Statistical analysis
All statistical analyses were conducted using SAS statistical software (SAS Institute 9.0 Inc. 2002). All traits were used in analyses (abbreviations see Table 1). All data were log-transformed except for petiole angles and branch angles (square root-transformed) to minimize variance heterogeneity before statistical analysis. Three-way ANOVA and ANCOVA were performed to evaluate the overall effects of growth stage, soil condition and population density and their interactions on all traits, with total mass nested in growth stage as a covariate in three-way ANCOVA. Within each soil condition at each stage, effects of density were analyzed by one-way ANOVA for total mass, and one-way ANCOVA for all the other traits with total mass as a covariate. Adjusted mean values of traits produced from multiple comparisons by LSD method of the General Linear Model (GLM) program in ANCOVAs were used in calculation of plasticity.
The plasticity in traits was evaluated with the revised simplified Relative Distance Plasticity Index (RDPIs), for its strong statistical power in tests of differences in plasticity (F. Valladares, Sanchez-Gomez, & Zavala, 2006). For a given trait, its RDPIs values in response to high and medium densities relative to low density (H-L RDPI­s and M-L RDPI­s) were calculated as:
RDPIs ­= (XY )/(X + Y )
where X was the adjusted mean trait value at high or medium density, and Y was the mean value at low density. Both the level and degree of plasticity (relative plasticity and absolute plasticity) in traits were calculated as REL RDPIs and ABS RDPIs respectively.
Phenotypic canalization was evaluated by coefficient of variation (CV) for a given trait, calculated as the standard deviation divided by mean value of the trait. Phenotypic integration was estimated with the number of significant correlations of a trait with other traits (NC; p < 0.05) and coefficient of integration (CI) (Cheverud, Rutledge, & Atchley, 1983). Correlations among traits were evaluated by Pearson Correlation Coefficients (PCC) produced by PROC CORR (Gianoli & Palacio-López, 2009). CI for traits was computed as:
I = [∑(λ -1)2/(n 2-n )]1/2
where n is the number of traits andλ is an eigenvalue of the correlation matrix of the normalized data.
Both Correlation and regression analyses were applied to qualify and quantify the relationships between phenotypic plasticity (RDPIs) and phenotypic canalization (CV) or integration (NC) at different densities for plants in each soil conditions at each stage. Results of correlations and regressions were also analyzed with three-way ANOVA to access effects of population density, soil conditions and growth stage and their interactions; and one-way ANOVA for effects of density on these relationships in each soil conditions at each stage.