Multilevel Nonlinear Mixed-Effect Crown Ratio Models for Individual Trees of Mongolian Oak (Quercus mongolica) in Northeast China

Abstract

In this study, an individual tree crown ratio (CR) model was developed with a data set from a total of 3134 Mongolian oak (Quercus mongolica) trees within 112 sample plots allocated in Wangqing Forest Bureau of northeast China. Because of high correlation among the observations taken from the same sampling plots, the random effects at levels of both blocks defined as stands that have different site conditions and plots were taken into account to develop a nested two-level nonlinear mixed-effect model. Various stand and tree characteristics were assessed to explore their contributions to improvement of model prediction. Diameter at breast height, plot dominant tree height and plot dominant tree diameter were found to be significant predictors. Exponential model with plot dominant tree height as a predictor had a stronger ability to account for the heteroskedasticity. When random effects were modeled at block level alone, the correlations among the residuals remained significant. These correlations were successfully reduced when random effects were modeled at both block and plot levels. The random effects from the interaction of blocks and sample plots on tree CR were substantially large. The model that took into account both the block effect and the interaction of blocks and sample plots had higher prediction accuracy than the one with the block effect and population average considered alone. Introducing stand density into the model through dummy variables could further improve its prediction. This implied that the developed method for developing tree CR models of Mongolian oak is promising and can be applied to similar studies for other tree species.

Development of a Mixed-Effects Individual-Tree Basal Area Increment Model for Oaks (Quercus spp.) Considering Forest Structural Diversity

“…As stated in our data and pre-analysis section, 20% of the plot data was used for model prediction. Using a mixed-effects model for prediction, two different predictive situations can be considered, i.e., the population average (PA) response, and the subject-specific (SS) response [47,67,72].…”

“…(a) Population average responses (also termed marginal mean responses) are predicted using only the fixed effects part of a mixed-effects model if no additional information on random effects is available [47,72]. In this case, the expected value of the EBLUP's for the random effects are zeroes, and the population average responses could hence be expressed as E(y) = Xβ.…”

Generalized Nonlinear Mixed-Effects Individual Tree Crown Ratio Models for Norway Spruce and European Beech

“…Considering the necessity of the crown models for decision-making in forestry, a number of studies that involved tree-based crown models have been carried out, including crown diameter models [39][40][41][42][43][44], crown shape models [45], crown rise models [46], crown spread ratio models [47], crown projection ratio models [48], and crown ratio models [8,[49][50][51][52][53]. The CR models are much more important than any other models of the crown measures for forest management [8,24].…”

“…Therefore, developing a precise species-specific CR model is necessary, which may be challenging, but possible with the use of a comprehensive and extensive dataset and application of a robust modeling method (mixed-effects modeling). Except for two existing CR modeling studies [20,53], all other studies have used the ordinary least square regression method-also known as the traditional regression method. This is not appropriate for modeling tree-or stand-level characteristics using the measurements taken from the same sampling unit, because measurements from the same sampling unit are most likely correlated to each other.…”

“…In this situation, the mixed-effects modeling approach, which allows to include both fixed and random effects into the model, may be the most appropriate [54]. The fixed parameters in the mixed-effects model account for the effects of main and covariate predictors as in the ordinary least square regression, while the random component accounts for variability caused by stochastic factors [41,[53][54][55][56][57].…”