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A friend or mine has performed a PCA and he asked me for help about interpretating a biplot.

In that biplot I found that the vector representing a variable, say A, forms a very wide angle, perhaps around 120º, with those of the rest of variables (say B and C). Furthermore, it has negative loadings for PC2. I interpreted this as PC2 separating individuals with high values of B and C and low values of A from individuals with high values of A and low values of B and C.

In addition, I thought that angles between variables, represented as vectors in biplot, were related with original variables correlations. So I thought A would be slightly but negatively correlated with both B and C because of the angle their vectors form.

But correlation matrix for A, B and C shows positive correlations in all cases. Is this possible? Should be interpreted angles between variables in biplot in some other way because we have constructed new reference axes (now our reference axes are PCs)?

Edit: I think it is not the same question as this one; this question is about particular values of angles in biplot, not about general interpretation of biplots; thanks @amoeba for editing the title.

mmv
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  • Possible duplicate of [Interpretation of biplots in principal components analysis in R](http://stats.stackexchange.com/questions/2038/interpretation-of-biplots-in-principal-components-analysis-in-r) – Firebug Jul 17 '16 at 19:54
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    I don't think it's a duplicate @Firebug, because this question is about a specific case. However, mmv, this question would be much easier to address if we could see the biplot and ideally also the correlation matrix. – amoeba Jul 17 '16 at 19:58
  • Was the PCA done on the covariance matrix or on the correlation matrix? – amoeba Jul 17 '16 at 20:00
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    Positively correlated variables can cast negative-angled arrows on a biplot. Don't forget that the vectors on the 2d plot are projections, just shadows, while true variables are out there in the p-dimensional space ([see](http://stats.stackexchange.com/a/119758/3277)). [This one](http://stats.stackexchange.com/q/219060/3277) is a similar to yours, just almost mirror-like symmetric question. – ttnphns Jul 17 '16 at 20:27
  • Thank you all for the comments. @amoeba, I will try to add some information – mmv Jul 17 '16 at 20:56
  • @ttnphns, I think my doubt is solved with those links, thank you very much. – mmv Jul 18 '16 at 01:28
  • @amoeba, the PCA was done on the correlation matrix. I saw the biplot but I haven't access to it, I'm sorry. I tried to simulate a data set but I've not been able to create a biplot with those characteristics. Thank you in any case. – mmv Jul 18 '16 at 01:35
  • I've been thinking about it, and actually it seems that it is *impossible* that A,B,C all have positive correlations but point at >90 degrees angles on the correlation-PCA biplot. I am not completely sure yet, but I doubt it is possible. @ttnphns, do you think you can construct such an example? – amoeba Jul 18 '16 at 11:28
  • @amoeba, upon reading the OP's last comment yesterday I felt just same doubt as you just expressed! The key meddling factors are: 1) the number of variables `p`, 2) their vector length (i.e. correlations restrict them to all same, unit length). I'm having no time to explore it but I expect that under `p=3` and `correlations, all +` it is impossible to receive negative angle between any two projections on the PC1-PC2 loading plot. But anyway, I would say that potential nuance is but of scholastic or "forensic" interest. If you appear to be checking on all that, please let know. – ttnphns Jul 18 '16 at 11:51
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    With `p>3` or under `covariances, all +` it is of course possible. – ttnphns Jul 18 '16 at 11:56
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    @amoeba, it looks that it is rarely possible, though. `r= 1, .247050670, .067976538; .247050670, 1, .327043818; .067976538, .327043818, 1`. Let A be the first 2 cols of the loading matrix, then `AA'= .9118253032, .4187243489, -.0643538513; .4187243489, .6657561283, .5846874005; -.0643538513, .5846874005, .8014018469`. See the negative restored (projected) corr b/w 1st and 3rd variables. – ttnphns Jul 18 '16 at 12:55
  • @amoeba The accepted answer in stats.stackexchange.com/questions/2038/… clearly states: "the angle formed by any two variables, represented here as vectors, reflects their actual pairwise correlation". This is only a question of subsetting the space, if you look at only two dimensions while in fact your space is multidimensional. – Firebug Jul 18 '16 at 14:32
  • Just like @ttnphns answered in stats.stackexchange.com/a/219084/60613 and http://stats.stackexchange.com/a/119758/60613. – Firebug Jul 18 '16 at 14:32
  • mmv, can you describe the biplot in a bit more detail? Were B and C very close? Was A around 120 degrees away from either of them? – amoeba Jul 18 '16 at 15:44
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    @amoeba In fact I tried to simplify my question but I think I complicated the whole matter... There were more than three variables, so as ttnphns says, it is possible. There were a group of four variables whose behaviour was as described; three of them (B, C and D) were close to each other and A was separated, with an angle clearly greater than 90 degrees away of all of them. Correlations between the four variables were positive; B, C and D showed moderate or high correlations with each other and A showed weak correlations with B, C and D. – mmv Jul 18 '16 at 16:07

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