The main idea...
Distance-based redundancy analysis (db-RDA) developed by Legendre and Anderson (1999) is a means to conduct RDA, a method which is intended to detect linear relationships, on (dis)similairties generated by measures which may be non-linear. A (dis)similarity matrix, calculated using a measure appropriate to the response data, is used as input to a principal coordinates analysis (PCoA). The result of this is a set of principal coordinates which represent the (dis)similarities in a Euclidean space, which is appropriate for analysis using standard RDA.
Figure 1: Schematic of a distance-based redundancy analysis. A (dis)similarity or distance matrix is taken as input to a principal coordinates analysis. The user will select informative principal coordinates to be used - along with explanatory variables associated with the (dis)similarity matrix - as input for a standard RDA.
While the original variables lose their individuality, the results of db-RDA can reveal whether a matrix of explanatory variables has some significant impact on the (dis)similarities derived from the response data as a whole. If the matrix of response variables from which the (dis)similarity matrix was calculated is available, they may be correlated with the PCoA axes to suggest which response variables contribute the most to the PCoA ordination.
Partial db-RDA is available in some implementations. As in partial RDA, this extension of db-RDA allows the influence of a matrix of conditioning variables to be partialled-out (approximately, "removed") prior to analysis.
If the analysis task can be addressed in ANOVA-like terms (i.e. the explanatory variable(s) define(s) the group membership of objects and the main concern is difference between groups) then dissimilarity or distance matrices may be directly tested using a non-parametric MANOVA (NPMANOVA). Click here for more information on NPMANOVA.
Implementations
The function capscale() in the package vegan performs constrained analysis of principal coordinates, or distance-based RDA
References
Legendre P, Anderson MJ (1999) Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecol Monogr. 69:1–24.