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In principal component analysis there is no prior restriction on which variables are included in a component.

I wonder if there is a method which has such a restriction. That is, if the original variables are $X_1,...,X_p$, I want to use only $X_1,...,X_i$ for the first component, $X_j,...X_m$ for the second, etc.

The purpose would be to give an "exact" interpretation of the components. E.g. having variables describing physical and cognitive performances, PCA may give components containing both physical and cognitive variables. However, I want components made of only one type of variables, so as to have a pure physical component and a pure cognitive one.

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    $\begingroup$ just perform two independent PCAs, one on each variable type $\endgroup$ – g3o2 Feb 11 '18 at 11:10
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As already commented by @g3o2, you can get what you want by running separate PCAs on your distinct groups of variables, and use the first PC from each. Make sure that you include precisely the same observations (cases, people?) in each one.

At the same time, it wouldn't be amiss to test your assumption of separate clusters of variables in different ways,

e.g.

  1. Look at the correlation between your new PCs from separate analyses. Plot the results and watch for any patterns.

  2. If latent variables or factors really are quite separate, that should also fall out of a PCA on all variables together.

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