Rotation in Principal Components Analysis

#1
Hi,

I was wondering if anybody could give me some general advice on a specific issue.

Is it possible to orthogonally (varimax) rotate when only one principal component is extracted?

SPSS does not allow you to rotate when only one principal component is extracted and after failing to find any literature on this issue we are struggling to defend our rationale for not rotating when only a single principal component meets the criteria for extraction. I was under the assumption that you need two axis (i.e. two principal components) in order to do any rotation.

Your help would be much appreciated. Many thanks.
 

spunky

Doesn't actually exist
#3
I was under the assumption that you need two axis (i.e. two principal components) in order to do any rotation.
you are correct. if anyone criticises you as for why you didn't rotate *one* factor, whack them in the head with a really, really hard and big book.
 
#4
Many thanks Spunky. This is what we thought.

Basically we have a manuscript under peer review which has multiple PCA's. We have a reviewer who has asked us for a strong methodological explanation as to why we didn't rotate two conditions in which only one PC was extracted during those conditions. Are you able to point us to any literature to respond with which explains why this cannot be done? I appreciate this may be difficult as it is quite simple logically why you cannot rotate when only one axis is present!

Many thanks again.
 
#5
That statement by the reviewer amused me.

We have a reviewer who has asked us for a strong methodological explanation as to why we didn't rotate ....
You could have used many different methods. Are you supposed to motivate why you did NOT choose any of them? It would be ambitious to motivate why you used PCA. But to motivate why not using any of the other methods. That seems very strange and absurd.

I guess people use PCA because it summarizes the data in a good way, (maximising the variance given the length of the coefficient vector).

Maybe a one component PCA can be said to be “rotated” if it is multiplied by -1. That would be like a 180 degrees rotation. Like: Ax = lambda x is equivalent to -Ax = -lambda x.
 

spunky

Doesn't actually exist
#6
Are you able to point us to any literature to respond with which explains why this cannot be done?
i cannot, unfortunately (although hopefully there's something out there). this similar to asking for a reference as why is it that 1+1=2.

as Greta said, sure. you can rotate one principal component all you want and still get a solution but it would be meaningless because it has no reference but itself.

think about it. you mentioned a classic method of orthogonal rotation (varimax). what is the reference against which said principal component is going to become orthogonal to? itself? so the pc becomes uncorrelated with itself?

i mean, it *is* doable (just choose an appropriate rotation matrix and that's it). but then you rotate it... and now what?
 
#7
Hi guys,

Many thanks for your replies to this issue. It has helped to confirm what we previously thought. Hopefully the reviewer understands what we are trying to explain! :)
 

noetsi

Fortran must die
#8
Unfortunately being a reviewer is no guarantee you understand the methods you are reviewing. A variety of studies have shown that elite medical journals commonly publish mistakes in logistic regression for example (such as treating relative risk and odds ratios as the same thing). Obviously had the reviewers understood the method this would not have occured.

As previous posters have noted, it makes absolutely no sense to rotate something when you only extracted one factor. If you are looking for support for this I suggest chosing a book on PCA and citing why you rotate from that (which should make it obvious why you would not rotate here).