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I have 10 measures of snakes that are strongly correlated (body length, tail length and 8 measures of head size). My dataset consists of different snake sizes (but excluded non-adults) so smaller (younger) snakes have smaller heads and shorter tails (because they depend on body length). Usually there are two approaches how to implement PCA on measures in biology in this case. One of them is to regress all measures on body length and then preform PCA on residuals + body length. The second approach is to perform PCA on original variables. I don´t know why should I regress all measures on body length, because in PCA it is not problem, when my varables are correlated. Only sense I can see in this procedure is in reducing potential multicollinearity.

If I always perform PCA using correlation matrix and oblique rotation in SPSS, does it make sense to do it on residuals? What is the difference between results of these two approaches? Aim of these PCAs is to reduce multidimensionality and then plot the graph of new components...

Can one think, that performing PCA on residuals is something like "size-standardizing" and performing PCA on original variables is not "size-standardized PCA"? Does it make sense?

Do I have to standardize all measures on the same size before performing PCA? I don´t think so, but I want to be sure.

Thanks

EDIT: some biological sources to this topic PCA on residuals: http://www.pnas.org/content/94/8/3828.full http://www.ispa.pt/ui/uie/pdf/OliveiraAlmada1995.pdf

PCA on original variables: http://swfsc.noaa.gov/uploadedFiles/Divisions/PRD/Publications/Jefferson_VW2004(82).pdf http://www.zool.uzh.ch/static/research/oekologie/literatur/pdf05_01/2001Evolution.pdf

These two approaches are most common in biology before PCA, few authors use for size-standardization ratio of dependent variable and body length. I don´t know if this question is unclear, or you don´t know answer so please, let me know how could I improve it.

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I think if you do PCA on all the variables, the first PC is very likely to be a "general size measure" so the idea of doing PCA on residuals may be to get rid of that. But, as far as I can see, the 2nd, 3rd etc. PCs on the original variables will be very similar to the 1st, 2nd, etc PCs on the residuals.

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  • $\begingroup$ Thanks for interesting answer, so you think (If I understand right what you are saying) that PC2 of orig.variables is similar to PC1 of residuals; PC3 of orig.var. is similar to PC2 of residuals, yes? So if I plot PC2 vs. PC3 of PCA conducted on original variables it wil be very similar to plot of PC1 and PC2 of residual PCA, yes? $\endgroup$ – Noro Mar 6 '12 at 8:58
  • $\begingroup$ I think so. Just my intuition and experience talking. $\endgroup$ – Peter Flom Mar 6 '12 at 11:33
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    $\begingroup$ Unfortunately there is great difference between these two graphs (conducted on same dataset)... My residual PCA show quite nice and separated groups and in my original data PCA the groups highly overlap $\endgroup$ – Noro Mar 6 '12 at 11:41
  • $\begingroup$ Oh well. Shows that intuition can be wrong! $\endgroup$ – Peter Flom Mar 6 '12 at 12:20
  • $\begingroup$ The first part of this answer is good, but the rest does not look generally true: the subsequent PCs can be totally different from the ones that are desired. To see this, consider the 2D case: take an elliptical cloud of points and translate it far from the origin in some direction between the two axes. The first PC will be that direction and the second necessarily is perpendicular to it: neither direction has anything to do with the axes of the cloud. $\endgroup$ – whuber Aug 27 '12 at 18:04
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As for a simple answer... with PCA, it is always a concern about interpretation. So the better option is to apply PCA on the original variables rather than the residuals.

"Size-standardizing": Standardizing is essential if the unit of measurements vary considerably (say 5-10 times). Most PCA algorithms/functions inherently do the standardization.

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  • $\begingroup$ I don´t know if we are talking about same "standardizing", because in statistics prePCA standardizing is centering and scaling, but I don´t know if I have to standardize size of my animals before doing this (Let´s call it biological standardizing). $\endgroup$ – Noro Mar 6 '12 at 9:04
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    $\begingroup$ By biological or size-standardizing I mean: What head dimensions and tail lengths would my animals have if they were the same body lenght? (so if snake 1 has 12 cm body lenght and 12 cm tail length and snake 2 has 24 body length and 18 cm tail length we can say after size-standardizig, that snake 2 has shorter tail - assuming same growth rate snake 1 will have in future body length 24 cm 24 cm long tail). Do I have to do something like this before PCA? $\endgroup$ – Noro Mar 8 '12 at 20:04

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