HeadB/HeadA

ERR_HeadB=ERR_HeadC=ERR_HeadB=ERR_HeadC/HeadB

ERR_HeadC=ERR_HeadD=ERR_HeadC=ERR_HeadD/HeadC

ERR_Extract3=ERR_HeadA=ERR_Extract1=ERR_HeadA

INRE_HeadA=INRE_HeadB=INRE_HeadA=INRE_HeadB/HeadA

INRE_HeadB=INRE_HeadC=INRE_HeadB=INRE_HeadC/HeadB

INRE_HeadC=INRE_HeadD=INRE_HeadC=INRE_HeadD/HeadC

INRE_Extract3=INRE_HeadA=INRE_Extract1=INRE_HeadA

SectionTitle=SectionTitle=SectionTitle=SectionTitle1

App_Head=App_HeadA=App_Head=App_HeadA/App_Head

Keywords=Text=Keywords=Text_First

applyparastyle “g//caption/p[1]” parastyle “FigCapt”

applyparastyle “body/p[1]” parastyle “Text_First”

1.5

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.52

1.55

1.60

1.65

1.70

1.75

1.80

1.85

1.90

1.95

1.100

1.104

2 Forest Science • XXXX 2019

ese newly presented equations can help answer an inter-

esting question of whether similar levels of predictive ability can be

obtained for H when using a calibrated mixed-eects H–D model

t using data of a species other than the one of interest as compared

to a calibrated model t using data of that species of interest. When

a species lacks an existing regionally applicable H–D model, using

predictions from an equation of another species would be very

useful if adequate estimates could be produced. Hence, in a sense,

if mixed-eects models can be calibrated across species, it can be

thought that all species are in the same population and that a single

model would be applicable to all species.

VanderSchaaf (2008) showed that using a loblolly pine plan-

tation mixed-eects H–D model when calibrated using sweetgum

(Liquidambar styraciua L.) data produced reasonable results to

predict H of sweetgum plantations. However, H was not of indi-

vidual trees but of arithmetic mean height, whereas D was quad-

ratic mean diameter. Asimilar concept was proposed by Huang

(2016) in which he used the term “universal” model to describe

calibrating models across species. He found that “composite” H–D

equations, or equations t using several species, following calibra-

tion, often produced similar predictions to species-specic mixed-

eects H–D models. is “universal” approach is similar in nature

to Zeide’s (1978, 1993, 1994) two-point principle. He suggested

for several stand-level variables that growth curves common across

many species could be tailored for an individual stand by obtaining

measurements of that variable in that stand at only two ages.

Although applicable operationally for many stand-level variables,

the two-point concept would be very dicult operationally to apply

to individual tree H–D relations because the two points would need

to be quantied for each individualtree.

Russell et al. (2014) showed that including species-specic

random eects when modeling the change in height in north-

eastern United States and eastern Canadian mixed-species stands

showed little dierence from species-specic models. However,

their study did not directly examine whether models t using one

species could be calibrated to produce adequate predictions for

another species. Crecente-Campo etal. (2013) conducted a sim-

ilar analysis to that carried out by Russell etal. (2014) for uneven-

aged mixed-species forests in the state of Durango, Mexico. ey

mentioned the potential to apply their models through the cali-

bration process in neighboring states but cautioned against it and

stated that the models needed to be tested prior to conducting such

an analysis. Although including species-specic random eects into

the model is a valid approach, their analyses did not address the

predictive ability of their models if a species was not included in the

model-tting dataset.

In addition to producing H–D models for the four pine species,

the aim was to determine the predictive ability of a calibrated H–D

model t for one species but calibrated for another. e previously

cited studies were limited in scope in that they addressed only a

small region of the world, and each only addressed H–D relations,

one at the stand-level (VanderSchaaf 2008) and the three others for

H–D relations of individual trees (Crecente-Campo etal. 2013,

Russell etal. 2014, Huang 2016). Calibrating across species may be

useful for many other variables such as basal area, volume, tree per

acre estimates, etc., and the ability to calibrate across species from

across the world may be applicable but of course must be tested.

is study examines the ability to calibrate across species from

throughout dierent parts of the United States. Other than than

VanderSchaaf (2008) and Huang (2016), no studies are known that

have examined the predictive ability of using mixed-eects models

of dierent species to predict for a species. e objective of this re-

search was to determine whether a linear mixed-eects H–D model

t using data of one species can be calibrated for another species to

produce adequate predictions.

Methods

Data Used in Model Fitting

e data used in model development were obtained from USDA

Forest Service Forest Inventory and Analysis (FIA) annual surveys

for Louisiana, Mississippi, and Texas (Figure 1). For Louisiana,

surveys completed between 2002 and 2015, for Mississippi,

surveys completed between 2009 and 2015, and for Texas, surveys

completed between 2009 and 2015, were included, respectively.

Data were obtained from the FIA database website (O’Connell

etal. 2017, US Forest Service 2017). Aplot consists of a cluster

of four points arranged such that point 1 is central, with points

2–4 located 120 ft from point 1 at azimuths of 0, 120, and 240°

(Bechtold and Scott 2005).

Each point is surrounded by a 24-ft xed-radius subplot where

trees 5.0 in. in diameter at breast height and larger are measured.

e four subplots combined total approximately 1/6th acre. Each

subplot contains a 6.8-ft xed-radius microplot where only saplings

(1.0–4.9 in. in diameter at breast height) are measured. Combined,

the four microplots total approximately 1/75th of an acre. e rel-

ative probabilities associated with a particular tree in relation to

diameter have no impact in terms of estimating H as a function

of D. For individual tree H–D measurements, they were simply

considered individual tree measurements when model tting and

calculating average tree-level summary statistics. However, the rel-

ative probabilities have some inuence when determining average

plot-level summary statistics and the number of trees of a particular

D used in model tting (Figure 2). e relative probabilities have

dierent associated-tree-per-acre expansion factors that will impact

summary statistics of per-unit-area estimates such as trees per acre

and basal area per acre estimates.

Individual tree mixed-eects models were developed for loblolly,

longleaf, shortleaf, and slash pine plantations in the Western Gulf,

USA region. Data from only those plots where a particular pine

species comprised at least 60 percent of the total basal area were

Management and Policy Implications

This paper presents mixed-effects height–diameter models for southern

yellow pine species in the Western Gulf region that can be applied during

forest inventories to help reduce inventory costs. Additionally, this study

demonstrates that a single mixed-effects height (H)–diameter (D) model can

be calibrated across a range of species. An example of where this could be

useful is for a species such as sand pine, Pinus clausa (Chapm. ex Engelm.)

Vasey ex Sarg., in the Lower Atlantic Coastal Plain, USA. Since it has less

commercial value and is located on far fewer acres relative to loblolly pine

(Pinus taeda L.) or slash pine (Pinus elliottii Engelm.), few models have been

developed for this species. This study shows that a mixed-effects individual

tree H–D model developed for loblolly pine could be calibrated for sand pine

if H–D measurements have been conducted in a sand pine plot/stand to pro-

duce adequate and plot/stand-speciﬁc H predictions.

2.5

2.10

2.15

2.20

2.25

2.30

2.35

2.40

2.45

2.50

2.55

2.59

2.60

2.65

2.70

2.75

2.80

2.85

2.90

2.95

2.100

2.105

2.110

2.115

2.118

Forest Science • XXXX 2019 3

included in the model-tting datasets. For the model of a partic-

ular species, only H and D measurements of that particular species

were modeled. Before model tting, all trees with broken stems

were removed from the model-tting data set, and only trees whose

heights were actually measured (as opposed to visually estimated or

predicted using equations) were used in model tting (only HTs

with HTCD= 1 within FIA database were included when mod-

eling, or those trees whose heights were actually measured in the

eld). Individual tree and plot-level summary data are presented in

Tables 1 and 2.

Model Development and Parameter Estimation

Total tree height was predicted as a function of diameter at

breast height:

lnHki =(β0+u0k)+(β1+u1k)lnDki +εki

(1)

where: ln=natural logarithm; Hki=individual tree total height (ft)

for tree i within plot (or stand) k; Dki=individual tree diameter at breast

height (in.) for tree i within plot (or stand) k; β

0, β

1=parameters to be

estimated; u0k, u1k=plot/stand-specic random eects, assumed to be

N(0, σ

2

0) and N(0, σ

2

1), respectively; (β

0+u0k)=plot/stand-specic

AQ2

Figure 1. Height–diameter models were created using Forest Inventory and Analysis data obtained from Louisiana, Mississippi, and Texas,

USA. Calibration data were obtained from South Carolina.

AQ9

Figure 2. Height–diameter relations by species for the model-ﬁtting and validation datasets. Sample sizes for the model-ﬁtting data:

loblolly pine, 64,415; longleaf pine, 1,049; shortleaf pine, 623; slash pine, 3,978. Sample sizes for the model validation data: loblolly

pine, 14,548; longleaf pine, 938; slash pine, 30. For the model-ﬁtting dataset, the gray curves are the “population average” estimates

from Equation 1 where individual plot-level factors creating correlations among individual trees from plots (created through the inclusion

of plot-speciﬁc random effects) were accounted for when model ﬁtting overlaid over the H–D measurements, and the black curve are

ordinary least-squares estimates overlaid over the H–D measurements where individual plot-level factors creating correlations among

individual trees from plots (created through the inclusion of plot-speciﬁc random effects) were ignored when model-ﬁtting.

3.5

3.10

3.15

3.20

3.25

3.30

3.35

3.40

3.45

3.50

3.55

3.59

3.60

3.65

3.70

3.75

3.80

3.85

3.90

3.95

3.100

3.105

3.110

3.115

3.118

4 Forest Science • XXXX 2019

intercept; (β

1+u1k)=plot/stand-specic slope; and ε

ki=random error

where it is assumed ε ~N(0, σ

2I).

Additionally, a covariance, σ

01, can be assumed to exist between

u0k and u1k, and thus it is assumed the plot/stand-specic random

eects have a bivariate normal distribution with mean 0.In this

particular case, linear mixed-eects models produce an ecient es-

timate of plot/stand-specic parameters because only six parameters

are estimated using the model-tting algorithm (β

0, β

1, σ

2

0, σ

2

1,

σ

01, σ

2). When using the variance and covariance estimates, plot/

stand-specic random eects (u0k, u1k) can be predicted and then

added to the “population average” intercept and slope (β

0, β

1)

estimates to obtain plot/stand-specic parameter estimates.

Plot/stand-specic random parameters produce a more localized

H–D equation, since the random eects account for local site

conditions such as soil type, genetic stock, site preparation,

midrotation silvicultural practices, spatial and time-specic cli-

matic conditions, etc. e prediction of plot/stand-specic random

eects is conducted outside the model-tting algorithm, and thus

degrees of freedom are not lost due to specic plot/stand random

eects. In terms of model tting, a less ecient method of obtaining

plot-specic parameter estimates would be to estimate parameters

separately for eachplot.

When compared to the traditional means of developing local

H–D equations, where H and D are measured and then a sepa-

rate equation is t for a stand (or in some cases a plot), a mixed-

eects model analysis is ecient because a model can be calibrated

without having to statistically t a model, and thus even small

sample sizes can be used (Lynch etal. 2005). In many cases, xed-

eects region-wide H–D equations are used to predict tree heights.

To account for growing-condition dierences among stands for the

same species, in addition to D, these models contain measures of

site quality and/or stand density, among others. Studies have shown

that a calibrated mixed-eects H–D model often produces better

predictions than a region-wide H–D model containing stand-

level regressors (e.g., Trincado et al. 2007, Temesgen etal. 2008,

Crecente-Campo etal. 2013).

e ln–ln transformation of the Power function was chosen be-

cause it (or its nonlinear untransformed form) has been shown to

provide reasonable predictions for a variety of species (e.g., Huang

et al. 1992, O’Brien et al. 1995, Lei etal. 2009, Sharma 2009,

Uzoh 2017, Subedi etal. 2018). eoretical and empirical studies

of the H–D relation suggest that it is an allometric function with

the power of diameter (or the slope in linear transformed form),

β

1, equal to 2/3 (Greenhill 1881, McMahon 1973, Norberg 1988,

O’Brien etal. 1995), and the slope is associated with the constant-

stress theory value, essentially that any diameter along the stem must

be of sucient size to support all weight above it, else the tree will

collapse under its own weight and other factors such as wind force

(e.g., Zeide and VanderSchaaf 2002). It is well known that the ln–

ln transformation often accounts for heterogeneity, and since it is

linear it will likely have fewer statistical convergence problems than

nonlinear equations, particularly when trying to include random

eects into the model. In addition, to determine whether this par-

ticular H–D mixed-eects model form (β

0+β

1lnDki+ε

ki) could be

calibrated across species, previously published mixed-eects models

using only the same model form and structure when tting this

model form for the other species were included in the analysis. is

congruence between the t and selected published models helps

avoid confounding issues between species and model form/struc-

ture when calibrating Equation 1 across species. If dierent equa-

tions or structures (e.g., the direct modeling of heterogeneity) are

used, then any ability or the lack of ability to calibrate cannot be

Table 1. Tree-level summary statistics of trees used in model ﬁtting and model calibration/validation.

Species No. of trees D (in.) H (ft)

Min Mean Max SD Min Mean Max SD

Fitting

Loblolly pine 64,415 1.0 8.0 59.1 3.5 6 53 148 18.17

Longleaf pine 1,049 1.0 9.7 28.1 4.8 8 65 112 22.34

Shortleaf pine 623 1.0 10.5 35.9 5.7 8 66 130 27.80

Slash pine 3,978 1.0 8.0 23.8 3.5 8 55 123 19.48

Validation

Loblolly pine 14,548 1.0 8.1 28.0 2.90 8 54 130 16.47

Longleaf pine 938 1.2 6.5 20.1 1.84 9 40 83 9.85

Slash pine 30 3.0 12.0 19.6 4.50 29 69 99 26.96

Note: Max, maximum; Min, minimum; SD, standard deviation.

Table 2. Model-ﬁtting and validation plot-levelmeans.

Species of interest No. of plots Species of interest Other species

TPA Dq (in.) BAA (sq ft) TPA Dq (in.) BAA (sq ft)

Fitting

Loblolly pine 2,289 332 6.6 78.8 363 3.0 18.0

Longleaf pine 70 181 7.9 62.2 119 5.0 16.0

Shortleaf pine 49 167 8.4 63.8 259 4.3 26.0

Slash pine 157 349 6.1 71.7 298 3.2 16.3

Validation

Loblolly pine 440 328 6.9 86.2 300 2.7 12.0

Longleaf pine 27 416 5.4 66.9 155 4.0 13.7

Slash pine 3 83 11.0 55.0 156 5.8 28.4

Note: “Species of interest” refers to plot-level summary statistics for the species used in tting the height–diameter equations. “Other species” refers to hardwoods as well

as other conifers. BAA, square ft of basal area per acre; Dq, quadratic mean diameter (in.); TPA, trees per acre.

4.5

4.10

4.15

4.20

4.25

4.30

4.35

4.40

4.45

4.50

4.55

4.59

4.60

4.65

4.70

4.75

4.80

4.85

4.90

4.95

4.100

4.105

4.110

4.115

4.118

Forest Science • XXXX 2019 5

separated exclusively into the eect of species. Aquestion would

then arise as to whether the ability or lack of ability to calibrate was

due to species, model form and/or structure, or the combination of

species and model form/structure.

Before model tting, all trees with broken stems were removed

from the model-tting data set, and only trees whose heights were

actually measured (as opposed to visually estimated or predicted

using equations) were used in model tting (only HTs with

HTCD=1 within FIA database were included when modeling,

or those trees whose heights were actually measured in the eld).

SAS Proc MIXED (Littell etal. 1996) was used to estimate the

parameters of Equation 1.No attempt was made to include spa-

tial correlation in the models to reduce complexity when applying

these models. Although heteroskedasticity (nonconstant variance)

likely exists, we are interested in predicting heights of trees equally

across all diameters. Hence, the random error variance–covari-

ance matrix was assumed to be σ

2Ink. Using weighted least squares

would put more weight on smaller-diameter trees when tting the

regression line that would reduce the predictive ability of larger-

diameter trees. Other studies have shown the tradeo between

biological and statistical considerations when tting models and

the impacts they can have on predictions and ultimately manage-

ment (VanderSchaaf and South 2009, VanderSchaaf etal. 2011).

Although it is well known that ln–ln transformations often ac-

count for heterogeneity, residuals were examined for trends in the

data (Figure 3).

Data Used in Model Calibration

If indeed models can be calibrated across species such that

adequate predictions are obtained, to gain insight into the op-

timal calibration sample size, independent FIA survey data from

the state of South Carolina were used. Surveys completed be-

tween 2012 and 2016 were included. Individual tree- and plot-

level summary data are presented in Tables 1 and 2. Data from

South Carolina were chosen because this is a meaningfully dif-

ferent population than those trees used in model tting from the

WesternGulf.

A minimum sample size of 10 trees per plot was selected, and so

a plot (both trees from the FIA subplot or microplot) had to con-

tain at least 12 trees of a particular pine species to be included. Ten

trees would be used to calibrate the model, the 11th and 12th trees

would be predicted, and at least two predicted trees would allow

calculation of variance. Calibration sample sizes of 1, 2, 3, 4, 5,

8, and 10 were examined. However, for slash pine, to have at least

three plots used in calibration, a minimum sample size of 5 was

used (a plot had to have at least seven trees to allow the calculation

of variance).

e tree used to calibrate the model for a sample size of one

was also the rst tree for a sample size of 2, the two trees used to

calibrate the model for a sample size of 2 were the rst two trees

for a sample size of 3, etc. To avoid the dependence of the calibra-

tion results on one particular sample for an individual calibration

sample size, calibrations were conducted 50 times for each calibra-

tion samplesize.

Validation analyses follow those presented in Trincado et al.

(2007). e dierence between the observed (Hobs) and predicted

height (Hpred) of all trees (i) whose heights were predicted for each

individual plot (k) and for each of the 50 replications (r) was cal-

culated (ekri= Hobs kri−Hpred kri), and trees used in calibration for a

particular plot and replication were not included in the validation

statistic calculations. For each plot (k) and replication (r) combina-

tion, the mean residual (ē) and the sample variance (v) of residuals

were computed and considered to be estimates of bias and preci-

sion; respectively. An estimate of mean square error (MSE) was

obtained for each combination by combining the bias and precision

measures using the following formula:

MSEkr

=¯

e2

kr

+

vkr

(2)

To obtain an average value of the three calibration sta-

tistics for each plot and calibration sample size, the 50 MSE,

ē, and v values of a particular plot for a particular calibration

sample size and species were then averaged. Average plot cali-

bration statistics were then averaged to obtain the nal calibra-

tion statistics for a given calibration sample size. e procedure

recommended by Baskerville (1972) was used to account for the

transformationbias.

Calibration of the same model form for several species

throughout the USA was tested. In addition to calibrating the

model developed for each of the four species (loblolly, longleaf,

shortleaf, and slash) for itself and the three other species, inde-

pendent models presented in other papers for loblolly pine across

the southeastern United States (Trincado etal. 2007), Douglas-r

(Pseudotsuga menziesii [Mirb.] Franco var. menziesii) and lodgepole

pine (Pinus contorta Douglas ex Loudon) from the inland north-

western United States (VanderSchaaf 2014), and quaking aspen

(Populus tremuloides Michx.) from Minnesota, USA (VanderSchaaf

2013), were also calibrated using the model validation data from

South Carolina, USA.

AQ3

Figure 3. Residuals from Equation 1 over standardized diameter at breast height for the model-ﬁtting datasets. Sample sizes for the

model-ﬁtting data: loblolly pine, 64,415; longleaf pine, 1,049; shortleaf pine, 623; slash pine, 3,978. Standardized diameter at breast

height is calculated as

D−A MD

SD

where AMD is the arithmetic mean diameter (inches), and SD is the standard deviation of D (diameter at

breast height).

5.5

5.10

5.15

5.20

5.25

5.30

5.35

5.40

5.45

5.50

5.55

5.59

5.60

5.65

5.70

5.75

5.80

5.85

5.90

5.95

5.100

5.105

5.110

5.115

5.118

6 Forest Science • XXXX 2019

Results and Discussion

Model-tting results (Table 3) for all t species (loblolly,

longleaf, shortleaf, and slash) showed that it is best to assume

that both β

0 and β

1 are random, or, essentially, that each plot

(or stand) has its own intercept and slope, and that a covariance

(σ

01) exists between u0k and u1k. Parameter estimates and model-

tting statistics are presented in Table 4. Residuals showed no de-

parture from model assumptions (Figure 3). When choosing an

optimal model calibration sample size at the plot or stand level

for a particular species, a suggested reasonable tradeo between

statistical measures (precision and accuracy) and sampling times

(e.g., costs) when using that species to calibrate is two or three

trees (Figures 4–6).

Example of Model Calibration

For clarity and ease of application, the methodology to predict

random eects at the plot or stand level is presented. Nomenclature

is based on Schabenberger and Pierce (2002, p.431). e expres-

sion used to predict random eects, estimated best linear unbiased

predictors, is:

ˆ

uk=

ˆ

DZ

T

k(

ˆ

Rk+Zk

ˆ

DZ

T

k)

−1

(yk−Xk

ˆ

β)

(3)

where:

ˆ

uk

= predicted random effects of plot/stand k,

a 2 × 1 vector where 2 is the number of predicted random

effects;

ˆ

D

= estimated variance–covariance matrix (2× 2) of

the random effects;

Zk

= matrix (nk×2) containing observed

values of D (natural log-transformed) from plot/stand k, and

a column of 1s;

ˆ

Rk

= estimated variance–covariance random

error matrix expressed as

σ2Ink

, since the variance is assumed

constant across all plots/stands and Hs are assumed tempo-

rally and spatially uncorrelated within a plot/stand;

yk

=nk×1

vector of observed Hs (natural log-transformed) from plot/

stand k;

Xk

=regressor matrix (nk×2) consisting of a column

of 1s and the observed values of D (natural log-transformed)

from plot/stand k; and

ˆ

β

= 2 × 1 vector of estimated fixed-

effects parameters.

To demonstrate model calibration, three individual longleaf

pine trees were randomly selected from an individual plot (or

stand) to be used in calibration. e heights for the three trees are

48, 50, and 44 ft, and the corresponding Ds are 5.8, 6.3, and 7.3

in., respectively.

Z=

1 ln

(

5.8 in .

)

1 ln(6.3 in .)

1 ln(7.3 in .)

=

1 1.757858

1 1.84055

1 1.987874

,

X=

1 ln(5.8 in .)

1 ln(6.3 in .)

1 ln(7.3 in .)

=

1 1.757858

1 1.84055

1 1.987874

where, besides the column of 1s, all numerical values are

lnD (naturally log-transformed value of D in inches) for the

three observations randomly selected from this particular plot

(or stand).

y

=

ln

(

48 ft

)

ln(50 ft )

ln(44 ft )

=

3.871201

3.9112023

3.78419

where all numerical values are lnH (naturally log-transformed

value of H in feet) for the three observations randomly selected

from this particular plot (or stand).

ˆ

β

=

ñ

β0

β1

ô

=

ñ3.0542

0.4990

ô

(

y−Xˆ

β)=

3.871201

−(

3.0542

+

0.4990

[

ln

(

48 ft

)]

3.912023 −(3.0542 +0.4990[ln (50 ft )]

3.78419 −(3.0542 +0.4990[ln (44 ft )]

=

3.871201 −3.931371]

3.912023 −3.972634]

3.78419 −4.046149]

=

−0.06017

−0.06061

−0.26196

ˆ

D

=

ñ

σ

2

0σ01

σ01 σ2

1ô

=

ñ0.2162

−

0.07319

−

0.07319 0.02621

ô

ˆ

R=

σ

200

0σ20

00σ2

=

0.01160 0 0

0 0.01160 0

0 0 0.01160

All numerical values for

ˆ

D

,

ˆ

R

, and

ˆ

β

were obtained from Table

4 for longleaf pine and are obtained from the model-tting results

of Equation 1.e dimensions of

ˆ

R

, Z, X, y, and

(y−Xˆ

β

)

will

change based on the number of observations used in the model

calibration.

When performing matrix operations as shown in Equation 3,

the following predictions of the random eects, estimated best

linear unbiased predictors, for β

0 and β

1 of this particular plot (or

stand) were obtained:

ˆ

u

=

ñu

0k

u1k

ô

=

ñ

−

0.2137

0.0562

ô

ese predicted random eects for this plot (or stand) are added

to the “population average” parameter estimates,

ˆ

β

, to obtain plot/

stand-specic parameter estimates for this particular plot/stand,

ˆ

βCalibrated

:

ˆ

β

Calibrated =

ñ

β0+

u

0k

β1+u1k

ô

=

ñ3.0542

−

0.2137

0.4990 +0.0562

ô

=

ñ2.8405

0.5552

ô

AQ4

AQ5

Table 3. –2Log likelihood (smaller is better) model-ﬁtting param-

eter estimates of the population average (β

0 and β

1) and random

effects variance (σ

2

0, σ

2

1) and covariance (σ

01) parameter estimates

by species.

Parameters –2Log likelihood

Loblolly Longleaf Shortleaf Slash

β

0, β

1–23,841.6 –586.7 –213.0 –2,466.6

β

0, β

1, σ

2

0–88,389.8 –1,175.3 –484.9 –5,418.5

β

0, β

1, σ

2

1–83,179.4 –1,001.7 –426.6 –5,114.3

β

0, β

1, σ

2

0, σ

2

1–96,291.8 –1,211.7 –491.0 –5,623.6

β0, β1, σ

2

0, σ

2

1, σ

01 –100,037.0 –1,390.6 –581.3 –5,909.1

n2,289 70 49 157

Note: n, number of clusters, or plots.

6.5

6.10

6.15

6.20

6.25

6.30

6.35

6.40

6.45

6.50

6.55

6.59

6.60

6.65

6.70

6.75

6.80

6.85

6.90

6.95

6.100

6.105

6.110

6.115

6.118

Forest Science • XXXX 2019 7

It is well known that logarithmic transformations often linearize

data and produce homogeneity of variances; however, a transfor-

mation bias occurs, since additive errors in ln–ln models become

multiplicative when transformed back to the original scale. To ac-

count for the transformation bias, the procedure recommended by

Baskerville (1972) should be used when predicting heights:

ln Hki

=(β

0

+

u0k

)+(β

1

+

u1k

)

ln Dki

+σ

2

/

2

(4)

where σ

2= mean square error (or residual variance) from the

model t (see Table 4).

To obtain H predictions in the original units (in feet), Equation

5 should beused:

Hki = exp

[β0k+β1kln Dki+σ

2

/2

]

(5)

where β

0k – (β

0+u0k) is the plot-/stand-specic intercept, and

β

1k – (β

1+u1k) is the plot-/stand-specicslope.

For the example above, Equation 5 would be expressedas:

Hki = exp[2.8405+0.5552l n D

ki

+0.01160/2]

(6)

Table 4. Population average (β

0 and β

1) and random effects variance (σ

2

0, σ

2

1) and covariance (σ

01) parameter estimates by species.

Loblolly pine Longleaf pine Shortleaf pine Slash pine

Est SE Est SE Est SE Est SE

β

03.0005 0.01027 3.0542 0.06358 2.9697 0.09099 2.9399 0.03758

β

10.4605 0.003897 0.4990 0.02299 0.5336 0.03127 0.5250 0.01443

σ

2

00.1997 – 0.2162 – 0.2869 – 0.1710 –

σ

2

10.02686 – 0.02621 – 0.03071 – 0.02345 –

σ

01 –0.06437 – –0.07319 – –0.08996 – –0.05754 –

–2LL –100,037 –1,390.60 –581.3 –5,909.1

AIC –100,029 –1,382.60 –573.3 –5,901.1

σ

20.009853 0.01160 0.01643 0.01059

n2,289 70 49 157

Note: σ

2, estimated mean square error; 2LL, twice the negative log-likelihood (smaller is better); AIC, Akaike’s Information Criterion (smaller is better); n, number of

clusters, or plots; SE, standard error of the estimate.

Figure 4. Model calibration mean square error results for longleaf pine plantations located in South Carolina, USA. Calibration sample

sizes are the number of trees used in calibration. n=27 plots.

7.5

7.10

7.15

7.20

7.25

7.30

7.35

7.40

7.45

7.50

7.55

7.59

7.60

7.65

7.70

7.75

7.80

7.85

7.90

7.95

7.100

7.105

7.110

7.115

7.118

8 Forest Science • XXXX 2019

Assuming a tree has a D of 7.44 in., the predicted H would be

52.5 ft. Use of the noncalibrated, “population average” curve, on

the other hand, provides a predicted height of 58.1 ft, a 9.6 percent

dierence.

Figure 7 clearly demonstrates that calibrating Equation 1 using

observed Hs and Ds from this particular plot (producing Equation

6) vastly improved predictive ability, consistent with many

other studies (e.g., Trincado et al. 2007, Temesgen et al. 2008,

VanderSchaaf 2014). Relative to the “population average” curve,

for a given D, most Hs for this particular plot are shorter (perhaps

because of genetic stock, stand-level species composition, aspect,

soil type, site productivity, etc.). Hence, Equation 6, through the

calibration process, provides an H–D curve that reects this be-

havior. However, using the so-called “population average” trend

fails to recognize the behavior of trees in this individual plot relative

to the average behavior across allplots.

Calibration Using Other Species

Figures 4–6 clearly demonstrate that mixed-eects models of a

variety of species can be calibrated to provide reasonable predictions

for a particular species. It is interesting to note for a particular cal-

ibration species that almost all curves regardless of model-tting

species converge after using just one tree in calibration. In general,

regardless of the model-tting species, for longleaf and slash pine

any model using two or more trees in calibration provided nearly

the same predictive ability. is is true even for model species such

as aspen. Loblolly pine showed more variability. However, the use

of three or more trees generally produced nearly the same predic-

tive ability among all the model species. For all three validation

species, in some cases models of other species actually produced

slightly better predictions. ese results indicate, at least for this

model form of a H–D equation, that when a species does not have

a species-specic model that models t using other species when

calibrated can provide similar, and in some cases even slightly

better, predictions.

Figure 7 also shows that the H–D mixed-eect model form

evaluated in this study can be calibrated for a particular species

regardless of the model-tting species. In fact, for this longleaf

pine plot represented by these three trees when calibrating, the

calibrated aspen model produced the best prediction statistics.

When calibrated using these three trees, the aspen, longleaf pine,

and Douglas-r models produced MSEs of 44.4419, 51.2244, and

59.0495, respectively. e aspen model may not always produce the

best predictions when applied to otherdata.

For this H–D model form, a calibrated model of another species

produces better predictions than an uncalibrated species-specic

model (Figures 4–7). Apparently, based on these model results,

the model calibration process exceeds the importance of species-

specic H–D relations, since, regardless in many cases of the partic-

ular species-specic parameter estimates used, improved predictive

Figure 5. Model calibration mean square error results for slash pine plantations located in South Carolina, USA. Calibration sample sizes

are the number of trees used in calibration. n=3 plots.

8.5

8.10

8.15

8.20

8.25

8.30

8.35

8.40

8.45

8.50

8.55

8.59

8.60

8.65

8.70

8.75

8.80

8.85

8.90

8.95

8.100

8.105

8.110

8.115

8.118

Forest Science • XXXX 2019 9

results were observed for a particular species using models t for

several other species. Often modelers and foresters believe that only

species-specic parameter estimates can apply to that specic spe-

cies. However, these results show that following calibration models

of many dierent species can be applied to predict variables of other

species; hence in many cases, for certain model forms, parameter

estimates and model forms may be universal across many species

rather than species-specic.

Conclusions

Mixed-eects H–D models were presented for loblolly, longleaf,

shortleaf, and slash pine plantations in the Western Gulf, USA.

By obtaining H–D measurements from plots/stands of interest,

Equation 1 can be calibrated to local site conditions. To calibrate

these models for specic plots/stands, an Excel spreadsheet is avail-

able upon request.

Mixed-eects models of a variety of species were calibrated for

individual species and were found to provide reasonable predictions

for that particular calibrated species. is was true, even for

groups of species as dierent as pines, rs, and in some cases even

hardwoods. Future work should concentrate on examining the pre-

dictive ability of other H–D model equationforms.

ese results, along with those of VanderSchaaf (2008) and

Huang (2016), suggest that models t for one species can be

calibrated across species. However, this process may not work for

all H–D model forms and for all variables. For instance, in order

Figure 6. Model calibration mean square error results for loblolly pine plantations located in South Carolina, USA. Calibration sample

sizes are the number of trees used in calibration. n=440 plots.

Figure 7. Observed longleaf pine total tree height and diameter at

breast height for n=22 trees. This plot resides in Charleston county,

South Carolina, USA. The bold solid lines are “population average”

estimates for three species (longleaf pine, aspen, and Douglas-ﬁr)

obtained by not calibrating Equation 1, and the lighter solid lines

are estimates for the three species following calibration of Equation

1.Black diamonds and black circles are paired height–diameter

observations. Black circles were trees used to calibrate Equation

1.Black lines are predictions using the longleaf pine model, dark

gray lines are predictions using the aspen model, and light gray

lines are predictions using the Douglas-ﬁr model.

9.5

9.10

9.15

9.20

9.25

9.30

9.35

9.40

9.45

9.50

9.55

9.59

9.60

9.65

9.70

9.75

9.80

9.85

9.90

9.95

9.100

9.105

9.110

9.115

9.118

10 Forest Science • XXXX 2019

to better constrain future predictions of basal area observed in

young stands for species with relatively longer rotations or of lower

value that do not justify the implementation of large-scale, com-

prehensive research studies, perhaps a mixed-eects model t but

calibrated using a species with a younger economic rotation or bio-

logical rotation age or that is valuable enough to justify large-scale,

comprehensive research studies, model calibration will be bene-

cial. By “borrowing” information from other species, expensive,

long-term, studies can be avoided for the species of interest, since

information on other species can be used to represent the temporal

growth dynamics of the calibrated species.

Future research should concentrate on determining whether

using parameters from a species where more complete data in terms

of stand ages, planting densities, site qualities, stand development,

etc., exist provides more reasonable extrapolations of stand develop-

ment for another species following calibration rather than using lim-

ited data of that desired species to t models (VanderSchaaf 2008).

is current analysis cannot denitively answer this question; how-

ever it is suggested, for this particular topic of determining if the

ability of calibrating mixed-eects models across a variety of species

improves model-tting ability, is a very important research topic

for this particular subject of study. Predicted plot-specic or stand-

specic random parameters depend on the amount of estimated

variability in the random eects and the “population average” pa-

rameter estimates. us, dierences in site requirements, structural

constraints, growth habits, etc., among species may not allow for

mixed-eects models t using one species to produce reasonable

predictions for another species.

Additionally, ecological and physiological studies can also be

conducted such as examining how calibrating across species and

determining its impact on the “population-average” and random

parameters (e.g., individual plot or stand parameters) of, say,

Equation 1 can provide an insight into the applicability of concepts

such as the constant-stress theory, etc. Perhaps mixed-eects model

calibration can be used to justify future research to determine the

inclusion or exclusion of certain species into various consistent

forms or functions of species that can lead to extensive amounts of

study to determine why the species have constant growth forms and

functions within categories or why they do not grow commonly

within categories by form and function.

LiteratureCited

B,G.L. 1972. Use of logarithmic regression in the estimation of

plant biomass. Can. J.For. Res. 2:49–53.

B,W.A., C.T.S. 2005. e forest inventory and anal-

ysis plot design. P. 27–42 in e enhanced forest inventory and anal-

ysis program—national sampling design and estimation procedures,

B,W.A., and P.L.P (eds.). US Forest Service Gen.

Tech. Rep. SRS-GTR-80.

B,C.B., T.B.L, J.M.G. 2008. A mixed-eects

model for the dbh–height relationship of shortleaf pine (Pinus echinata

Mill.). South. J.Appl. For. 32:5–11.

C,D.W., Y.L. 2011. A mixed-eects height–diameter model

for individual loblolly and slash pine trees in East Texas. South. J.Appl.

For. 35:12–17.

C-C, F., J.J. C-R, B. V-L,

C.W. 2013. Can random components explain dierences in

the height–diameter relationship in mixed uneven-aged stands? Ann.

For. Sci. 71:51–70.

F,G.M., N.M.L, J.H.W. 2004. Applied longitu-

dinal analysis. Wiley, New York. 506 p.

G,A.G. 1881. Determination of the greatest height consistent

with stability that a vertical pole or mast can be made, and of the

greatest height to which a tree of given proportions can grow. P. 65–

73 in Proceedings of the Conference of the Cambridge Philosophical

Society Vol. 4.

H,S. 2016. Population and plot-specic tree diameter and height pre-

diction models for major Alberta tree species. Forestry Division, Alberta

Agriculture and Forestry, Edmonton, AB. 59 p.

H,S., S.J.T, D.P.W. 1992. Comparison of nonlinear

height–diameter functions for major Alberta tree species. Can. J.For.

Res. 22:1297–1304.

L,J. 1991. Calibration of height and volume equations with random

parameters. For. Sci. 37:781–801.

L,X., C. P, H.W, X.Z. 2009. Individual height–di-

ameter models for young black spruce (Picea mariana) and jack pine

(Pinus banksiana) plantations in New Brunswick, Canada. For. Chron.

85:43–56.

L, R.C., G.A. M, W.W. S, R.D. W.

1996. SAS® system for mixed models. SAS Institute, Cary, NC. 633 p.

L, T.B., A.G. H, D.J. S. 2005. A random-

parameter height-dbh model for cherrybark oak. South. J. Appl. For.

29:22–26.

MM,T.A. 1973. Size and shape in biology. Science 179:1201–1204.

M, L., S. -M, T.G. G. 2015. Modeling

height–diameter curves for prediction. Can. J.For. Res. 45:826–837.

N, R.A. 1988. eory of growth geometry of plants and self-

thinning of plant populations: Geometric similarity, elastic similarity,

and dierent growth modes of plants. Am. Nat. 131:220–256.

O’B, S.T., S.P. H, P. S, R. C, R.B. F.

1995. Diameter, height, crown, and age relationships in eight neotrop-

ical tree species. Ecology 76:1926–l939.

O’C, B.M., B.L. C, A.M. W, E.A. B,

J.A. T, S.A. P, G. C, T. R,

J.M. 2017. e forest inventory and analysis database: Database

description and user guide version 7.0 for phase 2. USDA Forest Service.

830 p. [Online]. Available online at http://www.a.fs.fed.us/library/

databasedocumentation/.

R,M.B., A.R.W, J.A.K, J. 2014. Comparing

strategies for modeling individual-tree height and height-to-crown base

increment in mixed-species Acadian forests of northeastern North

America. Euro. J.For. Res. 133:1121–1135.

S, O., F.J. P. 2002. Contemporary statistical

models for the plant and soil sciences. CRC Press, Boca Raton, FL. 738 p.

S,R.P. 2009. Modelling height–diameter relationship for Chir pine

trees. Banko Janakari. 19:3–9.

S,M.R., B.N.O, S.S, S.C. 2018. Height–di-

ameter modeling of Cinnamomum tamala grown in natural forest in

mid-hill of Nepal. Intl. J.For. Res. 2018:1–11.

T,H., V.J.M, D.W.H. 2008. Analysis and com-

parison of nonlinear tree height prediction strategies for Douglas-r

forests. Can. J.For. Res. 38:553–565.

T,G., C.L.VS, H.E.B. 2007. Regional

mixed-eects height–diameter models for loblolly pine (Pinus taeda L.)

plantations. Euro. J.For. Res. 126:253–262.

US F S. 2017. FIA DataMart 1.6.1. Available online at

https://apps.fs.usda.gov/a/datamart/CSV/datamart_csv.html; last

accessed August 10, 2017.

U,F. 2017. Height–diameter model for managed even-aged stands of

ponderosa pine for the western United States using a hierarchical non-

linear mixed-eects model. Aust. J.Basic Appl. Sci. 11:69–87.

VS, C.L. 2008. Stand level height–diameter mixed eects

models: Parameters tted using loblolly pine but calibrated for

AQ6

AQ7

AQ8

10.5

10.10

10.15

10.20

10.25

10.30

10.35

10.40

10.45

10.50

10.55

10.59

10.60

10.65

10.70

10.75

10.80

10.85

10.90

10.95

10.100

10.105

10.110

10.115

10.118

Forest Science • XXXX 2019 11

sweetgum. P. 386–393 in Proceedings of the 16th Central Hardwood

Forest Conference, J, D.F., and C.H. M (eds.). USDA

Forest Service Gen. Tech. Rep. NRS-P-24, Northern Research Station,

Newtown Square, PA.

VS, C.L. 2013. Mixed-eects height–diameter models for

commercially and ecologically important hardwoods in Minnesota.

North. J.Appl. For. 30:37–42.

VS,C.L. 2014. Mixed-effects height–diameter models

for ten conifers in the inland Northwest, USA. South. For.

76:1–9.

VS,C.L., D. S. 2009. Biological, economic, and

management implications of weighted least squares. South. J.Appl. For.

33:53–57.

VS,C.L., Y.L, D.B.S. 2011. e weighted least

squares method aects the economic rotation age of loblolly pine—two

planting density scenarios. Open Forest Sci. J. 4:42–48.

Z,B. 1978. Standardization of growth curves. J. For. 76:289–292.

Z,B. 1993. A parsimonious number of growth curves. North. J.For.

10:132–136.

Z,B. 1994. To construct or not to construct more site index curves?

West. J.For. 9:37–40.

Z,B., C.L.VS. 2002. e eect of density on the

height–diameter relationship. P. 463–466 in Proceedings of the Eleventh

Biennial Southern Silvicultural Research Conference, K.W. O

(ed.). USDA Forest Service Gen. Tech. Rep. SRS-48, Southern

Research Station, Asheville, NC. 622 p.

11.5

11.10

11.15

11.20

11.25

11.30

11.35

11.40

11.45

11.50

11.55

11.59

11.60

11.65

11.70

11.75

11.80

11.85

11.90

11.95

11.100

11.105

11.110

11.115

11.118