Much beyond Mantel: bringing Procrustes association metric to the plant and soil ecologist's toolbox

Abstract

The correlation of multivariate data is a common task in investigations of soil biology and in ecology in general. Procrustes analysis and the Mantel test are two approaches that often meet this objective and are considered analogous in many situations especially when used as a statistical test to assess the statistical significance between multivariate data tables. Here we call the attention of ecologists to the advantages of a less familiar application of the Procrustean framework, namely the Procrustean association metric (a vector of Procrustean residuals). These residuals represent differences in fit between multivariate data tables regarding homologous observations (e.g., sampling sites) that can be used to estimate local levels of association (e.g., some groups of sites are more similar in their association between biotic and environmental features than other groups of sites). Given that in the Mantel framework, multivariate information is translated into a pairwise distance matrix, we lose the ability to contrast homologous data points across dimensions and data matrices after their fit. In this paper, we attempt to familiarize ecologists with the benefits of using these Procrustean residual differences to further gain insights about the processes underlying the association among multivariate data tables using real and hypothetical examples.

Figure 1. Papers published using Mantel and Procrustes for relating data matrices from soil or plant studies in the ten years since stated the advantages of Procrustes over the Mantel approach.

Data obtained using Thompson Reuters database (May, 12, 2014). We searched for papers using uniquely the Mantel approach, uniquely the Procrustes approach and papers using both approaches. The search was based on Procrust* (Procrustean or Procrustes) and PROTEST.

Figure 2. Roadmap for two alternative ways to reach the same dimensionality between matrices, and so relating it by Procrustes analysis.

a) Addition of columns containing zeros to the Y raw data matrix for matching the X raw data matrix dimension; b) Application of ordination to raw data matrices to make matrices have equal dimensionality prior to Procrustes analysis.

Figure 3. Roadmap for applying the Procrustes association metric (PAM) in the multivariate ordination context using data of .

a) Assembling matrices with different ordination axes, through Procrustes analysis, soil chemistry (SC) and plant community with soil microbial community (PLFA, and bacterial and fungal T-RFLP); b) Extraction of PAM from Procrustean relationships based on matrices with 6 ordination axes; c) Assembling of PAM based PCA matrices with 6 axes as rows in a single matrix (“effect matrix”), and using it in an ordination technique (e.g., PCA, PCoA, NMDS) to verify if the different effects diverge.

Figure 4. Results from PCA ordination of the Procrustes association metric matrix (“effect matrix”) gathering the interactions of soil chemistry and plant community with soil microbial matrices (PLFA, and bacterial and fungal T-RFLP).

The filled symbols are the Procrustes relationships between soil chemistry and soil microbial matrices, and the open symbols between plant community and soil microbial matrices. Data from three chronosequences (Craggan, Kerrow and Tulchan) obtained by .

Figure 5. Roadmap for using Procrustes Association Metric (PAM) in a multiple regression analysis framework (variation partitioning).

a) Soil microbial community (SMC) and soil microbial functioning (SMF) matrices are submitted to an ordination to reach the same dimensionality, and SMC and SMF matrices formed by 2, 3 and n axes related through Procrustes analysis in order to generate PAMs; b) PAMs generated were used as response variables in a variation partitioning to verify the individual contribution of soil properties and spatial information (PCNM eigenfunctions) on the SMC-SMF relationship; c) Venn diagram depicting the relative contribution of soil properties (niche processes [a]) and unmeasured spatial factors (neutral processes [c]).

Figure 6. Roadmap for using Procrustes association metric (PAM) in an ANOVA context.

a) PCA ordination of each SMC and SMF raw data matrices, and then Procrustes correlation from 2 axes-based PCA matrices in order to generate the PAM depicting the SMC-SMF relationship. b) Table showing results of a one-way ANOVA for using PAM as response and land use type as fixed factor. c) Multiple comparisons test (Tukey, 95%) for means of the Procrustean relationship between soil microbial structure and functioning (PAM in 2 axes) across land use types.