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I wonder what could be good examples of using scaling 1 and 2 for a principal component analysis biplot. By examples, I mean ecological examples or applied examples of the PCA scaling so that one can understand why it's preferable to use one scaling or another.

Here are the definitions of both scalings from Numerical Ecology by Legendre & Legendre (2012):

Distance biplot, scaling 1 (Fig. 9.3a). — The main features of a distance biplot are the following: (1) Distances among objects in the biplot are approximations of their Euclidean distances in multidimensional space. (2) Projecting an object at right angle on a descriptor approximates the position of the object along that descriptor. (3) Since descriptors have lengths of 1 in the full-dimensional space (eq. 9.7), the length of the projection of a descriptor in reduced space indicates how much it contributes to the formation of that space. (4) The angles among descriptor-axes are meaningless.

Correlation biplot, scaling 2 (Fig. 9.3b). — The main features of a correlation biplot are the following: (1) Distances among objects in the biplot are approximations of their Mahalanobis distances in multidimensional space; they are not approximations of their Euclidean distances. (2) Projecting an object at right angle on a descriptor approximates the position of the object along that descriptor. (3) Since descriptors have lengths sj in full-dimensional space (eq. 9.10), the length of the projection of a descriptor in reduced space is an approximation of its standard deviation. (4) The angles between descriptors in the biplot reflect their correlations. (5) When the distance relationships among objects are important for interpretation, this type of biplot is inadequate; a distance biplot should be used.

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Is there some kind of rule of thumb to choose a scaling in a particular situation? Wouldn't it be the same scaling between a PCA on species abundance data and a PCA on environmental variables?

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  • $\begingroup$ Related: stats.stackexchange.com/questions/141085. $\endgroup$ – amoeba says Reinstate Monica May 15 '16 at 22:47
  • $\begingroup$ Also explained here: stats.stackexchange.com/a/141755/3277. What you call "distance biplot" is the plotting of eigenvectors, or standard (standardized) coordinates. In "correlation biplot" magnitude of eigenvalues (called inertia) is completely given to "descriptors" or variables - the columns (I suppose) of the data, - to become the loadings or, synonymously, principal or unstandardized column coordinates. $\endgroup$ – ttnphns May 15 '16 at 23:59
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The general advice is

  • Use scaling 1 where you want a biplot focussed on the sites/samples and the (dis)similarity between them in terms of the species (or variables),
  • use scaling2 where you want to best represent the correlations between species (or variables).

As these numeric scaling codes are really a reflection of software implementations from the DOS era (or earlier), we recently implemented scaling selections via one of the following strings:

  1. sites (for samples)
  2. species (for variables)
  3. symmetric

for the various scalings in the vegan package for R.

In a PCA, if the variables were environmental or species, the interpretation is the same; arrows/species scores scaled with scaling 2 best represents correlations between species.

Which scaling you use really determines what values are preserved in the biplot and hence how you go about interpreting it and reading information off the plot.

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  • $\begingroup$ Do you have situations where we would prefer to use scaling 1 instead of scaling 2 and vice versa? $\endgroup$ – M. Beausoleil May 24 '16 at 15:06
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    $\begingroup$ Yes; if I'm interested in the distance between samples in terms of variable compositions I use the scaling that preserves those distance (1). If I'm interested in a plot that shows the relationships between variables then I'd choose a plot that preserved those correlations (2). In most studies I'm interested in both and so I draw several plots with different scalings. $\endgroup$ – Reinstate Monica - G. Simpson May 24 '16 at 15:37

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