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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 that 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).

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.

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Correlation Matrix

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.

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

Procedure Options
This section describes the options available in this procedure.

Variables Tab
Specify the variables on which to run the analysis.

Data Variables
Correlation Variables
Specify the variables whose correlations are to be formed. Only numeric data values are analyzed.
Partial Variables
An optional set of variables that are to be “partialled out” of the correlation matrix. The influence of these
variables is removed from the remaining variables using linear regression. The correlations that are formed are the
partial correlations.
For the Pearson-type correlations, the resulting matrix is the same that would be formed if the regular variables
were regressed on the partial variables, the residuals were stored, and the correlation matrix of these residuals was
formed.

Missing Values
Missing Value Removal
This option indicates how you want the program to handle missing values.

• Pair-wise
Pair-wise removal of missing values. Each correlation is based on all pairs of data values in which no missing
values occur. Missing values occurring in other variables do not influence this calculation. Note that although
this method appears to make full use of all your data, the resulting correlation matrix is difficult to analyze.
Mathematically, it may not have a positive determinant. Practically, each correlation may be based on a
different set of rows, making it difficult to compare correlations.

• Row-wise
Row-wise removal of missing values. In each row, if a missing value occurs in any of the variables specified,
that row of data is ignored in the calculation of all correlations.

Reports Tab
These options specify the reports.

Select Reports
Show Individual Tables
Check this option to display a separate matrix for each statistic checked. After activating this option, you must
specify which tables you would like to display.
Pearson Correlations…Count
Check all items you would like it to be displayed as separate tables.

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Show Combined Tables

Check this option to display a single table containing the selected statistics. After activating this option, you must
specify which items you would like to display in the table.
Pearson Correlations…Count
Check all items you would like it to be displayed in the combined table.
Add Cronbach’s Alpha beneath the Pearson Correlation matrix
Check this option to display Cronbach's Alpha right after the Pearson correlation table.

Report Options Tab

Variable Names
This option lets you select whether to display variable names, variable labels, or both.
Precision
Specify the precision of numbers in the report. Single precision will display seven-place accuracy, while double
precision will display thirteen-place accuracy.
Max Distance Items
This option specifies the maximum size of a distance matrix that will be displayed in the Distance Section report.
Distance matrices with more items than this will not be displayed.
This option is here because for large datasets, the distance matrix may be very large.
Max Linkage Clusters
The Linkage Report can be long if the results for all links are printed. This parameter allows you to limit the
number of links displayed so that only meaningful values are printed.

Decimal Places
Decimal Places (Correlation, …, Percentages)
These options specify the number of decimal places directly or using an Auto function.
If one of the Auto options is used, the ending zero digits are not shown.
Your choice here will not affect calculations; it will only affect the format of the output.

Auto
If one of the 'Auto' options is selected, the ending zero digits are not shown. For example, if 'Auto (Up to 7)' is
chosen,
0.0500 is displayed as 0.05
1.314583689 is displayed as 1.314584

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Table Formatting
Column Justification
Specify whether data columns in the tables will be left or right justified.
Column Widths
Specify how the widths of columns in the tables will be determined.
The options are

• Autosize to Minimum Widths

Each data column is individually resized to the smallest width required to display the data in the column. This
usually results in columns with different widths. This option produces the most compact table possible,
displaying the most data per page.

• Autosize to Equal Minimum Width

The smallest width of each data column is calculated and then all columns are resized to the width of the
widest column. This results in the most compact table possible where all data columns have the same with.
This is the default setting.

• Custom (User-Specified)
Specify the widths (in inches) of the columns directly instead of having the software calculate them for you.
Column Widths (Single Value or List)
Enter one or more values for the widths (in inches) of columns in the tables.

• Single Value
If you enter a single value, that value will be used as the width for all data columns in the table.

• List of Values
Enter a list of values separated by spaces corresponding to the widths of each column. The first value is used
for the width of the first data column, the second for the width of the second data column, and so forth. Extra
values will be ignored. If you enter fewer values than the number of columns, the last value in your list will
be used for the remaining columns.
Type the word "Autosize" for any column to cause the program to calculate it's width for you. For example,
enter "1 Autosize 0.7" to make column 1 be 1 inch wide, column 2 be sized by the program, and column 3 be
0.7 inches wide.
Wrap Column Headings onto Two Lines
Check this option to make column headings wrap onto two lines. Use this option to condense your table when
your data are spaced too far apart because of long column headings.

Heat Maps Tab

Specify the six heat maps that can be displayed.

Clustering Options
Max Clusters
Specify the maximum number of clusters allowed in the heat map and reports.

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Clustering Method
This option specifies which of the nine possible clustering techniques is used. These methods were described
earlier. Select ‘None’ if you want to omit clustering and display the matrix in its original order.
Alpha
Only displayed when the Flexible Strategy method is selected. Specifies the values of α i and α j . You may enter
a number or the letters “NI/NK.” The “NI/NK” will cause this constant to be calculated and used as it is in the
Centroid and Group Average methods.
Beta
Only displayed when the Flexible Strategy method is selected. Specifies the values of β . You may enter a
number between -1 and 1 or the letters “NIJ/NK.” The “NIJ/NK” will cause this constant to be calculated and
used as it is in the Centroid method.
Gamma
Only displayed when the Flexible Strategy method is selected. Specifies the values of γ . You may enter any
number.

Heat Maps
Diagonal Elements
Specify whether the diagonal elements of heat map are colored or missing.
The choices are

• Set as missing
The diagonal elements will be set to missing values. These elements are displayed using the missing value
color of the heat map.

• Set to 1
The diagonal elements will be shown with the color associated with a value of 1.
Select Heat Maps (Pearson Correlations, …, Squared Spearman Correlations)
Check the boxes corresponding to the heat maps to be displayed. For both the Pearson and the Spearman
correlation matrices, you can choose to display heat maps of the regular correlation matrix, the absolute values of
the correlation matrices, and the squared values of the correlations.
Heat Map Format Buttons
Click the format button to change the heat map settings of the two correlation matrices shown directly above.
Edit During Run
Checking this option will cause the clustered heat map format window to appear when the procedure is run. This
allows you to modify the format of the graph with the actual data.
Cluster Analysis Reports
These options let you select which cluster analysis reports you want displayed for each heat map selected.

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Eigenvectors Tab
These options control the eigenvector plots and reports.
Number of Eigenvectors
Specify the number of eigenvectors (principal components) displayed in the Eigenvector Report. This is also the
number of eigenvectors (principal components) plotted. Each unique pair of eigenvectors will be plotted on a
separate scatter plot.
Usually, 2 or 3 is all you will need.
Eigenvector Label
Specify the letters, word, or phrase to be used as the labels of the eigenvectors in the plots and reports.
For example, if you enter 'PC', the labels would be PC1, PC2, PC3, etc.
Eigenvector Plots
Select which eigenvector plots you want displayed. A separate plot is displayed for each pair of eigenvectors.
Eigenvector Plots
Select which eigenvector plots you want displayed. A separate plot is displayed for each pair of eigenvectors.
Show the eigenvalue percentage in the eigenvector labels
If checked, the eigenvalue percentage will be added to the eigenvector labels on the eigenvector plots.
For example, if checked, the labels would appear as PC1(61%), PC2(20%), PC3(11%).
Scatter Plot Format Button
Click the format button to change the scatter plot settings of the eigenvector plots.
Edit During Run
Checking this option will cause the corresponding format window to appear when the procedure is run. This
allows you to modify the format of the graph while viewing the actual data.
Eigenvalue and Eigenvector Reports
Select which eigenvalue and eigenvector reports you want displayed. This report includes Bartlett’s Sphericity
Test when the Missing Value Removal option has been set to ‘Row wise’.

Storage Tab
Specify if and where the correlation matrices are to be stored.

Data Storage Variables

Pearson Correlations
Specifies columns to receive the Pearson correlation (or partial correlation) matrix.
If you leave this option blank, the matrix is not saved.
If columns are specified, the correlation matrix will be automatically saved into these columns. You must specify
as many columns here as there are in your analysis.

Warning
Existing data will be replaced.

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Spearman Correlations
Specifies columns to receive the Spearman correlation (or partial correlation) matrix.
If you leave this option blank, the matrix is not saved.
If columns are specified, the correlation matrix will be automatically saved into these columns. You must specify
as many columns here as there are in your analysis.

Warning
Existing data will be replaced.

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.
You may follow along here by making the appropriate entries or load the completed template Example 1 by
clicking on Open Example Template from the File menu of the procedure window.

1 Open the IQ dataset.

• From the File menu of the NCSS Data window, select Open Example Data.
• Click on the file IQ.NCSS.
• Click Open.

2 Open the Correlation Matrix window.

• Using the Analysis menu or the Procedure Navigator, find and select the Correlation Matrix procedure.
• On the menus, select File, then New Template. This will fill the procedure with the default template.

3 Specify the variables.

• On the Correlation Matrix window, select the Variables tab.
• Double-click in the Correlation Variables text box. This will bring up the variable selection window.
• Select Test1, Test2, Test3, Test4, Test5, IQ from the list of variables and then click Ok. These variables
will appear in the Correlation Variables box.
• Enter Row Wise in the Missing Value Removal box.

4 Specify the reports.

• On the Correlation Matrix window, select the Reports tab.
• Check the Show Individual Tables box.
o Check the Pearson Correlations box.
o Check the Spearman Correlations box.
o Check the Difference box.
• Check the Show Combined Table box.
o Check the Pearson Correlations box.
o Check the Spearman Correlations box.
o Check the Pearson P-Value box.
o Check the Count box.
• Check the Add Cronbach’s Alpha… box.

5 Run the procedure.

• From the Run menu, select Run Procedure. Alternatively, just click the green Run button.

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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

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.
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.

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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.
 K 
K 
 ∑σ ii 

α= 1− i =1
K −1  K K 


∑∑
i =1 j =1
σ ij 


where K is the number of items (questions) and σ ij is the estimated covariance between items i and j. Note the σ ii
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.
Kρ
α=
1 + ρ( K − 1)

where K is the number of items (variables) and ρ 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.
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

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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

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.

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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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 are 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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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.
You may follow along here by making the appropriate entries or load the completed template Example 2 by
clicking on Open Example Template from the File menu of the procedure window.

1 Open the IQ dataset.

• From the File menu of the NCSS Data window, select Open Example Data.
• Click on the file IQ.NCSS.
• Click Open.

2 Open the Correlation Matrix window.

• Using the Analysis menu or the Procedure Navigator, find and select the Correlation Matrix procedure.
• On the menus, select File, then New Template. This will fill the procedure with the default template.

3 Specify the variables.

• Select the Variables tab.
• Double-click in the Correlation Variables text box. This will bring up the variable selection window.
• Select Test1, Test2, Test3, Test4, Test5, IQ from the list of variables and then click Ok. These variables
will appear in the Correlation Variables box.
• Select Row Wise for the Missing Value Removal method.

4 Specify the reports.

• Select the Reports tab.
• Check the Show Individual Tables box.
o Check the Pearson Correlations box.

5 Specify the Eigenvectors.

• Select the Eigenvectors tab.
• Check the Show Individual Tables box.
o Check the Pearson Correlations box.
• Check the Pearson Eigenvector Plot(s) box.
• Check the Show the eigenvalue percentages … box.
• Check the Eigenvalues and Eigenvectors of the Pearson Correlations … box.

6 Run the procedure.

• From the Run menu, select Run Procedure. Alternatively, just click the green Run button.

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.

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Pearson Eigenvector Plot

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 are 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 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

Eigenvector Individual Cumulative Scree

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 |

Log(Det|R) = -5.254947 Bartlett Sphericity Test = 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.

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Correlation Matrix

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.
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:

χ2 =
(11 + 2p - 6N ) Log R
e
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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