Skip to main content
PLOS ONE logoLink to PLOS ONE
. 2015 Apr 15;10(4):e0122257. doi: 10.1371/journal.pone.0122257

Linear Mixed-Effects Models to Describe Individual Tree Crown Width for China-Fir in Fujian Province, Southeast China

Xu Hao 1, Sun Yujun 1,*, Wang Xinjie 1, Wang Jin 1, Fu Yao 1
Editor: Rongling Wu2
PMCID: PMC4398382  PMID: 25876178

Abstract

A multiple linear model was developed for individual tree crown width of Cunninghamia lanceolata (Lamb.) Hook in Fujian province, southeast China. Data were obtained from 55 sample plots of pure China-fir plantation stands. An Ordinary Linear Least Squares (OLS) regression was used to establish the crown width model. To adjust for correlations between observations from the same sample plots, we developed one level linear mixed-effects (LME) models based on the multiple linear model, which take into account the random effects of plots. The best random effects combinations for the LME models were determined by the Akaike’s information criterion, the Bayesian information criterion and the -2logarithm likelihood. Heteroscedasticity was reduced by three residual variance functions: the power function, the exponential function and the constant plus power function. The spatial correlation was modeled by three correlation structures: the first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)], and the compound symmetry structure (CS). Then, the LME model was compared to the multiple linear model using the absolute mean residual (AMR), the root mean square error (RMSE), and the adjusted coefficient of determination (adj-R 2). For individual tree crown width models, the one level LME model showed the best performance. An independent dataset was used to test the performance of the models and to demonstrate the advantage of calibrating LME models.

Introduction

China-fir (Cunninghamia lanceolata (Lamb.) Hook) is the most commonly grown afforestation species in southeast China because of its fast growth and good wood qualities. It is widely used for buildings, furniture, bridge construction and many other purposes. According to the National Continuous Forest Inventory, approximately 11.26 million hectares and 734.09 million cubic meters of China-fir were distributed over 10 provinces in China in 2010.

Growth and yield models are commonly used for forest management planning because they can simulate stand development and production under various management alternatives [1; 2]. As an important tree variable, the crown width (CW) of individual trees is a fundamental component of forest growth and yield prediction frameworks [3; 4], and it is also crucial for assessing the competitive level, tree vigor, microclimate, biological diversity, mechanical stability, fire susceptibility and behavior under wind stress, amongst other features [5]. The tree crown displays the leaves to capture radiant energy for photosynthesis and is strongly correlated with tree growth [6]. Therefore, measurements of the tree crown are often made to aid the understanding and quantification tree growth [7]. However, it is excessively costly and time consuming to measure the crown width of trees [8; 9]. As a result, it is necessary to establish accurate crown width models for forest managers to predict crown width precisely based on the crown data from adequate numbers of sample trees within different sample plots.

Regression analysis, such as the Ordinary Linear least Squares (OLS) regression, is the most commonly used statistical method in forest modeling [10]. Most crown width models are simple linear or nonlinear functions of diameter at breast height (DBH), estimated using linear or nonlinear regression [8; 9]. The fitting data for crown width models are usually collected by measurements of trees within different plots, also known as cross-sectional data [11; 12]. The hierarchical nature of the data results in spatial correlation among measurements made in the same sampling unit (i.e., plot) [13]. However, the hierarchical structure is often ignored and independence of observations is assumed [810; 14; 15]. Furthermore, the data are autocorrelated and cannot be considered independent samples of the basic plot population [13]. The OLS regression assumption of independent residuals is therefore violated, biasing the estimates of the standard error of the parameter estimates [16].

Linear mixed-effects (LME) models that include both fixed-effects and random-effects provide an efficient means of analyzing some kinds of cross-sectional data [17; 18]. The fixed-effects parameters are associated with an entire population or with certain repeatable levels of experimental factors, and the random-effects parameters are related to individual experimental units drawn at random from a population. These parameters account for spatial correlation by defining the covariance structure of the model’s random component and by using this structure during parameter estimation. Because of their advantages, LME models provide an efficient statistical method for explicitly modeling hierarchical stochastic structure and are increasingly applied to forest growth and yield modeling [1923]. Use of LME models allows the models to be calibrated by predicting random components from plot-level covariates when a new subject is available and is not used in the fitting of the model by using the empirical best linear unbiased predictors (EBLUPs) [22; 2426].

The main purpose of this research was to develop an individual tree crown width model for C. lanceolata in Fujian province, southeast China, on the basis of data derived from 55 sample plots. A one-level (plots effects) linear mixed modeling approach was applied to the hierarchical structure of the data. This diminished the level of variance among the sampling units. Our preliminary analysis showed that the LME model effectively removed the heteroscedasticity and spatial correlation in the data and therefore could be an important tool for the sustainable management of China-fir within the study area. The predictive ability of the developed model and the applicability of the LME model were demonstrated using separate validation data.

Materials and Methods

Data

The pure China-fir stands are located in Jiangle County (117°05′-117°40′E, 26°26′-27°04′N), Fujian Province, southeast China. The soil type is red soil, the average annual precipitation is approximately 1699 mm, the annual mean frost-free season is 287 days, and the annual mean temperature is 18.7°C.

Data from four thousand one hundred ninety-nine trees were obtained from 55 single-species plots of plantation-grown China-fir on the Jiangle state-owned forest farm in Fujian Province, southeast China (Fig 1). The Jiangle state-owned forest farm issued permission for each location, and the field studies did not involve endangered or protected species. The sample plots were square and varied in size from 400 to 600 m2. All standing live trees (height > 1.3 m) on the plots were measured for DBH (outside bark), tree height, height to crown base (height above ground to crown base) and crown width. Three to five dominant trees on each plot were chosen to calculate plot dominant height and diameter. Crown width was taken as the arithmetic mean of two crown widths, obtained from measurements of four crown radii in four directions (from the east, west, south and north to the center of tree, respectively) representing two perpendicular azimuths [8]. The crown width data were randomly divided into two groups; 75% of the points were used for model fitting, and 25% were used for model validation, which can be claimed as independent. The fitting and validation data consisted of 2587 trees from 39 plots and 1613 trees from 16 plots, respectively. Summary statistics for both the fitting and validation data are shown in Table 1. The crown data are graphically depicted in Fig 2.

Fig 1. Fifty-five sample plots of pure China-fir plantation stands.

Fig 1

Table 1. Summary statistics for increments datasets.

Variables Fitting data Validation data
Mean Min Max sd Mean Min Max sd
CW (m) 2.53 0.4 7.4 0.95 2.42 0.3 8.2 0.93
DBH (cm) 14.44 2.0 44.4 6.75 13.54 1.5 44.2 5.84
H (m) 13.36 1.2 36.5 5.56 12.11 1.3 30.8 5.35
HCB (m) 7.68 0.1 21.5 4.08 6.98 0.3 19.8 4.52
DH (m) 18.01 6.9 30.3 4.93 15.78 6.5 26.2 5.78
DD (cm) 21.43 8.0 38.6 5.13 20.43 10.1 36.0 6.88
A (yr) 22.63 7.0 49.0 9.43 18.10 5.0 40.0 8.73
SI (m at 20 years) 17.13 12.0 24.0 3.69 16.89 12.0 22.0 2.86
SD (trees ha -1 ) 2311 617 4500 1044.85 2862 467 4400 907.91
QMD (cm) 15.03 4.8 25.2 4.64 13.80 9.9 26.9 3.50
MH (m) 14.01 4.2 21.3 4.00 12.06 5.0 21.4 4.25
BA (m 2 ·ha -1 ). 29.54 3.4 68.0 16.11 42.4 16.9 99.82 20.11

Fig 2. Plots of crown width against DBH for China-fir.

Fig 2

Methods

The covariate selection

DBH is an important tree characteristic and the variable that has the greatest correlation with crown width [27]. In addition to DBH, CW is explained by other tree and stand attributes, [9; 28] such as a reduction in growth from increases in stand density, SD (tree ha-1) and basal area, BA (m2·ha-1) [22; 29]. In addition, CW is also influence by tree size variables, such as sample tree height (H) and height to crown base (HCB), and stand variables, such as stand age (A), plot dominant height, DH (m), plot dominant diameter at breast height, DD (cm), plot quadratic mean diameter, QMD (cm) [28; 30; 31], plot mean height, MH (m), and site index, SI (m at 20 yr).

Crown width multiple linear model

Independent variables were identified and a backward stepwise linear regression routine that started with all candidate variables, tested the deletion of each variable using a chosen model comparison criterion, deleted the variable (if any) whose removal improved the model the most, and repeated this process until no further improvement was possible, was applied to reduce the number of chosen variables to avoid overfitting. Variance inflation factors (VIF<10), which provide an index that measures how much the variance of an estimated regression coefficient is increased because of collinearity, were also computed to reduce the number of chosen variables to avoid multicollinearity, which could result in numerically unstable estimates of the regression coefficients. Stepwise regression fits an observed dependent dataset using a linear combination of independent variables. The statistical methods were implemented in R, which is a free software environment for statistical computing and graphics [32]. The dependent variable is determined from a linear equation combining the values of the independent dataset with coefficients established by the regression. The statistical results were assessed in terms of the absolute mean residual (AMR), root mean square error (RMSE), and the adjusted coefficient of determination (adj-R 2), which accounts for the number of predictors. The calculation formulas of these statistics are listed as follows:

AMR=i=1Mj=1ni|yijy^ij|ni 1
RMSE=i=1Mj=1ni(yijy^ij)2nir 2
adj-R2=1(nij1)[i=1Mj=1ni(yijy^ij)2niri=1Mj=1ni(yijy¯)2] 3

where M is the number of plots, n i is the number of observations in plot i, r is the number of parameters in the model, y ij is the crown width of the jth tree taken from the ith plot, ŷ ij is the crown width prediction, and y¯ is the average of observations. The accuracy of the models was tested against the fitting data and against independent validation data from the same plot [23].

LME model method

Available data were from measurements of trees located in sample plots. Because of this nested structure, there is high correlation among observations taken from the same plot. To alleviate this issue, a linear mixed-effects model approach has been proposed by other authors [10; 33]. For a single level of grouping, a general expression for a LME model can be defined as [17; 20; 34]:

CWij=Xijβ+Zijbij+εij,i=1,...,M,j=1,...,nibijN(0,D),εijN(0,Rij) 4

where CWij is the crown width of the jth tree taken from the ith plot, β is the p-dimensional vector of fixed effects (where p is the number of fixed-effects parameters in the model), b ij is the q-dimensional vector of random effects associated with plot i that is assumed to follow a normal distribution with mean zero and a variance-covariance matrix D (where q is the number of random-effects parameters in the model), X ij (of size n i×p) and Z i (of size n i×q) are known fixed-effects and random-effects regressor matrices, and ε ij is the n i-dimensional within-group error vector with a spherical Gaussian distribution [35], which is assumed to be normally distributed with zero expectation and a positive-definite variance-covariance structure R ij, generally is a n i × 1 vector for the residual items [e i1, e i2, e i3,…, e ij,…, eini]T [36]. Both the random-effects b ij and the within-group errors ε ij are assumed to be independent for different groups and to be independent of each other for the same group.

Ascertainment of mixed parameters

To fit the mixed-effects models, the key question is which parameters in the model should be considered as random effects and which ones could be treated as purely fixed effects. Generally, an alternative model-building approach is to start with a model with random effects for all parameters and then examine the fitted object to decide which, if any, of the random effects can be eliminated from the model [18]. Therefore, different combinations of model parameters were tested to ascertain their importance with respect to crown width, and the best model was selected by Akaike’s information criterion (AIC) [37], Bayesian information criterion (BIC) [38] and -2 logarithm likelihood (-2 LL) [31]. The less criteria a model has, the better it performs. An appropriate variance function structure for LME models were determined by a likelihood ratio test (LRT) [18; 39]. All LME models presented in this paper were fitted using the LME function in the R statistical software environment.

Determining the structure of R

This special matrix structure R (which is allowed to depend on both random and fixed effects, as well as on a set of common but unknown parameters) can include both correlation effects and weighting factors to account for within-group heteroscedasticity and spatial correlation [35; 36; 40]. A general expression for the matrix is given by [40; 41]:

R=σ2G0.5IG0.5 5

where (in this case) for tree j in plot i, with n i increment, R is the n i×n i intraindividual variance- covariance matrix which defines within-group variability, G is a n i ×n i diagonal matrix of the within-group error variance structure (heteroscedasticity), I is a n i×n i matrix showing the within-group autocorrelation structure of error, and σ2 is a scaling factor for the error dispersion [10]. To remove variance heterogeneity, we used the power function, exponential function and constant plus power function as the variance functions to fit crown width models [18].

V(εij)=σ2DBHij2δ 6
V(εij)=σ2exp(2δDBHij) 7
V(εij)=σ2(δ1+DBHijδ2)2 8

Correlation structures were used to address the within-tree spatial correlations observed in the data [42; 43]. A method was selected from among three commonly used approaches: the first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)], and the compound symmetry structure (CS) [18].

AR(1)=σ2[1ρρ2ρ1ρρ2ρ1] 9
ARMA(1,1)=σ2[1γγργ1γγργ1] 10
CS=[σ2+σ1σ1σ1σ1σ2+σ1σ1σ1σ1σ2+σ1] 11

where ρ is the autoregressive parameter, γ is a moving average component, and σ1 is the residual covariance [44; 45].

Parameter estimation

The parameters in the equations were estimated by maximum likelihood (ML) using the Lindstrom and Bates (LB) algorithm implemented in the R LME function [17; 18]. The LB algorithm and LME function are detailed in several articles; see, for example, [17; 18].

A key question in fitting the LME models is to estimate the random effects parameters. In this study, they can be calculated with the information from measured trees, such as the measurements of CW and DBH, by the Empirical Best Linear Unbiased Predictors (EBLUPs) [34].

b^ijkD^Z^T(R^+Z^D^Z^T)-1ε^ijk 12

where D^ is the estimated variance-covariance matrix for the random-effects b^ijk, R^ is the estimated variance-covariance matrix for the error term, and Z^ is the estimated partial derivatives matrix with respect to random effects parameters.

Results

Selection of the basic crown width model

The following formula is the composition of individual tree size variables and stand variables for predicting crown width using OLS:

CWij=β0+β1DBHij+β2Hij+β3HCBij+β4DHi+β5DDi+β6Ai+β7SIi+β8SDi+β9QMDi+β10MHi+β11BAi+εij 13

where β 0 - β 11 are the formal parameters.

To avoid overfitting and multicollinearity between independent variables, the backward stepwise linear regression routine and the variance inflation factor were used to reduce the number of chosen variables. In addition, we took into account the biologically reasonable and the factors that exhibited significance (Pr value<0.05) between independent variables. The variable selection process involves a series of steps beginning with the stepwise regression method together with VIF control to identify those variables that may be useful in the model. DH, A and MH were removed from Eq 13 because their VIF>10 (VIFDH = 27.48, VIFA = 13.61, VIFMH = 17.72). As a result, the final diameter growth model for fir plantations can be expressed as:

CWij=β0+β1DBHij+β2Hij+β3HCBij+β4DDi+β5SIi+β6SDi+β7QMDi+β8BAi+εij 14

The statistics used for the selection of the basic model are shown with equations in Table 2.

Table 2. Comparison of fitting statistics and estimated variance components of the models with different alternatives of covariates inclusion, residual variance function and variance components estimation method.

Model Intercept DBH H HCB DD SI SD QMD BA
Eq 14 1.3450 0.1235 -0.0212 -0.0275 0.0236 -0.0077 2.31×10–5 -0.0127 -0.0105
(standard error) (0.0904***) (0.0032***) (0.0050***) (0.0045***) (0.0035***) (0.0037*) (4.4×10–5*) (0.0054**) (0.0009***)
Eq 14.4 1.1812 0.1103 0.0073 -0.0238 0.0152 -0.0115 9.07×10–5 -0.0182 -0.0060
(standard error) (0.5102) (0.0066**) (0.0080*) (0.0079) (0.0193) (0.0197) (7.68×10–5**) (0.0265) (0.0048)
Eq 14.4.8 0.6693 0.1090 0.0085 -0.0217 0.0231 -0.0134 0.0002 -6.93×10–3 -0.0075
(standard error) (0.4490*) (0.0061***) (0.0060*) (0.0062**) (0.0154*) (0.0163*) (0.0001**) (0.0217*) (0.0038**)

“*”means Pr value < 0.05

“**” means Pr value < 0.01

“***” means Pr values < 0.001.

Construction of LME models

There would be ninety different combinations of no more than four random-effects parameters for Eq 14 while simultaneously considering plots effects. The LME models with more than four random-effects parameters could not reach convergence.

LRT, AIC, BIC and -2 LL statistics were compared between the LME models with the best different combinations of random-effects parameters and are shown in Table 3. The model of Eq 14, incorporating plots effects on β 0, β 1, β 2 and β 3 (Eq 14.4), yielded the smallest AIC, BIC and -2 LL and had significant differences when compared to the other LME models (Eqs 14.114.3) (Pr<0.0001).

CWij=(β0+u0i)+β1DBHij+β2Hij+β3HCBij+β4DDi+β5SIi+β6SDi+β7QMDi+β8BAi+εij 14.1
CWij=(β0+u0i)+(β1+u1i)DBHij+β2Hij+β3HCBij+β4DDi+β5SIi+β6SDi+β7QMDi+β8BAi+εij 14.2
CWij=(β0+u0i)+(β1+u1i)DBHij+(β2+u2i)Hij+β3HCBij+β4DDi+β5SIi+β6SDi+β7QMDi+β8BAi+εij 14.3
CWij=(β0+u0i)+(β1+u1i)DBHij+(β2+u2i)Hij+(β3+u3i)HCBij+β4DDi+β5SIi+β6SDi+β7QMDi+β8BAi+εij 14.4

where β 0 - β 8 are the fixed effects parameters and u 0i, u 1i, u 2i and u 3i are the random-effects parameters generated by plots effects on β 0, β 1, β 2 and β 3, respectively.

Table 3. Performance criteria of LME models for combinations of random effects.

Equation Mixed parameters Number of parameters AIC BIC -2 LL LRT Pr values
14.1 β 0 10 7910.90 7980.64 7888.90
14.2 β 0, β 1 11 7820.25 7902.68 7794.25 94.64 <0.0001
14.3 β 0, β 1, β 2 12 7751.38 7852.83 7719.38 76.26 <0.0001
14.4 β 0, β 1, β 2 β 3 13 7723.20 7850.01 7683.20 34.80 <0.0001

LME model with heteroscedasticity and spatial correlation

We used the power function, the exponential function or the constant plus power function as the variance functions and AR, ARMA(1,1) or CS as the correlation structure to update Eq 14.4 to reduce heteroscedasticity and spatial correlation. The LME models with variance functions and correlation structures are shown in Table 4. In this study, Equation 14.4.1 is the same as Eq 14. The best models were chosen with the smallest AIC, BIC and -2 LL. Thus, the final models of plots effects are:

Equation14.4+Equation6 14.4.2
Equation14.4+Equation7 14.4.3
Equation14.4+Equation8 14.4.4
Equation14.4+Equation9 14.4.5
Equation14.4+Equation10 14.4.6
Equation14.4+Equation11 14.4.7
Equation14.4+Equation6+Equation10 14.4.8

Table 4. Comparisons of intercept effect mixed model performance for fir plantations diameter increment data with different within-tree correlation structures and different variance functions.

Equation Variance function Correlation structure Number of parameters AIC BIC -2LL LRT Pr values
14.4.1 Homogeneous Independent 13 7723.20 7850.01 7683.20
14.4.2 Power Independent 14 7458.74 7591.89 7416.74 266.46 a <0.001
14.4.3 Exponent Independent 14 7422.96 7556.11 7380.96 302.24 a <0.001
436.97 b <0.001
14.4.4 Const plus power Independent 15 7444.24 7573.73 7400.24 322.95 a <0.001
14.4.5 Homogeneous AR(1) 14 7473.63 7606.78 7431.63 251.56 a <0.001
368.08 b <0.001
14.4.6 Homogeneous ARMA(1,1) 14 7356.06 7495.55 7312.06 371.13 a <0.001
14.4.7 Homogeneous CS 14 7725.20 7858.35 7683.20 3.31×10–6 a 0.9985
14.4.8 Exponent ARMA(1,1) 16 6989.99 7135.82 6943.99 739.21 a <0.001

a Likelihood ratio is calculated with respect to Equation 14.4.1

b Likelihood ratio is calculated with respect to Eq 14.4.8

Parameter estimates

The LME CW model with plots effects is then defined by the following expression:

CWij=(0.8247+u0i)+(0.1093+u1i)DBHij+(0.0095+u2i)Hij+(0.0219+u3i)HCBij+0.0181DDi0.0105SIi+0.0004SDi0.0121QMDi0.0064BAi+εij 15

Where

ui=[u0iu1iu2iu3i]N{[0000],D=(0.53230.09160.53420.75420.09160.03600.57180.33280.53420.57180.02620.39870.75420.33280.39870.0323)}εijN(0,Rij=0.3592Gi0.5IiGi0.5)V(εij)=exp(0.4366DBHij)ARMA(1,1)=[10.96100.77310.961010.96100.77310.96101]

Model prediction

The predictive ability of Eq 14 was evaluated using prediction procedures and Eq 13 on both fitting and validation data. The performance of the LME models, with and without modeling the error structure, was evaluated using cross-validation procedures for both fitting and validation data; the random effects were predicted with the EBLUPs (Eq 12), using the measurement data.

Table 5 lists the three prediction statistics of Eq 14, Eq 15 and Eq 15 without random effects for both fitting and validation data. Compared with Eq 14, Eq 15 had a higher adj-R 2, 0.7226 compared to 0.4733, and lower RMSE, 0.4854 compared to 0.6688, and AMR, 0.3688 compared 0.4954, for the validation data. In Fig 3, the residuals of Eq 14 and 15 are plotted against the fitted values. The fitted values of these equations are plotted against the observed values in Fig 4. Based on the above analysis, we can conclude that Eq 15, incorporating the random effects plots, was better than Eq 14. The LME model provides a model for predicting the expected values of crown width for individual trees of China-fir in the single-species plantations of the study area.

Table 5. Evaluation indices of each model.

Model Effects Fitting data Validation data
AMR RMSE adj-R 2 AMR RMSE adj-R 2
Eq 14 0.5306 0.6914 0.4694 0.4954 0.6688 0.4733
Eq 15 Mixed effects 0.4027 0.5070 0.7147 0.3688 0.4854 0.7226

Fig 3. Distribution of residuals for two equations fitting crown width of China-fir trees.

Fig 3

Fig 4. Fitted values of two equations for crown width of China-fir trees against observed values.

Fig 4

Discussion

In this study, a backward stepwise linear regression was used to establish a multiple linear individual tree crown width model for China-fir. The relative importance of explanatory variables used to predict the crown width were assessed. Generally, DBH is the tree size variable most related to crown width [27]. In addition to DBH, the tree size variables (such as H and HCB in this study) and stand variables (such as DH in this study) are also obvious factors affecting the crown width [20; 22; 46; 47]. Both stand and tree development are linked to the DH because it is a measureable stand characteristic that indicates site quality in terms of the stand growth and yield capacity [20]. The variables H and HCB showed a significant effect on the crown width because they are closely related to tree size and have an important role in crown fire initiation and spread [47; 48]. Therefore, we selected the diameter at breast height, tree height, height to crown base, plot dominant height, plot dominant diameter at breast height, stand age, site index, stand density, plot quadratic mean diameter, plot mean height and basal area as the independent variables to establish an individual tree crown width model. However, variance inflation factors were used to avoid potential overfitting and multicollinearity.

Conclusions

Eleven variables were selected in this study to describe crown width (Eq 13) of China-fir in pure plantation stands in Fujian province, southeast China. Then, the backward stepwise linear regression routine and the variance inflation factor were used to reduce the number of chosen variables (Eq 14). The one-level (plot) LME model using the variance function structure and correlation structure approach were used to estimate the relationship of the chosen variables with crown width for individual trees. The results showed that the one-level LME models with mixed effects, considering variance function structure and correlation structure (Eq 15), provided better model fitting and more precise estimations than the LME models without mixed effects (Eq 14) (Table 5 and Figs 3 and 4). Therefore, we recommend using a linear mixed effects modeling approach to build an individual tree crown width model.

Data Availability

All relevant data are within the paper.

Funding Statement

This work was supported by the Special Public Interest Research and Industry Fund of Forestry (No.200904003-1) and the project of forestry science and technology research (No.2012-07). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

References

  • 1. Leites LP, Robinson AP, Crookston NL (2009) Accuracy and equivalence testing of crown ratio models and assessment of their impact on diameter growth and basal area increment predictions of two variants of the Forest Vegetation Simulator. Can J Forest Res 39:3. [Google Scholar]
  • 2. Canavan SJ, Ramm CW (2000) Accuracy and Precision of 10 Year Predictions for Forest Vegetation Simulator—Lake States. Northern Journal of Applied Forestry 17:2. [Google Scholar]
  • 3. Tahvanainen T, Forss E (2008) Individual tree models for the crown biomass distribution of Scots pine, Norway spruce and birch in Finland. Forest Ecol Manag 255:3. [Google Scholar]
  • 4. Peper PJ, McPherson EG, Mori SM (2001) Equations for predicting diameter, height, crown width, and leaf area of San Joaquin Valley street trees. Journal of Arboriculture 27:6. [Google Scholar]
  • 5. Crecente-Campo F, Alvarez Gonzalez JG, Castedo-Dorado F, Gomez-Garcia E, Dieguez-Aranda U (2013) Development of crown profile models for Pinus pinaster Ait. and Pinus sylvestris L. in northwestern Spain. Forestry 86:4. [Google Scholar]
  • 6. Dutilleul P, Herman M, Avella-Shaw T (1998) Growth rate effects on correlations among ring width, wood density, and mean tracheid length in Norway spruce (Picea abies). Can J Forest Res 28:1. [Google Scholar]
  • 7. Kjelgren RK, Clark JR (1992) Photosynthesis and leaf morphology of Liquidambar styraciflua L. under variable urban radiant-energy conditions. Int J Biometeorol 36:3. [Google Scholar]
  • 8. Bragg DC (2001) A local basal area adjustment for crown width prediction. Northern Journal of Applied Forestry 18:1. [Google Scholar]
  • 9. Sönmez T (2009) Diameter at breast height-crown diameter prediction models for Picea orientalis . African Journal of Agricultural Research 4:3. [Google Scholar]
  • 10. Grégoire TG, Schabenberger O, Barrett JP (1995) Linear modelling of irregularly spaced, unbalanced, longitudinal data from permanent-plot measurements. Can J Forest Res 1:25. [Google Scholar]
  • 11. Peugh JL, Enders CK (2005) Using the SPSS mixed procedure to fit cross-sectional and longitudinal multilevel models. Educ Psychol Meas 65:5. [Google Scholar]
  • 12. Waring RH, Schroeder PE, Oren R (1982) Application of the pipe model theory to predict canopy leaf area. Can J Forest Res 12:3. [Google Scholar]
  • 13. Fox JC, Ades PK, Bi H (2001) Stochastic structure and individual-tree growth models. Forest Ecol Manag 154:1. [Google Scholar]
  • 14. Biging GS (1985) Improved estimates of site index curves using a varying-parameter mode. Forest Sci 31:1. [Google Scholar]
  • 15. Keselman HJ, Algina J, Kowalchuk RK, Wolfinger RD (1999) A comparison of recent approaches to the analysis of repeated measurements. British Journal of Mathematical and Statistical Psychology:52 10613111 [Google Scholar]
  • 16. Schabenberger O, Gregoire TG (1995) A conspectus on estimating function theory and its application to recurrent modelling issues in forest biometry. Silva Fenn 29:1. [Google Scholar]
  • 17. Lindstrom MJ, Bates DM (1990) Nonlinear mixed effects models for repeated measures data. Biometrics 46. [PubMed] [Google Scholar]
  • 18.Pinheiro JC, Bates DM (2000) Mixed Effects Models in S and S-Plus. New York: 291–342 p.
  • 19. Matos LA, Lachos VH, Balakrishnan N, Labra FV (2012) Influence diagnostics in linear and nonlinear mixed-effects models with censored data. Comput Stat Data An 57:1. [Google Scholar]
  • 20. Fu L, Sun H, Sharma RP, Lei Y, Zhang H, Tang S (2013) Nonlinear mixed-effects crown width models for individual trees of Chinese fir (Cunninghamia lanceolata) in south-central China. Forest Ecol Manag 302. [Google Scholar]
  • 21. Timilsina N, Staudhammer CL (2013) Individual Tree-Based Diameter Growth Model of Slash Pine in Florida Using Nonlinear Mixed Modeling. Forest Sci 59:1. [Google Scholar]
  • 22. Lhotka JM, Loewenstein EF (2011) An individual-tree diameter growth model for managed uneven-aged oak-shortleaf pine stands in the Ozark Highlands of Missouri, USA. Forest Ecol Manag 261:3. [Google Scholar]
  • 23. Adame P, Hynynen J, Canellas I, Del Rio M (2008) Individual-tree diameter growth model for rebollo oak (Quercus pyrenaica Willd.) coppices. Forest Ecol Manag 255:3–4. [Google Scholar]
  • 24. Nigh G (2012) Calculating empirical best linear unbiased predictors (EBLUPs) for nonlinear mixed effects models in Excel/Solver. Forest Chron 88:3 doi: doi:10.2223/JPED.2181 [DOI] [PubMed] [Google Scholar]
  • 25. Adame P, Del Rio M, Canellas I (2008) A mixed nonlinear height-diameter model for pyrenean oak (Quercus pyrenaica Willd.). Forest Ecol Manag 256:1–2. [Google Scholar]
  • 26. Calama R, Montero G (2005) Multilevel linear mixed model for tree diameter increment in stone pine (Pinus pinea): a calibrating approach. Silva Fenn 39:1. [Google Scholar]
  • 27.Warbington R, Levitan J (1993) How to estimate canopy over using maximum crown width/DBH relationships, Stand Inventory Technologies '92, Portland, pp. 319–328.
  • 28. Uzoh FCC, Oliver WW (2008) Individual tree diameter increment model for managed even-aged stands of ponderosa pine throughout the western United States using a multilevel linear mixed effects model. Forest Ecol Manag 256:3. [Google Scholar]
  • 29. Wykoff WR (1990) A basal area increment model for individual conifers in the northern Rocky Mountains. Forest Sci 36:4. [Google Scholar]
  • 30. Adame P, Ca N Ellas I, Roig S, Del R I O M (2006) Modelling dominant height growth and site index curves for rebollo oak (Quercus pyrenaica Willd.). Ann Forest Sci 63:8. [Google Scholar]
  • 31. Zhao L, Li C, Tang S (2012) Individual-tree diameter growth model for fir plantations based on multi-level linear mixed effects models across southeast China. Journal of Forest Research 18:4. [Google Scholar]
  • 32. Ihaka R, Gentleman R (2004) R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. [Google Scholar]
  • 33. Palmer MJ, Phillips BF, Smith GT (1991) Application of nonlinear models with random coefficients to growth data. Biometrics 47. [Google Scholar]
  • 34.Vonesh E, Chinchilli VM (1997) Linear and nonlinear models for the analysis of repeated measurements. New York: 61–84 p.
  • 35.Davidian M, Giltinan DM (1995) Nonlinear models for repeated measurement data. New York: 58–74 p.
  • 36. Meng SX, Huang S (2009) Improved calibration of nonlinear mixed-effects models demonstrated on a height growth function. Forest Sci 55:3. [Google Scholar]
  • 37. Akaike H (1974) A new look at the statistical model identification. Automatic Control, IEEE Transactions on 19:6. [Google Scholar]
  • 38.Weiss RE (2005) Modeling longitudinal data. New York: 19–21 p.
  • 39. Fang Z, Bailey RL (2001) Nonlinear mixed effects modeling for slash pine dominant height growth following intensive silvicultural treatments. Forest Sci 47:3. [Google Scholar]
  • 40. Calama R, Montero G (2004) Interregional nonlinear height-diameter model with random coefficients for stone pine in Spain. Can J Forest Res 34:1. [Google Scholar]
  • 41. Crecente-Campo F, Tome M, Soares P, Dieguez-Aranda U (2010) A generalized nonlinear mixed-effects height-diameter model for Eucalyptus globulus L. in northwestern Spain. Forest Ecol Manag 259:5. [Google Scholar]
  • 42. Lappi J, Malinen J (1994) Random parameter height-age models when stand parameters and stand age are correlated. Forest Sci 40:4. [Google Scholar]
  • 43. Omule SAY, MacDonald RN (1991) Simultaneous curve fitting for repeated height-diameter measurements. Can J Forest Res 21:9. [Google Scholar]
  • 44. Leak W (1996) Analysis of multiple systematic remeasurement. Forest Sci 1:12. [Google Scholar]
  • 45. Gregorie TG (1987) Generalized error structure for forestry yield models. Forest Sci 33:2. [Google Scholar]
  • 46. Gonzalez-Benecke CA, Gezan SA, Samuelson LJ, Cropper WP Jr, Leduc DJ, et al. (2014) Estimating Pinus palustris tree diameter and stem volume from tree height, crown area and stand-level parameters. Journal of Forestry Research 25:1. [Google Scholar]
  • 47.Gómez-Vázquez I, Fernandes PM, Arias-Rodil M, Barrio-Anta M, Castedo-Dorado F (2013) Using density management diagrams to assess crown fire potential in Pinus pinaster Ait. stands. Ann Forest Sci.
  • 48. Alexander ME, Cruz MG (2011) Crown fire dynamics in conifer forests. Synthesis of knowledge of extreme fire behavior 1. [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Data Availability Statement

All relevant data are within the paper.


Articles from PLoS ONE are provided here courtesy of PLOS

RESOURCES