Beyond bivariate correlations: three-block partial least squares illustrated with vegetation, soil, and topography

Ecologists, particularly those engaged in biogeomorphic studies, often seek to connect data from three or more domains. Using three-block partial least squares regression, we present a procedure to quantify and define bi-variance and tri-variance of data blocks related to plant communities, their soil parameters, and topography. Bi-variance indicates the total amount of covariation between these three domains taken in pairs, whereas tri-variance refers to the common variance shared by all domains. We characterized relationships among three domains (plant communities, soil properties, topography) for a salt marsh, four coastal dunes, and two temperate forests spanning several regions in the world. We defined the specific bi- and tri-variances for the ecological systems we included in this study and addressed larger questions about how these variances scale with each other looking at generalities across systems. We show that a system tends to exhibit high bi-variance and tri-variance (tight coupling among d...