What is the difference between using the PCA method with and without the rotate command ?

[Michael Wegmuller]

Every tutorial I find online implement PCA that way

pca varlist
rotate
predict varname

I showed this to my professor who asked me to use a PCA index for an analysis and they asked me why I used the rotate command, telling me they just used pca and then predict in their own analysis. To be honest I don't fully understand what rotate is and what it does, but by testing with and without it I can confirm it changes the results.

So what is that rotate command and how does it affect my results ? which would be the correct way to use PCA to construct an index out of a list of variables ?

Thanks !

[Clyde Schechter]

PCA is used to take a set of variables and create a new set of variables, having the same number of variables as the original, that captures all of the variance of the original variables, but the new variables are independent of each other. Mathematically, there are an infinite number of ways to do this. The particular way that PCA does it has the additional property that the first variable in the new set has the largest possible amount of the variance--and except for changing signs, there is usually only one solution that satisfies this additional restriction. So if the only reason for going down this road is to reduce a set of variables to a single variable that does the best possible job of capturing the variance in the original set of variables, then you just run PCA and use the first component.

Rotate doesn't come into play there. What the -rotate- command does is drop that additional requirement that the first variable have the largest possible variance, and looks at other solutions to the more general problem--remember, there are infinitely many solutions to the general problem. There are a number of different options available for the -rotate- command, and they produce solutions with different properties. When, as in your post, no option is specified, you get the varimax rotation. It has the additional property that the correlations of the original variables with the (rotated) components tend to be close to 0 or close to 1. You may wonder why anybody would want that. Well, if the original variables are people's responses to different items in a questionnaire, often the goal is not just to reduce the number of variables you need to work with, but you would like the new variables to "mean something." That is, you don't want the new variables to just be some arbitrary hodge-podge of the originals. You would prefer the new variables to representing something like picking subsets of the original items, each component representing a subset of the original items that "goes together" in terms of being about a common theme or topic. Rotated components are usually better than the unrotated results of PCA for this purpose.

Anyway, if the purpose of getting an "index" is to just capture as much information from the original group of variables as you can into a single variable and you don't care if the single variable looks like a selection of a subset of the original variables that "make sense going together" then -rotate- does you no good, and in fact reduces the amount of variation you can capture with just a single variable.

If you're not just looking to boil everything down to one variable but rather you are looking for a way of grouping the variables together into "coherent" subsets, then -rotate- is a useful tool.

If you're wondering why it's called "rotate," I can only tell you that if you have studied linear algebra, it's obvious: the principal components are a different basis for the space spanned by the original variables, and the rotated principal components are rotations of that different basis. If you haven't studied linear algebra, it's much too complicated to explain here.

[Michael Wegmuller]

Thank you very much for the response and sorry for looking at it only now. That really helped