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Calculating Principal Components |

The **Eigenvalues (CORR)** table illustrated in
Figure 19.7 contains all the eigenvalues of
the correlation matrix, differences between
successive eigenvalues, the proportion of
variance explained by each eigenvalue, and the
cumulative proportion of the variance explained.
Eigenvalues correspond to each of the principal
components and represent a partitioning of the
total variation in the sample.
Because correlations are used, the sum of all the
eigenvalues is equal to the number of variables.
The first row of the table corresponds to the
first principal component, the second row to
the second principal component, and so on.
In this example, the first two principal components
account for over 97% of the variation.

The **Eigenvectors (CORR)** table illustrated in
Figure 19.7 contains the first two
eigenvectors of the correlation matrix.
Eigenvectors correspond to each of the eigenvalues
and associated principal components and are
used to form linear
combinations of the Y variables.
The first column of the table corresponds to the
first principal component, and the second column to
the second principal component.
Now examine the coefficients making up the eigenvectors.
The first component (**PCR1**) appears to be a measure of
the player's overall performance as is evidenced
by approximately the same magnitude of the coefficients
corresponding to all six variables.

Next examine the coefficients making up the eigenvector
for the second principal component (**PCR2**).
Only the coefficients associated with the
variables **CR_HOME** and **CR_RBI**
are positive, and the remaining coefficients are negative.
The coefficient with the variable **CR_HOME**
is considerably larger than any of the other coefficients.
This indicates a measure of career home runs
performance versus other performance for 1986.

One way to quantify the strength of the linear relationship
between the original Y variables and principal components
is through the **Correlations (Structure)** table,
as shown in Figure 19.7.
This correlation matrix contains the correlations between
the Y variables and the principal components.
Eigenvector coefficients of a relatively large magnitude
translate into larger correlations and vice versa.
For example, **PCR2** has one coefficient
substantially larger than other coefficients in
the same eigenvector, **CR_HOME**.
The correlation of the variable with
this **PCR2** is also large.

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