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.