As we discussed previously, the point behind rotation is to maximize the loadings of the variables on a specific component, which helps in simplifying the interpretation by reducing/eliminating the correlation among these components. The method to conduct orthogonal rotation is known as "varimax". There are other non-orthogonal rotation methods that allow correlation across factors/components. The choice of the rotation methodology that you will use in your profession should be based on the pertinent literature, which exceeds the scope of this chapter. Feel free to experiment with this dataset. I think that when in doubt, the starting point for any PCA should be orthogonal rotation.

For this process, we will simply turn back to the principal() function, slightly changing the syntax to account for 5 components and orthogonal rotation, as follows:

    > pca.rotate <- principal(train.scale, nfactors = 5, rotate = 
      "varimax")    

    > pca.rotate
    Principal Components Analysis
    Call: principal(r = train.scale, nfactors = 5, rotate = "varimax")
    Standardized loadings (pattern matrix) based upon correlation 
      matrix
                    RC1   RC2    RC5   RC3    RC4    h2    u2  com
    Goals_For     -0.21  0.82   0.21  0.05  -0.11  0.78  0.22  1.3
    Goals_Against  0.88 -0.02  -0.05  0.21   0.00  0.82  0.18  1.1
    Shots_For     -0.22  0.43   0.76 -0.02  -0.10  0.81  0.19  1.8
    Shots_Against  0.73 -0.02  -0.20 -0.29   0.20  0.70  0.30  1.7
    PP_perc       -0.73  0.46  -0.04 -0.15   0.04  0.77  0.23  1.8
    PK_perc       -0.73 -0.21   0.22 -0.03   0.10  0.64  0.36  1.4
    CF60_pp       -0.20  0.12   0.71  0.24   0.29  0.69  0.31  1.9
    CA60_sh        0.35  0.66  -0.25 -0.48  -0.03  0.85  0.15  2.8
    OZFOperc_pp   -0.02 -0.18   0.70 -0.01   0.11  0.53  0.47  1.2
    Give          -0.02  0.58   0.17  0.52   0.10  0.65  0.35  2.2
    Take           0.16  0.02   0.01  0.90  -0.05  0.83  0.17  1.1
    hits          -0.02 -0.01   0.27 -0.06   0.87  0.83  0.17  1.2
    blks           0.19  0.63  -0.18  0.14   0.47  0.70  0.30  2.4

                       RC1  RC2  RC5  RC3  RC4
SS loadings           2.69 2.33 1.89 1.55 1.16
Proportion Var        0.21 0.18 0.15 0.12 0.09
Cumulative Var        0.21 0.39 0.53 0.65 0.74
Proportion Explained  0.28 0.24 0.20 0.16 0.12
Cumulative Proportion 0.28 0.52 0.72 0.88 1.00

There are two important things to digest here in the output. The first is the variable loadings for each of the five components that are labeled RC1 through RC5. We see with component one that Goals_Against and Shots_Against have high positive loadings, while PP_perc and PK_perc have high negative loadings. The high loading for component two is Goals_For. Component five has high loadings with Shots_For, ff, and OZFOperc_pp. Component three seems to be only about the variables take while component four is about hits. Next, we will move on to the second part for examination: the table starting with the sum of square, SS loadings. Here, the numbers are the eigenvalues for each component. When they are normalized, you will end up with the Proportion Explained row, which as you may have guessed, stands for the proportion of the variance explained by each component. You can see that component one explains 28 percent of all the variance explained by the five rotated components. Remember above I mentioned the heuristic rule that your selected components should account for a minimum of about 70 of the total variation. Well, if you look at the Cumulative Var row, you see that these five rotated components account for 74% of the total and we can feel confident we have the right number to go forward with our modeling.