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I am wanting to know if I can use the ratio of Constrained/Total Inertia in my CCA to describe 'The variability explained by my constraining variables'. I am asking because I've seen different interpretations and thought I would get another opinion.

I believe this question was asked more or less here -but without a reply

I also found this code-interpretation question on SO

G. Simpson suggested in the link above that Inertia could be used in this way (Constrained/Total = amount of variance explained by CCA). I've seen other tutorials suggesting the same thing.

But, in this helpful Vegan tutorial, J. Oksanen suggests that "Total inertia does not have a clear meaning in CCA and the meaning of this proportion is just as obscure...total inertia may be random noise. It may be better to concentrate on results instead of these proportions"

So to recap my question: " Is it valid to report the proportion of inertia the constrained variables account for as 'variance explained'? Or is there a better way to report how a CCA performed?

Example of output:

          Inertia Proportion Rank
Total          4.5922     1.0000     
Constrained    0.5126     0.1116    3
Unconstrained  4.0796     0.8884   17
Inertia is mean squared contingency coefficient 

Eigenvalues for constrained axes:
   CCA1    CCA2    CCA3 
0.29170 0.17300 0.04792 
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Jari's comments aside, whatever we call "variance" (inertia in CA and CCA models) it is certainly acceptable to treat this as a measure of the "stuff" in a data set and the amount of that "stuff" that is explained by the constrained axes of the CCA. This has been done in Canoco for a long time, at least since the version 3.x days when it only ran in MS DOS, so it's use is certainly acceptable as such in some quarters.

I suspect Jari's comment stems from the fact that the quantity measured by inertia is not the same thing as that measured by the usual variance (or the thing we typically think of as variance) in statistics. So inertia $\neq$ variance but you can decompose inertia into inertia explained and residual inertia and interpret it as such, just don't call it variance explained.

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  • $\begingroup$ Thank you again @GavinSimpson, This helps clarify the issue - I keep running into the problem of folks really wanting to use the phrase 'Variance explained' with statistics that do not truly measure variance, and at time have been stumped on the appropriate way to describe my stats. $\endgroup$ – Kodiakflds Jul 8 '17 at 0:23
  • $\begingroup$ @Kodiakflds Variance is a strictly defined narrow definition. In particular, variance implies centered and interval-level data. Inertia (term used mostly within CA/biplot literature) is broader term, actually it is "magnitude" or "scale" in terms of sums of squares. Variance is a particular example of inertia. In CA, inertia is what eigenvalues represent. $\endgroup$ – ttnphns Jul 8 '17 at 10:35

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