This page shows an example of a principal components analysis with footnotes
explaining the output.  The data used in this example were collected by
Professor James Sidanius, who has generously shared them with us.  You can
download the data set here.

Overview:  The “what” and “why” of principal components analysis

Principal components analysis is a method of data reduction.  Suppose that
you have a dozen variables that are correlated.  You might use principal
components analysis to reduce your 12 measures to a few principal components.
In this example, you may be most interested in obtaining the component
scores (which are variables that are added to your data set) and/or to look at
the dimensionality of the data.  For example, if two components are extracted
and those two components accounted for 68% of the total variance, then we would
say that two dimensions in the component space account for 68% of the variance.
Unlike factor analysis, principal components analysis is not usually used to
identify underlying latent variables.  Hence, the loadings onto the components
are not interpreted as factors in a factor analysis would be.  Principal
components analysis, like factor analysis, can be preformed on raw data, as
shown in this example, or on a correlation or a covariance matrix.  If raw data
is used, the procedure will create the original correlation matrix or covariance
matrix, as specified by the user.  If the correlation matrix is used, the
variables are standardized and the total variance will equal the number of
variables used in the analysis (because each standardized variable has a
variance equal to 1).  If the covariance matrix is used, the variables will
remain in their original metric.  However, one must take care to use variables
whose variances and scales are similar.  Unlike factor analysis, which analyzes
the common variance, the original matrix in a principal components analysis
analyzes the total variance.  Also, principal components analysis assumes that
each original measure is collected without measurement error.

In this example we have included many options, including the original
correlation matrix and the scree plot.  While you may not wish to use all of
these options, we have included them here to aid in the explanation of the
analysis.  We have also created a page of annotated output for a factor analysis
that parallels this analysis.  For general information regarding the
similarities and differences between principal components analysis and factor
analysis, please see our FAQ entitled What are some of the similarities and
differences between principal components analysis and factor analysis?.

proc factor data = "d:m255_sas" corr scree ev method = principal;
var item13 item14 item15 item16 item17 item18 item19 item20 item21 item22 item23 item24 ;
run;

                                           Correlations

                                                                  ITEM13       ITEM14       ITEM15

ITEM13   INSTRUC WELL PREPARED                                   1.00000      0.66146      0.59999
ITEM14   INSTRUC SCHOLARLY GRASP                                 0.66146      1.00000      0.63460
ITEM15   INSTRUCTOR CONFIDENCE                                   0.59999      0.63460      1.00000
ITEM16   INSTRUCTOR FOCUS LECTURES                               0.56626      0.50003      0.50535
ITEM17   INSTRUCTOR USES CLEAR RELEVANT EXAMPLES                 0.57687      0.55150      0.58664
ITEM18   INSTRUCTOR SENSITIVE TO STUDENTS                        0.40898      0.43311      0.45707
ITEM19   INSTRUCTOR ALLOWS ME TO ASK QUESTIONS                   0.28632      0.32041      0.35869
ITEM20   INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS      0.30418      0.31481      0.35568
ITEM21   INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING              0.47553      0.44896      0.50904
ITEM22   I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION      0.33255      0.33313      0.36884
ITEM23   COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS       0.56399      0.56461      0.58233
ITEM24   COMPARED TO OTHER COURSES THIS COURSE WAS               0.45360      0.44281      0.43481

                                           Correlations

                                                                  ITEM16       ITEM17       ITEM18

ITEM13   INSTRUC WELL PREPARED                                   0.56626      0.57687      0.40898
ITEM14   INSTRUC SCHOLARLY GRASP                                 0.50003      0.55150      0.43311
ITEM15   INSTRUCTOR CONFIDENCE                                   0.50535      0.58664      0.45707
ITEM16   INSTRUCTOR FOCUS LECTURES                               1.00000      0.58649      0.40479
ITEM17   INSTRUCTOR USES CLEAR RELEVANT EXAMPLES                 0.58649      1.00000      0.55474
ITEM18   INSTRUCTOR SENSITIVE TO STUDENTS                        0.40479      0.55474      1.00000
ITEM19   INSTRUCTOR ALLOWS ME TO ASK QUESTIONS                   0.33540      0.44930      0.62660
ITEM20   INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS      0.31676      0.41682      0.52055
ITEM21   INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING              0.45245      0.59526      0.55417
ITEM22   I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION      0.36255      0.44976      0.53609
ITEM23   COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS       0.45880      0.61302      0.56950
ITEM24   COMPARED TO OTHER COURSES THIS COURSE WAS               0.42967      0.52058      0.47382

                                           Correlations

                                                                  ITEM19       ITEM20       ITEM21

ITEM13   INSTRUC WELL PREPARED                                   0.28632      0.30418      0.47553
ITEM14   INSTRUC SCHOLARLY GRASP                                 0.32041      0.31481      0.44896
ITEM15   INSTRUCTOR CONFIDENCE                                   0.35869      0.35568      0.50904
ITEM16   INSTRUCTOR FOCUS LECTURES                               0.33540      0.31676      0.45245
ITEM17   INSTRUCTOR USES CLEAR RELEVANT EXAMPLES                 0.44930      0.41682      0.59526
ITEM18   INSTRUCTOR SENSITIVE TO STUDENTS                        0.62660      0.52055      0.55417
ITEM19   INSTRUCTOR ALLOWS ME TO ASK QUESTIONS                   1.00000      0.44647      0.49921
ITEM20   INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS      0.44647      1.00000      0.42479
ITEM21   INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING              0.49921      0.42479      1.00000
ITEM22   I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION      0.48404      0.38297      0.50651
ITEM23   COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS       0.44401      0.40962      0.59751
ITEM24   COMPARED TO OTHER COURSES THIS COURSE WAS               0.37383      0.35722      0.49977


                                           Correlations

                                                                  ITEM22       ITEM23       ITEM24

ITEM13   INSTRUC WELL PREPARED                                   0.33255      0.56399      0.45360
ITEM14   INSTRUC SCHOLARLY GRASP                                 0.33313      0.56461      0.44281
ITEM15   INSTRUCTOR CONFIDENCE                                   0.36884      0.58233      0.43481
ITEM16   INSTRUCTOR FOCUS LECTURES                               0.36255      0.45880      0.42967
ITEM17   INSTRUCTOR USES CLEAR RELEVANT EXAMPLES                 0.44976      0.61302      0.52058
ITEM18   INSTRUCTOR SENSITIVE TO STUDENTS                        0.53609      0.56950      0.47382
ITEM19   INSTRUCTOR ALLOWS ME TO ASK QUESTIONS                   0.48404      0.44401      0.37383
ITEM20   INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS      0.38297      0.40962      0.35722
ITEM21   INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING              0.50651      0.59751      0.49977
ITEM22   I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION      1.00000      0.49317      0.44440
ITEM23   COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS       0.49317      1.00000      0.70464
ITEM24   COMPARED TO OTHER COURSES THIS COURSE WAS               0.44440      0.70464      1.00000

The table above was included in the output because we included the keyword
corr on the proc factor statement.  This table gives the correlations
between the original variables (which are specified on the var
statement).  Before conducting a principal components analysis, you want to
check the correlations between the variables.  If any of the correlations are
too high (say above .9), you may need to remove one of the variables from the
analysis, as the two variables seem to be measuring the same thing.  Another
alternative would be to combine the variables in some way (perhaps by taking the
average).  If the correlations are too low, say below .1, then one or more of
the variables might load only onto one principal component (in other words, make
its own principal component).  This is not helpful, as the whole point of the
analysis is to reduce the number of items (variables).

Initial Factor Method: Principal Components

Prior Communality Estimates: ONE

Eigenvalues of the Correlation Matrix: Total = 12  Average = 1

        Eigenvaluea   Differenceb   Proportionc   Cumulatived

   1    6.24914661    5.01966832        0.5208        0.5208
   2    1.22947829    0.51048923        0.1025        0.6232
   3    0.71898906    0.10585957        0.0599        0.6831
   4    0.61312949    0.05196458        0.0511        0.7342
   5    0.56116491    0.05817383        0.0468        0.7810
   6    0.50299107    0.03172750        0.0419        0.8229
   7    0.47126357    0.08244834        0.0393        0.8622
   8    0.38881523    0.02091149        0.0324        0.8946
   9    0.36790373    0.03970330        0.0307        0.9252
  10    0.32820043    0.01082277        0.0274        0.9526
  11    0.31737767    0.06583773        0.0264        0.9790
  12    0.25153994                      0.0210        1.0000

2 factors will be retained by the MINEIGEN criterion.

a.  Eigenvalue – This column contains the eigenvalues.  The first
component will always account for the most variance (and hence have the highest
eigenvalue), and the next component will account for as much of the left over
variance as it can, and so on.  Hence, each successive component will account
for less and less variance.

b.  Difference – This column gives the differences between the
current and the next eigenvalue.  For example, 6.24 – 1.22 = 5.02.
This gives you a sense of how much change there is in the eigenvalues from one
component to the next.

c.  Proportion – This column gives the proportion of variance
accounted for by each component.  In this example, the first component
accounts for just over half of the variance (approximately 52%).

d.  Cumulative – This column sums up to proportion column, so
that you can see how much variance is accounted for by, say, the first five
components, .7810.

Initial Factor Method: Principal Components

Scree Plot of Eigenvalues
    |
  7 +
    |
    |
    |
    |
    |           1
  6 +
    |
    |
    |
    |
    |
  5 +
    |
    |
E   |
i   |
g   |
e 4 +
n   |
v   |
a   |
l   |
u   |
e 3 +
s   |
    |
    |
    |
    |
  2 +
    |
    |
    |
    |
    |                  2
  1 +
    |
    |                         3      4
    |                                       5      6      7
    |                                                            8      9      0      1      2
    |
  0 +
    -----+------+------+------+------+------+------+------+------+------+------+------+------+----
         0      1      2      3      4      5      6      7      8      9     10     11     12

                                                Number

Initial Factor Method: Principal Components

The scree plot graphs the eigenvalue against the component number.  You can
see these values in the first two columns of the table immediately above.  From
the third component on, you can see that the line is almost flat, meaning the
each successive component is accounting for smaller and smaller amounts of the
total variance.  In general, we are interested in keeping only those principal
components whose eigenvalues are greater than 1.  Components with an eigenvalue
of less than 1 account for less variance than did the original variable (which
had a variance of 1), and so are of little use.  Hence, you can see that the
point of principal components analysis is to redistribute the variance in the
correlation matrix (using the method of eigenvalue decomposition) to
redistribute the variance to first components extracted.

                                         Eigenvectors

                                                                             1e              2e

ITEM13      INSTRUC WELL PREPARED                                      0.29093        -0.40510
ITEM14      INSTRUC SCHOLARLY GRASP                                    0.28953        -0.36765
ITEM15      INSTRUCTOR CONFIDENCE                                      0.29851        -0.27789
ITEM16      INSTRUCTOR FOCUS LECTURES                                  0.27406        -0.25376
ITEM17      INSTRUCTOR USES CLEAR RELEVANT EXAMPLES                    0.32261        -0.09492
ITEM18      INSTRUCTOR SENSITIVE TO STUDENTS                           0.30207         0.33002
ITEM19      INSTRUCTOR ALLOWS ME TO ASK QUESTIONS                      0.25641         0.44823
ITEM20      INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS         0.23709         0.34083
ITEM21      INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING                 0.30536         0.12133
ITEM22      I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION         0.26057         0.32871
ITEM23      COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS          0.32768        -0.03634
ITEM24      COMPARED TO OTHER COURSES THIS COURSE WAS                  0.28550         0.00421

e.  Eigenvectors – These columns give the eigenvectors for each
variable in the principal components analysis.  An eigenvector is a linear
combination of the original variables.  The two components that have been
extracted are orthogonal to one another, and they can be thought of as weights.
These weights are multiplied by each value in the original variable, and those
values are then summed up to yield the eigenvector.  The eigenvectors tell
you about the strength of relationship between the variables and the components.

                                        Factor Pattern

                                                                       Factor1         Factor2

ITEM13      INSTRUC WELL PREPARED                                      0.72729        -0.44919
ITEM14      INSTRUC SCHOLARLY GRASP                                    0.72378        -0.40766
ITEM15      INSTRUCTOR CONFIDENCE                                      0.74622        -0.30813
ITEM16      INSTRUCTOR FOCUS LECTURES                                  0.68511        -0.28137
ITEM17      INSTRUCTOR USES CLEAR RELEVANT EXAMPLES                    0.80647        -0.10525
ITEM18      INSTRUCTOR SENSITIVE TO STUDENTS                           0.75512         0.36593
ITEM19      INSTRUCTOR ALLOWS ME TO ASK QUESTIONS                      0.64098         0.49700
ITEM20      INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS         0.59269         0.37792
ITEM21      INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING                 0.76335         0.13454
ITEM22      I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION         0.65138         0.36448
ITEM23      COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS          0.81914        -0.04029
ITEM24      COMPARED TO OTHER COURSES THIS COURSE WAS                  0.71371         0.00467

f.  Factor1 and Factor2 – This is the component matrix.
This table contains component loadings, which are the correlations between the
variable and the component.  Because these are correlations, possible values
range from -1 to +1.  The columns under these headings are the principal
components that have been extracted.  As you can see, two components were
extracted (the two components that had an eigenvalue greater than 1).  You
usually do not try to interpret the components the way that you would factors
that have been extracted from a factor analysis.  Rather, most people are
interested in the component scores, which are used for data reduction (as
opposed to factor analysis where you are looking for underlying latent
continua).

Variance Explained by Each Factor

   Factor1         Factor2

 6.2491466       1.2294783

                       Final Communality Estimates: Total = 7.478625

    ITEM13          ITEM14          ITEM15          ITEM16          ITEM17          ITEM18

0.73071411      0.69004215      0.65179276      0.54854615      0.66147090      0.70412023

Initial Factor Method: Principal Components

    ITEM19          ITEM20          ITEM21          ITEM22          ITEM23          ITEM24

0.65786784      0.49410612      0.60081090      0.55713785      0.67261205      0.50940384