Nonlinear mixed-effects height-diameter model of Pinus koraiensis

The Chapman-Richards function was used to construct individual height-diameter model for Pinus koraiensis. The data were collected from mixed deciduous stands of Mongolian oak-deciduous (Quercus mongolica) stands and mixed stands of spruce-fir and deciduous trees in Wangqing Forestry Bureau, Jilin Province of northeastern China. A total of 2 598 trees in 24 permanent plots were used for model development. Simple and generalized height-diameter model with and without random effect parameters were tested. Model evaluation criteria included adjusted determination coefficient (R2a), relative mean absolute error (RMA) and root mean square error (RMSE). For mixed-effects models, different sample sizes of four sampling designs, i.e., random sampling, the largest diameter tree sampling, the smallest diameter tree sampling and intermediate diameter tree sampling for random parameter estimation were compared and the relationship between sample size and predicted accuracy was analyzed. The results showed that the goodness-of-fit of individual height-diameter models based on mixed-effects model (R2a of simple models ranged between 0.753-0.886, RMA between 11.3%-15.1%, RMSE between 1.38-2.01m; R2a of generalized models ranged between 0.754-0.886, RMA between 11.1%-15.0%, RMSE between 1.38-2.01m) was better than that of individual height-diameter model without random effect parameters (R2a of simple models ranged between 0.502-0.868, RMA between 12.2%-17.8%, RMSE between 1.42-2.65m; R2a of generalized models ranged from 0.711-0.877, RMA between 11.6%-17.2%, RMSE between 1.41-2.10m). Furthermore, the difference of goodness-of-fit between simple and generalized height-diameter models with mixed-effects was not significant, which may lead us to a conclusion that individual height-diameter simple model with mixed-effects can describe the variation of height-diameter relationship in different forest types and plots, thus additional variables were not be necessary in height-diameter model. In addition, intermediate diameter tree sampling was better than others and predicted accuracy can be improved obviously when four mean diameter trees were sampled per plot. Taking predicted accuracy and investigation cost into account, we recommend sampling of four intermediate diameter trees per plot for practical application of the mixed-effects height-diameter model of Pinus koraiensis.