This research was funded by the Fundamental Research Funds for the Central Universities, grant number 2019GJZL04. We thank Professor Weisheng Zeng at the Academy of Forest Inventory and Planning, National Forestry and Grassland Administration, China for providing access to data.

1. Data preparation
<ol>
	<li>Prepare modeling data, which includes individual-tree information (species and diameter at breast height at 1.3 m) and plot information (slope, aspect, and elevation). In this study, the data were obtained from the 8th (2009) and 9th (2014) Chinese National Forest Inventory in Xinjiang Province, northwest China, which includes 21,898 observations of 779 sample plots. These sample plots are square-shaped with a size of 1 Mu (Chinese unit of area equivalent to 0.067 ha) and are systematicall...

<ol>
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The authors have nothing to disclose.

A crucial issue for the development of mixed-effects models is to determine which parameters can be treated as random effects and which should be considered fixed effects34,35. Two methods have been proposed. The most common approach is to treat all parameters as random effects and then have the best model selected by AIC, BIC, Loglik, and LRT. This was the method employed by our study35. An alternative is to fit basal area increment model...

Compared with even-aged monoculture, uneven-aged mixed-species forest management with multiple objectives has received increased attention recently1,2,3. Prediction of different management alternatives is necessary for formulating robust forest management strategies, especially for complex uneven-aged mixed-species forest4. Forest growth and yield models have been used extensively to forecast tree or stand development and harvest under various management schemes5,6,7. Forest growth and yield models are classified into individual-tree models, size-class models, and whole-stand growth models6,7,8. Unfortunately, size-class models and whole-stand models are not appropriate for uneven-aged mixed-species forests, which require a more detailed description to support the forest management decision-making process. For this reason, individual-tree growth and yield models have received increased attention throughout the last few decades because of their ability to make predictions for forest stands with a variety of species compositions, structures, and management strategies9,10,11.
Ordinary least squares (OLS) regression is the most commonly used method for the development of individual-tree growth models12,13,14,15. The datasets for individual-tree growth models collected repeatedly over a fixed length of time on the same sampling unit (i.e., sample plot or tree) have a hierarchical stochastic structure, with a lack of independence and high spatial and temporal correlation among observations10,16. The hierarchical stochastic structure violates the fundamental assumptions of OLS regression: namely independent residuals and normally distributed data with equal variances. Therefore, the use of OLS regression inevitably produces biased estimates of the standard error of parameter estimates for these data13,14.
Mixed-effects models provide a powerful tool for analyzing data with complex structures, such as repeated measures data, longitudinal data, and multi-level data. Mixed-effects models consist of both fixed components, common to the complete population, and random components, which is specific to each sampling level. In addition, mixed-effects models take into account heteroscedasticity and autocorrelation in space and time by defining non-diagonal variance-covariance structure matrices17,18,19. For this reason, mixed-effects models have been extensively used in forestry, such as in diameter-height models20,21, crown models22,23, self-thinning models24,25, and growth models26,27.
Here, the main objective was to develop an individual-tree basal area increment model using a linear mixed-effects approach. We hope that the mixed-effects approach could be broadly applied.

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development of an individual-tree basal area increment model using a linear mixed-effects approach

Here, we developed an individual-tree model of 5-year basal area increments based on a dataset including 21898 Picea asperata trees from 779 sample plots located in Xinjiang Province, northwest China. To prevent high correlations among observations from the same sampling unit, we developed the model using a linear mixed-effects approach with random plot effect to account for stochastic variability. Various tree- and stand-level variables, such as indices for tree size, competition, and site condition, were included as fixed effects to explain the residual variability. In addition, heteroscedasticity and autocorrelation were described by introducing variance functions and autocorrelation structures. The optimal linear mixed-effects model was determined by several fit statistics: Akaike&#8217;s information criterion, Bayesian information criterion, logarithm likelihood, and a likelihood ratio test. The results indicated that significant variables of individual-tree basal area increment were the inverse transformation of diameter at breast height, the basal area of trees larger than the subject tree, the number of trees per hectare, and elevation. Furthermore, errors in variance structure were most successfully modeled by the exponential function, and the autocorrelation was significantly corrected by first-order autoregressive structure (AR(1)). The performance of the linear mixed-effects model was significantly improved relative to the model using ordinary least squares regression.

The basic basal area increment model for P. asperata was expressed as Equation (7). The parameter estimates, their corresponding standard errors, and the lack-of-fit statistics are shown in Table 2. The residual plot is shown in Figure 1. Pronounced heteroscedasticity of the residuals was observed. 
<img alt="Equation 7" src="/files/ftp_upload/60827/60827eq7.jpg" />&#160; &#160;(7)
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Watch this Scientific Journal Video about Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach at JoVE.com

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Mixed-effects models are flexible and useful tools for analyzing data with a hierarchical stochastic structure in forestry and could also be used to significantly improve the performance of forest growth models. Here, a protocol is presented that synthesizes information relating to linear mixed-effects models.

Here, we developed an individual-tree model of 5-year basal area increments based on a dataset including 21898 Picea asperata trees from 779 sample plots ...

This protocol provides the key procedures of developing an individual-tree basal area increment model using a linear mixed-effects approach. The main feature of this technique is that it can powerfully analyze data with complex structures in forestry and significantly improve the performance of forest growth models. Begin by reading the model development data set and loading the package nlme"in R software.Select sample plots as random effects to develop the mixed-effects model. Fit all possible combinations of random effects with the maximum likelihood method and output the results. Set the intercept to random parameters, then change the random statements until all combinations are fitted.In the process of fitting, the codes may report errors due to the nonconvergence of the fitted model. Select the best model by Akaike's information criterion, the Bayesian information criterion, the logarithm likelihood, and the likelihood ratio test. Observe whether the residuals have heteroscedasticity from the residual plot.If there is heteroscedasticity, introduce the constant plus power function, the power function, and the exponential function to model the errors variance structure. Determine the best variance function for the model according to Akaike's information criterion, the Bayesian information criterion, the logarithm likelihood, and the likelihood ratio test. Next, introduce the compound symmetry structure, first-order autoregressive structure, and a combination of first-order autoregressive and moving average structures to account for autocorrelation.Determine the best autocorrelation structure according to Akaike's information criterion, the Bayesian information criterion, the logarithm likelihood, and the likelihood ratio test. Output the final results of the mixed-effects model using the restricted maximum likelihood method. The basic basal area increment model for P.asperata is expressed with this equation.The parameter estimates, their corresponding standard errors, and the lack-of-fit statistics are shown here. Pronounced heteroscedasticity of the residuals was observed. There were 31 possible combinations of random effects parameters for the basic basal area increment model.After fitting, 300 combinations reached convergence. Among these 30 combinations, Model 30 was selected because it yielded the lowest AIC, the lowest BIC, and the largest Loglik. Furthermore, the LRT was significantly different when compared with the other models.The linear mixed-effects model with variance functions and correlation structures are shown here. According to the AIC, BIC, Loglik, and LRT, the exponential function and AR(1)were selected as the best variance function and autocorrelation structure, respectively. The final linear mixed-effects individual-tree basal area increment model was proposed using the REML method.The estimated fixed parameters, their corresponding standard errors, and the lack-of-fit statistics are shown here. A significant improvement was observed in the residuals. Prediction statistics of the two models show that the performance of the linear mixed-effects model was significantly improved compared to the basic model.When the model comparisons are completed, remember to use the restricted maximum likelihood method to output the final results.

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3.2K Görüntüleme.  Beijing Forestry University. This protocol provides the key procedures of developing an individual-tree basal area increment model using a linear mixed-effects approach. The main feature of this technique is that it can powerfully analyze data with complex structures in forestry and significantly improve the performance of forest growth models. Begin by reading the model development data set and loading the package nlme"in R software.Select sample plots as random effects to develop the mixed-effects model. Fit all...