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

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.