Arboriculture & Urban Forestry 35(2): March 2009 transplantation data, 15 City of Montreal nursery trees of ball and burlap transplanting size were measured for every species and mean values were calculated. Other incremental indices were similarly constructed. Theoretically, such indices should permit differentiation between trees of the same morphological stature but of different ages, thus identifying differential growth. METHODOLOGICAL ASPECTS OF MULTIVARIATE ANALYSIS Evaluation of the Explanatory Potential of Qualitative and Quantitative Parameters Qualitative parameters are rapid appraisals of tree growth and they can be appealing for tree inventory, but they are nec- essarily biased as their estimation is based on a subjective evaluation that can vary from one observer to another. On the contrary, quantitative measurements are accurate but time- consuming. To assess the respective weight and explanatory potential of both types of variables, the experimental data set was used as input for a multidimensional statistical proce- dure that allows the simultaneous comparison of data that are of different mathematical formats. First, the records of each sampled tree were transformed into similarities, by us- ing the Gower’s similarity coefficient. For instance, when bi- nary variables such as presence or absence of chlorosis were used to compare the physiological status of individual trees, the similarity value (s) between two trees was computed as follows: if both trees were chlorotic, then s = 1, otherwise s = 0. Semi-quantitative and qualitative variables such as intensity of mechanical damages, crown development and crown den- sity were treated following the simple matching rule: s = 1 when the variable took the same value for both trees. Quan- titative variables such as DBH, height, crown diameter, and crown volume were evaluated by dividing the difference between the states of the two trees by the largest difference found for a given variable across all pairs of trees. Eventu- ally, the overall similarity between two trees was assessed by computing the average value of calculated similarities for all types of variables. These computations were neces- sary steps prior to the use of principal coordinates analysis (PCO), a nonparametric statistical procedure that can gener- ate reduced-space ordinations and estimate the variance ex- plained when testing different hypothetical models. In this research, in order to identify which inventory parameters adequately expressed urban tree growth patterns, two mod- els were compared. The first one tested the concurrent use of qualitative and quantitative parameters, while the second one restricted computations to quantitative measurements only. Selection of the Most Important Quantitative Variables and Assessment of Model Integrity In this study, the morphological and composite parameters that were used are not independent. For example, DBH and annual DBH increment, crown diameter and crown diameter incre- ment, height and height increment, are just different expres- sions of trunk, crown and height growth. In addition, crown volume values were computed using crown diameter. Such affinities can induce collinearity that may severely bias any urban tree growth model. To test this hypothesis, six differ- 55 ent scenarios were elaborated to investigate interrelation- ships between variables and identify which combination of parameters is best suited for inventory-data analysis (Table 1). Table 1. Scenarios designed to test for collinearity between quantitative variables. Quantitative variables Diameter at breast height (DBH) Annual DBH increment Crown diameter Annual crown diameter increment Height Annual height increment Crown diameter / DBH Crown volume / DBH Height / DBH Crown volume Crown volume increment • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • In scenario A, which is the main hypothesis, all selected par- ameters were used at once in the multivariate statistical analyses. Scenario B excluded ratio parameters from the analyses to esti- mate a possible redundancy effect. The results of this test were compared to those of scenario A. The third scenario (C) left out every parameter directly linked to crown diameter, to verify if their use in scenario A inflated the weight of crown inputs and biased the model. Scenario D challenged a key concept in street tree growth: small trees are not necessarily the youngest ones and may be older, poorly growing trees. Accordingly, inclusion of only incremental and ratio parameters in the growth model may have given the best representation of time span and vigor when compared to scenario A. Finally, scenarios E and F were models with the lowest possible collinearity between parameters: only crown volume and diameter could directly be linked. However, we do not believe that crown volumes are systematically related to particular crown diameters. For example, a small cylindrical tree near street curb may have the same mean crown diameter as an asymmetrically pruned old tree transplanted close to a nearby building but the corresponding crown volumes will be significant- ly different. Scenarios E and F were the simplest models challen- ging the complexity of scenario A, but they were also designed to estimate the relative explanatory power of incremental param- eters over standard morphological variables in any growth model. Correspondence analysis (CA) was used to test every scenario. Correspondence analysis can be designated as a nonparametric principal component analysis that can reveal the principal axes of a high-dimensional space, enabling projection into a subspace of low dimensionality that accounts for the main portion of variance in the data. This particular property of CA can reveal relationships that would not be detected in a series of pairwise comparisons of variables, and can simplify remarkably complex data (Theodorou et al. 2007). Another procedure, the broken-stick random model, was used concomitantly to enhance the interpretation of CA re- sults. That method compares the computed distribution of vari- ance in CA multidimensional axes to that of a random model. If the CA procedure has distributed the total variance at random among the principal axes, the portions of variance explained by the various axes would be about or less than the broken-stick ©2009 International Society of Arboriculture • • A • B • Scenarios C • D E F •
March 2009
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