VP free

Variance partitioning
In an attempt to re-understand VP, I’m going to attempt to explain it to myself in word-form…

Variance, in regressions, is always present. R2 values are given to illustrate the amount of variance the model (regression) explains. Multiple regressions use several variables (x1, x2, etc) in an attempt to explain the response variable (y); each x contributes some “explanatory power” to the model. That’s stuff we know already…

How much does each x contribute to the explanatory power of the model? Well, first note that the order of insertion of independent variables into the model doesn’t change the final result (x1b + x2b + x3b…). Intermediate results will differ, but the final result is the same. Secondly, the contribution (in terms of increasing r2, if that’s our scorecard) of the second variable entered will change depending on what was entered first. Not only that, some variables which don't add anything to r2 might still be important, as they could mediate the effect of another variable in some important way. Overlapping disturbances would be a candidate.

Basically, it appears that you look at the change in r2 when each independent variable is added last, and then call that difference the “unique variance.”
Nothing super new so far, it’s the commonality analysis which seems like it’ll be useful- is it possible to look at the influence of A on B even if A adds nothing to an explanatory model of B, but mediates the effect of variable C on B?

Perhaps it works this way: A -> B <- C


Or this: C -> A -> B <- C

In the second case, A would appear, in a regression, to add nothing to a predictive model of B, but perhaps it’s mediating the effect of C in some way. Since they are correlated, it would be difficult (impossible?) to separate the two influences specifically, but perhaps it would be possible to get a measure of the importance of C via subtraction…

VARIATION PARTITIONING OF SPECIES DATA MATRICES: ESTIMATION AND COMPARISON OF FRACTIONS
Pedro R. Peres-Neto1, Pierre Legendre, Stéphane Dray, and Daniel Borcard
Ecology 87(10)