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Chapter 401

Correlation Matrix

Introduction
This program calculates matrices of Pearson product-moment correlations and Spearman-rank
correlations. It allows missing values to be deleted in a pair-wise or row-wise fashion.
When someone speaks of a correlation matrix, they usually mean a matrix of Pearson-type correlations.
Unfortunately, these correlations are unduly influenced by outliers, unequal variances, nonnormality, and
nonlinearities. One of the chief competitors of the Pearson correlation coefficient is the Spearman-rank
correlation coefficient. The Spearman correlation is calculated by applying the Pearson correlation formula
to the ranks of the data. In so doing, many of the distortions that infect the Pearson correlation are reduced
considerably.
A matrix of differences can be displayed to compare the two types of correlation matrices. This allows you
to determine which pairs of variables require further investigation.

Partial Correlation
This program lets you specify an optional set of partial variables. The linear influence of these variables is
removed from the correlation matrix. This provides a statistical adjustment to the correlations among the
remaining variables using multiple regression. Note that in the case of Spearman correlations, this
adjustment occurs after the complete correlation matrix has been formed.

Heat Maps
Using heat maps to display the features of a correlation matrix was the topic of Friendly (2002) and Friendly
and Kwan (2003). This program generates a heat map for various correlation matrices.

Plots of Eigenvectors
Friendly (2002) and Friendly and Kwan (2003) discuss the strengths of plotting the eigenvectors of a
correlation matrix. They imply that such a plot is more informative than a heat map. This program generates
a plot of the eigenvectors for various correlation matrices.
Another plot that is similar to the eigenvector plot is the map which is provided by a metric multidimensional
scaling analysis (see the Multidimensional Scaling procedure for details).

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Correlation Matrix

Discussion
When there is more than one independent variable, the collection of all pair-wise correlations are succinctly
represented in a matrix form. In regression analysis, the purpose of examining these correlations is two-fold: to
find outliers and to identify collinearity. In the case of outliers, there should be major differences between the
parametric measure, the Pearson correlation coefficient, and the nonparametric measure, the Spearman rank
correlation coefficient. In the case of collinearity, high pair-wise correlations could be indicators of collinearity
problems.
The Pearson correlation coefficient is unduly influenced by outliers, unequal variances, nonnormality, and
nonlinearities. As a result of these problems, the Spearman correlation coefficient, which is based on the ranks
of the data rather than the actual data, may be a better choice for examining the relationships between
variables.
Finally, the patterns of missing values in multiple regression and correlation analysis can be very complex. As a
result, missing values can be deleted in a pair-wise or a row-wise fashion. If there are only a few observations
with missing values, it might be preferable to use the row-wise deletion, especially for large data sets. The row-
wise deletion procedure omits the entire observation from the analysis.
On the other hand, if the pattern of missing values is randomly dispersed throughout the data and the use of
the row-wise deletion would omit at least 25% of the observations, the pair-wise deletion procedure for missing
values would be a safer way to capture the essence of the relationships among the variables. While this method
appears to make full use of all your data, the resulting correlation matrix may have mathematical and
interpretation difficulties. Mathematically, this correlation matrix may not have a positive determinant. Since
each correlation may be based on a different set of rows, practical interpretations could be difficult, if not
illogical.
The Spearman correlation coefficient measures the monotonic association between two variables in terms of
ranks. It measures whether one variable increases or decreases with another even when the relationship
between the two variables is not linear or bivariate normal. Computationally, each of the two variables is ranked
separately, and the ordinary Pearson correlation coefficient is computed on the ranks. This nonparametric
correlation coefficient is a good measure of the association between two variables when outliers, nonnormality,
nonconstant variance, and nonlinearity may exist between the two variables being investigated.

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Correlation Matrix

Data Structure
The data are entered as two or more variables. An example of data appropriate for this procedure is shown
in the table below. It is assumed that each row gives measurements on the same individual.
Test Scores

Test 1 Test 2 Test 3

45 54 78
87 92 58
55 77 88
44 46 53
73 45
75 66 66
93 46 85
57 78 91
66 58 77
68 53 73
45 68
54 65 65
65
59 66 72
54 83
75 53 82

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Correlation Matrix

Example 1 – Creating a Correlation Matrix

This section presents an example of how to run an analysis of the data contained in the IQ dataset.

Setup
To run this example, complete the following steps:

1 Open the IQ example dataset

• From the File menu of the NCSS Data window, select Open Example Data.
• Select IQ and click OK.

2 Specify the Correlation Matrix procedure options

• Find and open the Correlation Matrix procedure using the menus or the Procedure Navigator.
• The settings for this example are listed below and are stored in the Example 1 settings file. To load
these settings to the procedure window, click Open Example Settings File in the Help Center or File
menu.

Variables Tab
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Correlation Variables ....................................... Test1, Test2, Test3, Test4, Test5, IQ

Missing Value Removal ................................... Row-Wise

Reports Tab
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Show Individual Tables .................................... Checked

Pearson Correlations ....................................... Checked
Spearman Correlations .................................... Checked
Difference ........................................................ Checked
Show Combined Table .................................... Checked
Pearson Correlations ....................................... Checked
Spearman Correlations .................................... Checked
Pearson P-Value.............................................. Checked
Count ............................................................... Checked
Add Cronbach’s Alpha… ................................. Checked

3 Run the procedure

• Click the Run button to perform the calculations and generate the output.

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Correlation Matrix

Individual Reports

Pearson Correlation Report

─────────────────────────────────────────────────────────────────────────
Row-Wise Missing Value Deletion
─────────────────────────────────────────────────────────────────────────
Variables Test1 Test2 Test3 Test4 Test5 IQ
───────────────────────────────────────────────────────────────────────────────────────────
Test1 1.0000 0.1000 -0.2608 0.7539 0.0140 0.2256
Test2 0.1000 1.0000 0.0572 0.7196 -0.2814 0.2407
Test3 -0.2608 0.0572 1.0000 -0.1409 0.3473 0.0741
Test4 0.7539 0.7196 -0.1409 1.0000 -0.1729 0.3714
Test5 0.0140 -0.2814 0.3473 -0.1729 1.0000 -0.0581
IQ 0.2256 0.2407 0.0741 0.3714 -0.0581 1.0000
───────────────────────────────────────────────────────────────────────────────────────────

Coefficient Alpha
────────────────────────────────────────────────────
Cronbach's Alpha 0.4519
Standardized Cronbach's Alpha 0.4785
────────────────────────────────────────────────────
─────────────────────────────────────────────────────────────────────────

Spearman Correlation Report

─────────────────────────────────────────────────────────────────────────
Row-Wise Missing Value Deletion
─────────────────────────────────────────────────────────────────────────
Variables Test1 Test2 Test3 Test4 Test5 IQ
───────────────────────────────────────────────────────────────────────────────────────────
Test1 1.0000 0.0098 -0.3539 0.6517 0.0000 0.2202
Test2 0.0098 1.0000 0.0430 0.6971 -0.3118 0.2303
Test3 -0.3539 0.0430 1.0000 -0.2143 0.3982 0.1238
Test4 0.6517 0.6971 -0.2143 1.0000 -0.1577 0.3772
Test5 0.0000 -0.3118 0.3982 -0.1577 1.0000 -0.0125
IQ 0.2202 0.2303 0.1238 0.3772 -0.0125 1.0000
─────────────────────────────────────────────────────────────────────────

Difference (Pearson - Spearman) Report

─────────────────────────────────────────────────────────────────────────
Row-Wise Missing Value Deletion
─────────────────────────────────────────────────────────────────────────
Variables Test1 Test2 Test3 Test4 Test5 IQ
──────────────────────────────────────────────────────────────────────────────────────────
Test1 0.0000 0.0902 0.0931 0.1022 0.0140 0.0054
Test2 0.0902 0.0000 0.0142 0.0225 0.0304 0.0104
Test3 0.0931 0.0142 0.0000 0.0734 -0.0509 -0.0497
Test4 0.1022 0.0225 0.0734 0.0000 -0.0152 -0.0058
Test5 0.0140 0.0304 -0.0509 -0.0152 0.0000 -0.0455
IQ 0.0054 0.0104 -0.0497 -0.0058 -0.0455 0.0000
─────────────────────────────────────────────────────────────────────────

The above tables display the Pearson Correlation Report, Spearman Correlation Report, and the Difference
Report. Cronbach’s Alpha is displayed at the bottom of the first report.
The Difference report displays the difference between the Pearson and the Spearman correlation
coefficients. The report lets you find those variable pairs for which these two correlation coefficients are
very different. A large difference indicates the presence of outliers, nonlinearity, nonnormality, and the like.
You should investigate scatter plots of pairs of variables with large differences.

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Correlation Matrix

Reliability
Because of the central role of measurement in science, scientists of all disciplines are concerned with the
accuracy of their measurements. Item analysis is a methodology for assessing the accuracy of
measurements that are obtained in the social sciences where precise measurements are often hard to
secure. The accuracy of a measurement may be broken down into two main categories: validity and
reliability. The validity of an instrument refers to whether it accurately measures the attribute of interest.
The reliability of an instrument concerns whether it produces identical results in repeated applications. An
instrument may be reliable but not valid. However, it cannot be valid without being reliable.
The methods described here assess the reliability of an instrument. They do not assess its validity. This
should be kept in mind when using the techniques of item analysis since they address reliability, not validity.
An instrument may be valid for one attribute but not for another. For example, a driver’s license exam may
accurately measure an individual’s ability to drive. However, it does not accurately measure that individual’s
ability to do well in college. Hence the exam is reliable and valid for measuring driving ability. It is reliable
and invalid for measuring success in college.
Several methods have been proposed for assessing the reliability of an instrument. These include the retest
method, alternative-form method, split-halves method, and the internal consistency method. We will focus
on internal consistency here.

Cronbach’s Alpha
Cronbach’s alpha (or coefficient alpha) is the most popular of the internal consistency coefficients. It is
calculated as follows.

𝐾𝐾 ∑𝐾𝐾
𝑖𝑖=1 𝜎𝜎𝑖𝑖𝑖𝑖
𝛼𝛼 = �1 − 𝐾𝐾 �
𝐾𝐾 − 1 ∑𝑖𝑖=1 ∑𝐾𝐾
𝑗𝑗=1 𝜎𝜎𝑖𝑖𝑖𝑖

where K is the number of items (questions) and 𝜎𝜎𝑖𝑖𝑖𝑖 is the estimated covariance between items i and j. Note
the 𝜎𝜎𝑖𝑖𝑖𝑖 is the variance (not standard deviation) of item i.
If the data are standardized by subtracting the item means and dividing by the item standard deviations
before the above formula is used, we obtain the standardized version of Cronbach’s alpha. A little algebra
will show that this is equivalent to the following calculations based directly on the correlation matrix of the
items.
𝐾𝐾𝜌𝜌̅
𝛼𝛼 =
1 + 𝜌𝜌̅ (𝐾𝐾 − 1)

where K is the number of items (variables) and 𝜌𝜌̅ is the average of all the correlations among the K items.
Cronbach’s alpha has several interpretations. It is equal to the average value of alpha coefficients obtained
for all possible combinations of dividing 2K items into two groups of K items each and calculating the two-
half tests. Also, alpha estimates the expected correlation of one instrument with an alternative form
containing the same number of items. Furthermore, alpha estimates the expected correlation between an
actual test and a hypothetical test which may never be written.
Since Cronbach’s alpha is supposed to be a correlation, it should range between -1 and 1. However, it is
possible for alpha to be less than -1 when several of the covariances are relatively large, negative numbers.
In most cases, alpha is positive, although negative values arise occasionally.

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Correlation Matrix

What value of alpha should be achieved? Carmines (1990) stipulates that as a rule, a value of at least 0.8
should be achieved for widely used instruments. An instrument’s alpha value may be improved by either
adding more items or by increasing the average correlation among the items.

Combined Report

Combined Correlation Report

─────────────────────────────────────────────────────────────────────────
Row-Wise Missing Value Deletion
─────────────────────────────────────────────────────────────────────────
Variables Test1 Test2 Test3 Test4 Test5 IQ
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Test1 Pearson Correlation 1.0000 0.1000 -0.2608 0.7539 0.0140 0.2256
Spearman Correlation 1.0000 0.0098 -0.3539 0.6517 0.0000 0.2202
Pearson P-Value 0.7228 0.3478 0.0012 0.9606 0.4187
Count 15 15 15 15 15 15

Test2 Pearson Correlation 0.1000 1.0000 0.0572 0.7196 -0.2814 0.2407

Spearman Correlation 0.0098 1.0000 0.0430 0.6971 -0.3118 0.2303
Pearson P-Value 0.7228 0.8395 0.0025 0.3095 0.3876
Count 15 15 15 15 15 15

Test3 Pearson Correlation -0.2608 0.0572 1.0000 -0.1409 0.3473 0.0741

Spearman Correlation -0.3539 0.0430 1.0000 -0.2143 0.3982 0.1238
Pearson P-Value 0.3478 0.8395 0.6164 0.2046 0.7931
Count 15 15 15 15 15 15

Test4 Pearson Correlation 0.7539 0.7196 -0.1409 1.0000 -0.1729 0.3714

Spearman Correlation 0.6517 0.6971 -0.2143 1.0000 -0.1577 0.3772
Pearson P-Value 0.0012 0.0025 0.6164 0.5378 0.1729
Count 15 15 15 15 15 15

Test5 Pearson Correlation 0.0140 -0.2814 0.3473 -0.1729 1.0000 -0.0581

Spearman Correlation 0.0000 -0.3118 0.3982 -0.1577 1.0000 -0.0125
Pearson P-Value 0.9606 0.3095 0.2046 0.5378 0.8371
Count 15 15 15 15 15 15

IQ Pearson Correlation 0.2256 0.2407 0.0741 0.3714 -0.0581 1.0000

Spearman Correlation 0.2202 0.2303 0.1238 0.3772 -0.0125 1.0000
Pearson P-Value 0.4187 0.3876 0.7931 0.1729 0.8371
Count 15 15 15 15 15 15
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────

Coefficient Alpha
────────────────────────────────────────────────────
Cronbach's Alpha 0.4519
Standardized Cronbach's Alpha 0.4785
────────────────────────────────────────────────────
─────────────────────────────────────────────────────────────────────────

The above report displays the Pearson and Spearman correlations, the significance level of a test of the
Pearson correlation (Pearson P-Value) and count for each pair of variables.

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Correlation Matrix

Heat Map of the Pearson Correlation Matrix

Heat Map of the Pearson Correlation Matrix

─────────────────────────────────────────────────────────────────────────

This report displays a heat map of the correlation matrix. Note that the rows and columns are sorted in the
order suggested by the hierarchical clustering.
This plot allows you to discover various subsets of the variables that seem to be highly correlated within the
subset. You can see that Test1, Test4, and Test2 seem to be highly related. Similarly, Test3 and Test5 seem
to be related.
This plot was suggested by Friendly (2002) and Friendly and Kwan (2003).

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Correlation Matrix

Pearson Eigenvectors Plot(s)

Pearson Eigenvectors Plot(s)

─────────────────────────────────────────────────────────────────────────

This plot displays a scatter plot of PC1 (the first eigenvector) on the horizontal axis and PC2 (the second
eigenvector) on the vertical axis. The number within the parentheses is the percentage of the sum of the
eigenvalues that that is accounted for by the corresponding eigenvector. For example, in this plot, 40% of
the variability in the correlation matrix is accounted for by the first eigenvector and 22% of the variability is
accounted for by the second eigenvector. Thus, the two eigenvectors in this plot account for 62% of the
variation among the correlations.
Note that this plot lets you see which variables to be clustered. In this case, Test3 and Test5 are related as
are Test1, Test2, and Test4. The IQ variable seems to be by itself, although it is somewhat similar to the
second three variables.
This is the same interpretation that we obtained from the heat map, but perhaps it is easier to see subtleties
in this plot.
This plot was suggested by Friendly (2002) and Friendly and Kwan (2003).

Storing the Correlations on the Database

When you specify variables in either the Pearson Correlations or the Spearman Correlations boxes, the
correlation matrix will be stored in those variables during the execution of the program.

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Correlation Matrix

Example 2 – Bartlett’s Sphericity Test

This section presents an example of how to run Bartlett’s Sphericity test of the data contained in the IQ
dataset. Note that Bartlett’s test is only available when Missing Value Removal is set to Row Wise.

Setup
To run this example, complete the following steps:

1 Open the IQ example dataset

• From the File menu of the NCSS Data window, select Open Example Data.
• Select IQ and click OK.

2 Specify the Correlation Matrix procedure options

• Find and open the Correlation Matrix procedure using the menus or the Procedure Navigator.
• The settings for this example are listed below and are stored in the Example 2 settings file. To load
these settings to the procedure window, click Open Example Settings File in the Help Center or File
menu.

Variables Tab
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Correlation Variables ....................................... Test1, Test2, Test3, Test4, Test5, IQ

Missing Value Removal ................................... Row-Wise

Reports Tab
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Show Individual Tables .................................... Checked

Pearson Correlations ....................................... Checked

Eigenvectors Tab
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Pearson Eigenvector Plot(s) ............................ Checked

Show the eigenvalue… .................................... Checked
Eigenvalues and Eigenvectors of… ................. Checked

3 Run the procedure

• Click the Run button to perform the calculations and generate the output.

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Correlation Matrix

Individual Reports

Pearson Correlation Report

─────────────────────────────────────────────────────────────────────────
Row-Wise Missing Value Deletion
─────────────────────────────────────────────────────────────────────────
Variables Test1 Test2 Test3 Test4 Test5 IQ
───────────────────────────────────────────────────────────────────────────────────────────
Test1 1.0000 0.1000 -0.2608 0.7539 0.0140 0.2256
Test2 0.1000 1.0000 0.0572 0.7196 -0.2814 0.2407
Test3 -0.2608 0.0572 1.0000 -0.1409 0.3473 0.0741
Test4 0.7539 0.7196 -0.1409 1.0000 -0.1729 0.3714
Test5 0.0140 -0.2814 0.3473 -0.1729 1.0000 -0.0581
IQ 0.2256 0.2407 0.0741 0.3714 -0.0581 1.0000
─────────────────────────────────────────────────────────────────────────

The above table displays the Pearson Correlation Report.

Pearson Eigenvector Plot

Pearson Eigenvectors Plot(s)

─────────────────────────────────────────────────────────────────────────

This plot displays a scatter plot of PC1 (the first eigenvector) on the horizontal axis and PC2 (the second
eigenvector) on the vertical axis. The number within the parentheses is the percentage of the sum of the
eigenvalues that that is accounted for by the corresponding eigenvector. For example, in this plot, 40% of
the variability in the correlation matrix is accounted for by the first eigenvector and 22% of the variability is
accounted for by the second eigenvector. Thus, the two eigenvectors in this plot account for 62% of the
variation among the correlations.

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Correlation Matrix

Note that this plot lets you see which variables could be clustered. In this case, Test3 and Test5 are related
as are Test1, Test2, and Test4. The IQ variable seems to be by itself, although it is somewhat similar to the
second three variables.

Eigenvalues Report

Eigenvalues of Pearson Correlation Matrix

─────────────────────────────────────────────────────────────────────────
Row-Wise Missing Value Deletion
─────────────────────────────────────────────────────────────────────────
Individual Cumulative Scree
Eigenvector Eigenvalue Percent Percent Plot
─────────────────────────────────────────────────────────────────────────────────
PC1 2.374012 39.57 39.57 ||||||||
PC2 1.297129 21.62 61.19 |||||
PC3 1.109029 18.48 79.67 ||||
PC4 0.779485 12.99 92.66 |||
PC5 0.435845 7.26 99.92 ||
PC6 0.004500 0.08 100.00 |
─────────────────────────────────────────────────────────────────────────────────

Matrix Summary Measures

────────────────────────────────
Log(Det|R) -5.254947
────────────────────────────────

Bartlett Sphericity Test

──────────────────────────────────
Test Statistic 58.68
DF 15
Prob Level 0.000000
──────────────────────────────────
─────────────────────────────────────────────────────────────────────────

The above report displays the Pearson and Spearman correlations, the significance level of a test of the
Pearson correlation (Pearson P-Value) and count for each pair of variables.

Eigenvector
This column gives the label of the eigenvector whose eigenvalue is displayed. Note that you can modify the
label.

Eigenvalue
The eigenvalues. Often, these are used to determine how many eigenvectors to retain. (In this example, we
would retain the first three.)
One rule-of-thumb is to retain those eigenvectors whose eigenvalues are greater than one. The sum of the
eigenvalues is equal to the number of variables. Hence, in this example, the first eigenvector retains the
information contained in 2.37 of the original variables.

Individual and Cumulative Percents

The first column gives the percentage of the total variation in the variables accounted for by this
eigenvector. The second column is the cumulative total of the percentage. Some authors suggest that the
user pick a cumulative percentage, such as 80% or 90%, and keep enough factors to attain this percentage.

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Scree Plot
This is a rough bar plot of the eigenvalues. It enables you to quickly note the relative size of each eigenvalue.
Many authors recommend it as a method of determining how many eigenvectors to plot.
The word scree, first used by Cattell (1966), is usually defined as the rubble at the bottom of a cliff. When
using the scree plot, you must determine which eigenvalues form the “cliff” and which form the “rubble.”
You keep the eigenvectors that make up the cliff. Cattell and Jaspers (1967) suggest keeping those that make
up the cliff plus the first eigenvector of the rubble.

Log(Det|R|)
This is the log (base e) of the determinant of the correlation matrix.

Bartlett Test, DF, Prob Level

This is Bartlett’s sphericity test (Bartlett, 1950) for testing the null hypothesis that the correlation matrix is an
identity matrix (all correlations are zero). If you get a significance level (Prob Level) greater than 0.05, there is
no evidence that any of the correlations are different from zero. The test is valid for large samples (N>150). It
uses a Chi-square distribution with p(p-1)/2 degrees of freedom.
Note that this test is only available when the Missing Value Removal option is set to Row Wise.
The formula for computing this test is:
(11 + 2𝑝𝑝 − 6𝑁𝑁)
𝜒𝜒 2 = Log 𝑒𝑒 |𝑅𝑅|
6

Eigenvectors Report

Eigenvectors of Pearson Correlation Matrix

─────────────────────────────────────────────────────────────────────────
Row-Wise Missing Value Deletion
─────────────────────────────────────────────────────────────────────────
Eigenvectors
─────────────
Variables PC1 PC2
────────────────────────────────────────
Test1 -0.4608 -0.0060
Test2 -0.4575 0.1575
Test3 0.1720 0.7261
Test4 -0.6263 0.1161
Test5 0.2251 0.5656
IQ -0.3253 0.3386
─────────────────────────────────────────────────────────────────────────

The eigenvectors show the direction of each factor (principal component) after the correlation matrix is
suitably scaled and rotated. These are the values that are plotted in the Eigenvector plots shown above.

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