# Angles between variables on a PCA biplot and correlations between the variables

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 120ยบ, 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.

• 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
• 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). This one is a similar to yours, just almost mirror-like symmetric question. – ttnphns Jul 17 '16 at 20:27
• With p>3 or under covariances, all + it is of course possible. – ttnphns Jul 18 '16 at 11:56
• @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 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