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I am searching for an explanation what the x and y axis within a PCA analysis really means? I could not find any explanation on this topic, can somebody help me with it?

So, just as an example (output of the psych R-package, available on CRAN):

Principal Components Analysis
Call: principal(r = Thurstone, nfactors = 3, rotate = "Promax", n.obs = 213) Standardized loadings (pattern matrix) based upon correlation matrix
                  PC1   PC2   PC3   h2   u2 com
Sentences        0.92  0.01  0.01 0.86 0.14 1.0
Vocabulary       0.90  0.10 -0.05 0.86 0.14 1.0
Sent.Completion  0.91  0.04 -0.04 0.83 0.17 1.0
First.Letters    0.01  0.84  0.07 0.78 0.22 1.0
4.Letter.Words  -0.05  0.81  0.17 0.75 0.25 1.1
Suffixes
Letter.Series
Pedigrees
Letter.Group
 0.18  0.79 -0.15 0.70 0.30 1.2
 0.03 -0.03  0.88 0.78 0.22 1.0
 0.45 -0.16  0.57 0.67 0.33 2.1
-0.19  0.19  0.86 0.75 0.25 1.2
       PC1  PC2  PC3
      2.83 2.19 1.96
      0.31 0.24 0.22
      0.31 0.56 0.78
SS loadings
Proportion Var
Cumulative Var
Proportion Explained  0.41 0.31 0.28
Cumulative Proportion 0.41 0.72 1.00
 With component correlations of
     PC1  PC2  PC3
PC1 1.00 0.51 0.53
PC2 0.51 1.00 0.44
PC3 0.53 0.44 1.00
Mean item complexity =  1.2
Test of the hypothesis that 3 components are sufficient.
The root mean square of the residuals (RMSR) is  0.06
 with the empirical chi square  56.17  with prob <  1.1e-07
Fit based upon off diagonal values = 0.98

So, if I plot this, we have the Sentences at 0.92 on the X axis and at 0.01 on the Y axis (I skipped the 3rd dimension).
What would a value of 1 or 0.5 actually mean?

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  • $\begingroup$ This is not PCA. This looks like PCA followed by promax rotation of the first 3 components. That's complicated analysis; don't use it unless you understand really well what you are doing. $\endgroup$
    – amoeba
    Jul 31, 2015 at 10:10
  • $\begingroup$ This was actually copied from psych package overview shown on page 46. You are right, they where using a promax rotation and this is not what I did. I used the standard setting (varimax) which might also be complicated but I thought I might be on the save side... I actually only wonder, why some PCA axis range further then 1 and I really try to understand, what those dimensions are. $\endgroup$
    – drmariod
    Jul 31, 2015 at 11:26
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    $\begingroup$ Roughly speaking, it means that the first principal component is constructed mainly by $Sentences$, $Vocabulary$, and $Sent.Completion$ variables, and the second component is constructed mainly by $First.Letters$, $4.Letter.Words$, and $Suffixes$. I would explain it as the $PC_1$ is mainly related to sentence construction and the $PC_2$ related to some short patterns in a word. And I think that's pretty good grouping. There is a very good overview from H. Abdi regarding PCA: onlinelibrary.wiley.com/doi/10.1002/wics.101/epdf $\endgroup$
    – Kirill
    Jul 31, 2015 at 14:38

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