Collinearity does not affect Procrustes analysis outputs: directions for plant and soil ecologists

Background . The Procrustean residual vector (or PAM, an acronym for the alternative equivalent term Procrustean association metric) derived from Procrustes analysis can be seen as the univariate form of relationship between two or more data tables, which provides an interesting way for ecologists to place multivariate relationships as the central object of investigation in more familiar statistical approaches such as ANOVA and post hoc tests. However, many aspects need to be elucidated to make ecologists more confident in using Procrustes in their studies going beyond the simple comparisons. We attempted to address two questions: 1) How does the increasing number of correlated columns within an entire data table affect the Procrustes results? 2) Can the PAM be used for detecting how the correlation is partitioned across treatment levels within the original data table? Methods. Question 1) two data tables, X and Y , from a previous research were used to conduct the study. Four levels of correlation between variables (0.9, 0.7, 0.5, and 0.2) within the X data table were imposed to an increasing number of variables (6, 9, 12, and 15) to assess their effects on Procrustes relationship and its significance. Question 2) two simulated data tables covering four hypothetical categorical predictors (A, B, C, D) were created varying the relationship between them regarding the treatment A (0.2, 0.5, 0.7, 0.9) in order to assess the association between Procrustes and multiple mean comparisons method. Results. for the first question, we found that increasing the number of correlated variables across different imposed correlation levels (0.9, 0.7, 0.5, and 0.2) in the data table not subject to Procrustean linear transformation (translation and rotation), i.e. the X data table, had no effects either on the classical Procrustes outcomes related to the fit between data tables ( R statistic and its P value), or on the significance of the ANOVA using the Procrustes association metric (PAM), which summarizes the multivariate correlation between two data tables, as the response variable. For the second question, increasing the between correlation levels between X and Y data tables for a specific set of rows in these tables corresponding to a hypothetical treatment A resulted in PAMs that, when used in mean multiple comparisons, did show this treatment A as different from all others treatments B, C, and D from which X and Y were not related above (0.1). Discussion . Our results support that the Procrustes fit is only dependent on the information between data tables instead of within a data table. Finally, we showed that PAM, in fact, reflects the differences in multivariate correlation across data tables which can be useful for ecological questions addressing the partitioning of the multivariate correlation among different categorical levels (e.g. plots, time, land use type, etc.).