Statistical analyses
Due to the data structure (i.e. several sample plots in each study site), a nested two-level linear mixed-effects model (Equation 2) was fitted using Restricted Maximum Likelihood included in package nlme (Pinheiro et al. 2020) of the R-software to assess the effects of thinning treatment on box dimension.
\(y_{\text{ij}}=\ {\beta_{1}\text{Moderate\ below}}_{i}+{\beta_{2}\text{Moderate\ above}}_{i}+{\beta_{3}\text{Moderate\ systematic}}_{i}+{\beta_{4}\text{Intensive\ below}}_{i}+{\beta_{5}\text{Intensive\ above}}_{i}+{\beta_{6}\text{Intensive\ systematic}}_{i}+{\beta_{7}\text{No\ treatment}}_{i}+a_{i}+c_{\text{ij}}+\epsilon_{\text{ij}},\)(2)
where \(y_{\text{ij}}\) is box dimension, \(\beta_{1},\ldots\beta_{7}\)are fixed parameters, i, i = 1, …, M, refers to study site, j, j = 1, …, \(n_{i}\), to a plot, \(a_{i}\) and \(c_{\text{ij}}\) are normally distributed random effects for sample plot j and for sample plot j within study site i , respectively, with mean zero and unknown, unrestricted variance-covariance matrix, and\(\epsilon_{\text{ij}}\) is a residual error with a mean zero and unknown variance. The random effects are independent across study sites and sample plots as well as residual errors are independent across trees. The effects of a study site and a sample plot within the study sites on box dimension, crown and stem attributes, surface-to-crown ratio, growth, as well as light competition were assessed through their variances.
The analysis of variance utilizing the results from the nested two-level linear mixed-effects model was applied in testing the statistically significant difference in the box dimension affected by the thinning treatments, the study sites as well as the plots within the study sites. Furthermore, to reveal the possible statistically significant difference in the box dimension between a thinning treatment against other treatments, Tukey’s honest significance test was applied.
To assess the relationship between structural complexity and stem and crown dimensions, benefit-to-cost ratio, as well as growth and light availability, similar approach was applied but box dimension was added as a continuous predictor variable into Equation 2. Then the response variable was a single stem and crown attribute, benefit-to-cost ratio, and growth attributes (i.e. DBH, height, stem volume, and ΔH/DBH) at a time.
The analysis of variance was applied to investigate the significance of the relationship between box dimension and thinning treatment, whereas Tukey’s honest significance test was used for revealing difference in architectural attributes (stem and crown dimensions, benefit-to-cost-ratio, growth, and light availability) between thinning intensity.
Finally, Pearson’s correlation coefficient and coefficient of determination (R2) were calculated between box dimension and stem and crown attributes as well as benefit-to-cost ratio, growth attributes, and competition index for each thinning treatment to assess their relationships.