# When doing PCA, is it possible or correct to find the correlation between variables and the first two components combined?

So I saw a correlation matrix between variables and components like below:

      comp1&2   comp1   comp2   comp3   comp4   comp5
var1    0.93     0.87   -0.34    0.16   -0.29    0.14
var2    0.91    -0.90    0.10   -0.35   -0.12    0.11
var3    0.46    -0.02    0.46   -0.43    0.04    0.56
var4    0.97     0.80    0.55    0.13    0.08   -0.02
var5    0.92     0.85    0.35   -0.07    0.30   -0.01
var6    0.69    -0.66    0.20   -0.52   -0.13   -0.07
var7    0.55    -0.43    0.35    0.52   -0.51   -0.31
var8    0.78    -0.73   -0.28    0.54    0.26    0.13
var9    0.89    -0.89   -0.04    0.21    0.02   -0.18
var10   0.90     0.52   -0.74   -0.35    0.16   -0.14
var11   0.61    -0.60   -0.11   -0.11    0.14   -0.21
var12   0.43     0.20   -0.38   -0.32    0.09   -0.40


I know how to calculate the correlation matrix between variables and component scores but i have never seen the correlation between the first two components combined and the variables in the first column.

So how is the first column calculated and what does it mean?

• If your question is how comp1&2 s computed, you should have the answer or give more details where did you saw that. – rapaio Oct 7 '16 at 7:45
• the first two components combined Please define how they are 'combined". – ttnphns Oct 7 '16 at 8:59
• This correlation matrix comes from a software called Brandmap after running a PCA. I have no idea how it defines "comp1&2". I have thought of comp1+comp2 and comp1*comp2 but it is obviously not the case. Btw, @ttnphns, thank you for your help last time. I am able to run the same results of component coordinates from Brandmap using R. – bbbbbliu Oct 7 '16 at 9:19
• most likely it is the squareroot of the communality over the two first pc's per item: so var1(comp1&2) : $0.93 = \sqrt{0.87^2+(-0.34)^2}$ with some error in the third decimal (likely because the numbers are all cutted to two digits only) – Gottfried Helms Oct 7 '16 at 19:14

I computed $$\small b(r) = \sqrt{\small\text{comp}_1(r)^2 + \text{comp}_2(r)^2}$$ and $$\small \text{err}(r)= \small \text{comp}_{1\&2}(r) - b(r)$$ where $r$ indicates the row-index, so comp1&2 $\approx b$ very likely means the squareroot of the communality up to the two first pc's.
In a biplot with the two pc's as axes the values comp1&2 give the lengthes of the graphed vectors for var1,var2, ... .

    *   comp1&2 comp1   comp2   comp3   comp4   comp5       b      err
var1      0.93  0.87   -0.34    0.16   -0.29    0.14   0.934   -0.004
var2      0.91 -0.90    0.10   -0.35   -0.12    0.11   0.906    0.004
var3      0.46 -0.02    0.46   -0.43    0.04    0.56   0.460    0.000
var4      0.97  0.80    0.55    0.13    0.08   -0.02   0.971   -0.001
var5      0.92  0.85    0.35   -0.07    0.30   -0.01   0.919    0.001
var6      0.69 -0.66    0.20   -0.52   -0.13   -0.07   0.690    0.000
var7      0.55 -0.43    0.35    0.52   -0.51   -0.31   0.554   -0.004
var8      0.78 -0.73   -0.28    0.54    0.26    0.13   0.782   -0.002
var9      0.89 -0.89   -0.04    0.21    0.02   -0.18   0.891   -0.001
var10      0.90  0.52   -0.74   -0.35    0.16   -0.14   0.904   -0.004
var11      0.61 -0.60   -0.11   -0.11    0.14   -0.21   0.610    0.000
var12      0.43  0.20   -0.38   -0.32    0.09   -0.40   0.429    0.001

• +1. Nice guess what it was. If the two components or factors are orthogonal then it is indeed the "combined loading" of them, the length of the arrow in their plane, sq. root of the projected there communality. – ttnphns Oct 7 '16 at 19:49
• Thank you for your answer!! I tested a few more datasets and it is exactly that. – bbbbbliu Oct 8 '16 at 2:44