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

Abstract

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 domains) when subjected to the effects of frequent and widespread (i.e., broadly acting) hydrogeomorphic disturbance. When major disturbance events are uncommon, bi‐variance and tri‐variance decrease, because the formation of vegetation, soil, and topographic patterns is primarily localized, and the couplings of these properties diverge over space, contingent upon site‐specific disturbance history and/or fine‐grained environmental heterogeneity. We also demonstrate that the bi‐variance and tri‐variance of a whole system are not consistently either greater or smaller than those of the associated sub‐zones. This point implies that the overall correlation structure among vegetation, soil, and topography is conserved across spatial scales. This paper addresses a critical aspect of ecology: the conceptual and analytical integration of data across multiple domains. By example, we show that bi‐variances and tri‐variances provide useful insight into how the strength of couplings among vegetation, soil, and topography data blocks varies across scales and disturbance regimes. Though we describe the simplest case of multi‐variance beyond the usual two‐block linear statistical model, this approach can be extended to any number of data domains, making integration tractable and more supportive of holistic inferences.

Tri-Variate Relationships among Vegetation, Soil, and Topography along Gradients of Fluvial Biogeomorphic Succession

“…Furthermore, after the devastative eruption of Mount St. Helens, Washington, USA in 1980, researchers found little correlation of recovering vegetation with environment in the early stages of primary succession, presumably because stochastic recruitment and chance survival were playing the leading roles [ 70 , 71 ]. All of these empirical, consistent evidences lead us to generalize that patterns of vegetation–environment relationships are contingent upon the magnitude, frequency, and timing of disturbance during the course of system dynamics over a wide range of ecological settings (e.g., river floodplain, tidal marsh creek, mountain forest, and post-volcanism regenerative slope; see also [ 46 , 72 – 74 ]). We propose that bi-variance and tri-variance provide useful insights into changing relationships among different data domains across scales and disturbance regimes.…”

“…These new unit vectors represented the summation of “predictions of the individual variables using combinations of the scores s with the correlations r as weights” ([ 41 ], p. 185). We then returned to Eq 2 in order to re-iterate these optimization procedures until we have acquired stable correlation coefficients ( r vs , r st , and r tv ) that did not vary between iterations (see Fig 5 of [ 46 ] for an example). At such convergence, the final vectors, U v , U s , and U t , were designated as the first singular axes representing the data blocks of vegetation, soil, and topography, respectively.…”

“…The strength of vegetation–soil–topography couplings along the FBS gradient was conceptualized and quantified as bi-variance and tri-variance (for more details, see [ 46 ]). Bi-variance indicates the amount of the covariation between any two data blocks, that is, between any two singular axes of vegetation, soil, and topography (β+γ+δ+ζ in Fig 2 ).…”

“…For example, the unexplained variance of topography (α) was estimated by designating the singular axis of topography as the dependent variable and those of soil and vegetation as the independent variables. To calculate β and γ, we employed the concept of extra sums of squares (ESS) that measured the marginal reduction in the residual sum of squares that occurred due to the addition of an independent variable to the existing regression model [ 46 ]. Put another way, if vegetation was already included as an independent variable in a regression model, ESS computed how much additional variation of topography would be explained after adding soil as a new independent variable.…”

“…Therefore, the components originating from different blocks possess the greatest mutual linear predictive power [ 42 – 44 ]. Although PLS has been underrepresented in ecological literature, there is a growing body of researchers who demonstrate the usefulness of the approach, given its inherent design to overcome the typical problems in ecological data: (1) the number of independent variables is greater than the number of observations and/or (2) environmental variables are significantly correlated with one another (e.g., [ 45 , 46 ]).…”