Predictive Ability of Mixed-Effects Height–Diameter Models Fit Using One Species but Calibrated for Another Species

Abstract

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.

Full text

Journal : FORSCI
Article Doi : 10.1093/forsci/fxz058
Article Title : Predictive Ability of Mixed-Eects Height–Diameter Models Fit Using One Species but Calibrated for Another Species
First Author : CurtisL. VanderSchaaf
Corr. Author : CurtisL. VanderSchaaf
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Journal : FORSCI
Article Doi : 10.1093/forsci/fxz058
Article Title : Predictive Ability of Mixed-Eects Height–Diameter Models Fit Using One Species but Calibrated for Another
Species
First Author : CurtisL. VanderSchaaf
Corr. Author : CurtisL. VanderSchaaf
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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
CurtisL. 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-eects 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-eects models often is the ability to calibrate the model using
data independent of those used in model tting. Plot/stand-specic
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 specic 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 etal. 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-eects model better captures
the individual behavior of the H–D curve within a particular stand
(or plot). Mixed-eects 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-eects models were developed.
Mixed-eects H–D models have been developed for loblolly
pine plantations across the southeastern United States (Trincado
etal. 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-eects H–D
model developed for longleaf pine and shortleaf pine plantations in
the Western Gulf. Mixed-eects H–D models have been developed
for naturally regenerated shortleaf pine stands for western Arkansas
and eastern Oklahoma (Budhathoki etal. 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
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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-eects 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-eects 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-eects H–D model when calibrated using sweetgum
(Liquidambar styraciua 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. Asimilar 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-specic mixed-
eects 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 dicult operationally to apply
to individual tree H–D relations because the two points would need
to be quantied for each individualtree.
Russell et al. (2014) showed that including species-specic
random eects when modeling the change in height in north-
eastern United States and eastern Canadian mixed-species stands
showed little dierence from species-specic 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 etal. (2013) conducted a sim-
ilar analysis to that carried out by Russell etal. (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-specic random eects 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 etal. 2013,
Russell etal. 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 dierent 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-eects models
of dierent species to predict for a species. e objective of this re-
search was to determine whether a linear mixed-eects 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
etal. 2017, US Forest Service 2017). Aplot 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 inuence 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
dierent 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-eects 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.
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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-specic random eects, assumed to be
N(0, σ
2
0) and N(0, σ
2
1), respectively; (β
0+u0k)=plot/stand-specic
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.
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4 Forest Science • XXXX 2019
intercept; (β
1+u1k)=plot/stand-specic 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-specic random
eects have a bivariate normal distribution with mean 0.In this
particular case, linear mixed-eects models produce an ecient es-
timate of plot/stand-specic 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-specic random eects (u0k, u1k) can be predicted and then
added to the “population average” intercept and slope (β
0, β
1)
estimates to obtain plot/stand-specic parameter estimates.
Plot/stand-specic random parameters produce a more localized
H–D equation, since the random eects account for local site
conditions such as soil type, genetic stock, site preparation,
midrotation silvicultural practices, spatial and time-specic cli-
matic conditions, etc. e prediction of plot/stand-specic random
eects is conducted outside the model-tting algorithm, and thus
degrees of freedom are not lost due to specic plot/stand random
eects. In terms of model tting, a less ecient method of obtaining
plot-specic parameter estimates would be to estimate parameters
separately for eachplot.
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-
eects model analysis is ecient because a model can be calibrated
without having to statistically t a model, and thus even small
sample sizes can be used (Lynch etal. 2005). In many cases, xed-
eects region-wide H–D equations are used to predict tree heights.
To account for growing-condition dierences 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-eects 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 etal. 2008,
Crecente-Campo etal. 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 etal. 2009, Sharma 2009,
Uzoh 2017, Subedi etal. 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 etal. 1995), and the slope is associated with the constant-
stress theory value, essentially that any diameter along the stem must
be of sucient 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
eects into the model. In addition, to determine whether this par-
ticular H–D mixed-eects model form (β
0+β
1lnDki+ε
ki) could be
calibrated across species, previously published mixed-eects 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 dierent 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-levelmeans.
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 eect of species. Aquestion 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 etal. 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 etal. 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
WesternGulf.
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 samplesize.
Validation analyses follow those presented in Trincado et al.
(2007). e dierence 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
transformationbias.
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 etal. 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 eects 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 eects, 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
lnD (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 lnH (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 eects, 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 eects for this plot (or stand) are added
to the “population average” parameter estimates,
ˆ
β
, to obtain plot/
stand-specic 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 beused:
Hki = exp
[β0k+β1kln Dki+σ
2
/2
]
(5)
where β
0k – (β
0+u0k) is the plot-/stand-specic intercept, and
β
1k – (β
1+u1k) is the plot-/stand-specicslope.
For the example above, Equation 5 would be expressedas:
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
dierence.
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 reects 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 allplots.
Calibration Using Other Species
Figures 4–6 clearly demonstrate that mixed-eects 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-specic 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-eect 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 otherdata.
For this H–D model form, a calibrated model of another species
produces better predictions than an uncalibrated species-specic
model (Figures 4–7). Apparently, based on these model results,
the model calibration process exceeds the importance of species-
specic H–D relations, since, regardless in many cases of the partic-
ular species-specic 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-specic parameter estimates can apply to that specic spe-
cies. However, these results show that following calibration models
of many dierent 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-specic.
Conclusions
Mixed-eects 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 specic plots/stands, an Excel spreadsheet is avail-
able upon request.
Mixed-eects 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 dierent as pines, rs, and in some cases even
hardwoods. Future work should concentrate on examining the pre-
dictive ability of other H–D model equationforms.
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-eects 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 denitively answer this question; how-
ever it is suggested, for this particular topic of determining if the
ability of calibrating mixed-eects models across a variety of species
improves model-tting ability, is a very important research topic
for this particular subject of study. Predicted plot-specic or stand-
specic random parameters depend on the amount of estimated
variability in the random eects and the “population average” pa-
rameter estimates. us, dierences in site requirements, structural
constraints, growth habits, etc., among species may not allow for
mixed-eects 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-eects 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.
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