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I would like to introduce two supplementary variables into a PCA I'm conducting on a set of data measuring concentration in different material phases.

However I'm unclear as to how to interpret the correlation of the "supplementary" variables to the principal component (as opposed to how I would interpret active variable loadings, and eventually the relationship between the PC and the scores).

Clearly the factor loadings and sq. cosines of the variables are different depending on whether I treat them as active or supplementary. Specifically when variable "A" and "B" are included as active variables they have a strong correlation with PC2, but when treated as supplementary they do not link well to the first few PCs (and explain very little of the total variance). Based on my current understanding, the two different scenarios impact my interpretation of the scores.

I realize this is likely a simple question, but would appreciate any help to clarify my understanding so I can make sure I proceed correctly.

UPDATE

To follow up then, one is investigating the relationships of the passive variables (eg: w1,w2,w3) with each other in the context of the PC structure. So if the loadings of all active variable (eg: v1, v2, v3) is high on PC1, but the passive variables have minimal calculated loadings on PC1 - how would one interpret the relationship between the passive variable and the active variable? There is no relationship between active and passive variables?

more info here are poor quality charts - I will try and improve them. The first shows the red points as the active variables, and the second poor quality image shows the 3 variables treated as supplementary variables. I believe the scales are similar. The factor loadings and sq. cosines are different of course.

active variables active and supplementary

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    $\begingroup$ Depending on the context of your study and the number of variables, I would say that variables A and B 'contribute' to a certain extent to the 2nd dimension of your PCA, or maybe drive that particular component. When set as illustrative variables, the first PCs are linear combinations of other variables that are poorly correlated with either A or B. I would suggest to look at PC2 (with A and B) more carefully to see what this dimension might mean, or share with us a reproducible example or illustrative sample dataset. $\endgroup$ – chl Jul 5 '13 at 7:49
  • $\begingroup$ Supplementary variables, unlike active ones, do not influence the results (eigenvectors/eigenvalues and hence loadings), but they still find their place in those results: they are projected on the results drawn from active variables. $\endgroup$ – ttnphns Jul 5 '13 at 8:11
  • $\begingroup$ @ttnphns if they do not influence the results, then my question is how should they be interpreted and what is the meaning of sq.cosine of the supplementary variable on a PC? $\endgroup$ – Oleic Jul 5 '13 at 8:41
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    $\begingroup$ They are kept from influencing loadings of other variables. Loadings of active variables are the same regardless whether supplementary variables were introduced or not. So, if v1, v2, v3 are active and w1, w2, w3 are supplementary (passive) then it means that you investigate how well interrelations between the w's fit into the PC structure defined by the v's. $\endgroup$ – ttnphns Jul 5 '13 at 9:07
  • $\begingroup$ In the mid-90ies I've constructed a program specifically to study the whereabouts of PCA on sets of variables, where subsets can be set active or passive. I'd like to try your data with that program (or you do it yourself, it's easy), perhaps to answer substantially your question. $\endgroup$ – Gottfried Helms Jul 5 '13 at 19:39

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