title: Principal component analysis (PCA) in R: why are the scores not orthogonal? (using Psych package)
body: <p>I ran PCA in R using the principal() function in the "psych" package. I made the argument "rotate="none"", which asks for orthogonal rotation method. From what I understand, the scores of PC1 and PC2 should be orthogonal (i.e. there should be zero correlation between (raw data)<em>(loading of PC1)and (raw data)</em>(loading of PC2). However, I got 90% correlation. Why is that?</p>
  
  <pre><code>&gt; #load the package
  &gt; library(psych)
  
  &gt; #calculate the correlation matrix
  &gt; corMat  &lt;- cor(data)
  
  &gt; #run PCA
  &gt; pca.results &lt;- principal(r = corMat,**rotate ="none"**, nfactors = 20,covar=FALSE,scores=TRUE)
  &gt; pca.results`enter code here`
  Principal Components Analysis
  Call: principal(r = corMat, nfactors = 20, rotate = "none", covar = FALSE,
      scores = TRUE)
  Standardized loadings (pattern matrix) based upon correlation matrix
                                  **PC1   PC2**   PC3   PC4   PC5   PC6   PC7   PC8   PC9
  payroll.chg                   -0.30  0.85  0.21  0.35 -0.03  0.02  0.07 -0.11 -0.02
  HH.empl.chg                   -0.26  0.62  0.64 -0.35  0.01 -0.06  0.06  0.00  0.01
  pop.empl.ratio                -0.92 -0.34  0.13  0.04  0.06 -0.03 -0.04  0.03 -0.04
  u.rate                         0.99  0.10  0.02  0.04  0.01  0.04  0.04  0.04  0.01
  median.duration.unempl         0.88  0.44 -0.02  0.02 -0.04  0.06  0.02  0.13 -0.05
  LT.unempl.unempl.ratio         0.86  0.49 -0.04  0.01 -0.07  0.02  0.00  0.08 -0.02
  U4                             0.99  0.13  0.01  0.03  0.01  0.04  0.04  0.05  0.01
  U6                             0.98  0.13 -0.05 -0.02  0.00  0.06  0.04  0.03  0.04
  vacancy.rate                  -0.87  0.35 -0.18 -0.11 -0.01  0.22  0.10  0.03 -0.01
  hires.rate                    -0.92  0.08  0.24  0.21 -0.16  0.06  0.00  0.05  0.09
  unemployed.to.employed         0.89  0.17  0.21 -0.02  0.05  0.24 -0.25 -0.05  0.00
  Layoff.rate..JOLT.             0.23 -0.86  0.19 -0.03 -0.40  0.09  0.03 -0.02 -0.05
  Exhaustion.rate                0.95  0.19  0.14  0.14  0.00 -0.07  0.01  0.06 -0.04
  Quits.rate..JOLT.             -0.98  0.01  0.04  0.04  0.01  0.02 -0.06  0.10  0.13
  participation.rate            -0.67 -0.61  0.31  0.14  0.16 -0.01 -0.03  0.11 -0.08
  insured.u.rate                 0.88 -0.40  0.17  0.08  0.12  0.05  0.09 -0.03  0.02
  Initial.jobless.claims         0.78 -0.60  0.04 -0.06  0.06  0.05  0.07  0.02  0.07
  Continuing.claims              0.86 -0.44  0.15  0.06  0.14  0.08  0.09 -0.05  0.03
  Jobs.plentiful.jobs.hardtoget -0.98  0.00 -0.02  0.01  0.08  0.13  0.04 -0.02 -0.04
  vacancy.unempl.ratio          -0.97  0.04 -0.05 -0.03  0.08  0.18  0.07  0.03 -0.03
  
                                 PC10  PC11  PC12  PC13  PC14  PC15  PC16  PC17  PC18
  payroll.chg                   -0.06  0.02 -0.02  0.00  0.03  0.00  0.00  0.00  0.00
  HH.empl.chg                    0.01  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00
  pop.empl.ratio                -0.02  0.00 -0.01  0.01  0.00  0.00  0.00  0.01  0.01
  u.rate                        -0.01  0.00  0.03 -0.03  0.02  0.00  0.00 -0.01 -0.01
  median.duration.unempl         0.02  0.05 -0.06 -0.01 -0.03  0.01 -0.02  0.00  0.00
  LT.unempl.unempl.ratio         0.01  0.02 -0.01  0.02  0.00  0.00  0.05  0.00  0.00
  U4                            -0.01  0.00  0.04 -0.02  0.02  0.00 -0.01 -0.01  0.01
  U6                            -0.01  0.01  0.03 -0.03  0.02 -0.02  0.00  0.03  0.00
  vacancy.rate                  -0.08 -0.06  0.01  0.01 -0.01  0.04  0.00  0.00  0.00
  hires.rate                     0.01  0.00  0.04  0.00 -0.06 -0.01  0.00  0.00  0.00
  unemployed.to.employed        -0.01  0.00  0.00  0.01  0.00  0.00  0.00  0.00  0.00
  Layoff.rate..JOLT.             0.01  0.00 -0.01 -0.01  0.03  0.00  0.00  0.00  0.00
  Exhaustion.rate                0.05 -0.07  0.02  0.06  0.01 -0.01 -0.02  0.00  0.00
  Quits.rate..JOLT.              0.04 -0.01 -0.04  0.00  0.05  0.02  0.00  0.00  0.00
  participation.rate            -0.06  0.00  0.02 -0.02  0.01  0.01  0.01  0.00  0.00
  insured.u.rate                 0.04 -0.02 -0.02  0.00 -0.02  0.02  0.01  0.00  0.02
  Initial.jobless.claims        -0.09  0.06  0.00  0.06  0.01 -0.01 -0.01  0.00  0.00
  Continuing.claims              0.05 -0.02 -0.02 -0.02 -0.01  0.01  0.01  0.01 -0.02
  Jobs.plentiful.jobs.hardtoget  0.11  0.07  0.05  0.02  0.01  0.02  0.00  0.00  0.00
  vacancy.unempl.ratio           0.03 -0.01 -0.03  0.00  0.01 -0.06  0.00  0.00  0.00
  
                                 PC19  PC20 h2       u2
  payroll.chg                    0.00  0.00  1  5.6e-16
  HH.empl.chg                    0.00  0.00  1 -2.9e-15
  pop.empl.ratio                 0.01  0.01  1 -1.6e-15
  u.rate                        -0.01  0.01  1  1.1e-16
  median.duration.unempl         0.00  0.00  1 -4.4e-16
  LT.unempl.unempl.ratio         0.00  0.00  1 -6.7e-16
  U4                             0.01  0.00  1 -4.4e-16
  U6                             0.00  0.00  1  2.2e-16
  vacancy.rate                   0.00  0.00  1  0.0e+00
  hires.rate                     0.00  0.00  1  4.4e-16
  unemployed.to.employed         0.00  0.00  1 -2.2e-16
  Layoff.rate..JOLT.             0.00  0.00  1 -2.2e-15
  Exhaustion.rate                0.00  0.00  1 -4.4e-16
  Quits.rate..JOLT.              0.00  0.00  1  1.1e-16
  participation.rate             0.00 -0.01  1  5.6e-16
  insured.u.rate                -0.01  0.00  1 -6.7e-16
  Initial.jobless.claims         0.00  0.00  1 -2.0e-15
  Continuing.claims              0.01  0.00  1 -6.7e-16
  Jobs.plentiful.jobs.hardtoget  0.00  0.00  1  2.2e-16
  vacancy.unempl.ratio           0.00  0.00  1 -2.2e-16
  
                          PC1  PC2  PC3  PC4  PC5  PC6  PC7  PC8  PC9 PC10 PC11 PC12
  SS loadings           14.23 3.73 0.83 0.37 0.28 0.20 0.12 0.07 0.05 0.05 0.02 0.02
  Proportion Var         0.71 0.19 0.04 0.02 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00
  Cumulative Var         0.71 0.90 0.94 0.96 0.97 0.98 0.99 0.99 0.99 1.00 1.00 1.00
  Proportion Explained   0.71 0.19 0.04 0.02 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00
  Cumulative Proportion  0.71 0.90 0.94 0.96 0.97 0.98 0.99 0.99 0.99 1.00 1.00 1.00
  
                        PC13 PC14 PC15 PC16 PC17 PC18 PC19 PC20
  SS loadings           0.01 0.01 0.01    0    0    0    0    0
  Proportion Var        0.00 0.00 0.00    0    0    0    0    0
  Cumulative Var        1.00 1.00 1.00    1    1    1    1    1
  Proportion Explained  0.00 0.00 0.00    0    0    0    0    0
  Cumulative Proportion 1.00 1.00 1.00    1    1    1    1    1
  
  Test of the hypothesis that 20 components are sufficient.
  
  The degrees of freedom for the null model are  190  and the objective function was  68.46
  The degrees of freedom for the model are -20  and the objective function was  0
  
  Fit based upon off diagonal values = 1
  </code></pre>