Principal Components
Principal components is a method of dimension 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. Unlike factor analysis, principal components
analysis is not usually used to identify underlying latent
variables.
1
Wednesday, 08 April 2015
11:13 AM
Principal Components
Principal components is a technique that requires a
large sample size.
Principal components is based on the correlation
matrix of the variables involved, and correlations
usually need a large sample size before they
stabilize.
2
Principal Components
As a rule of thumb, a bare minimum of 10 observations
per variable is necessary to avoid computational
difficulties.
Number of Cases
Prospects
50
very poor
100
poor
200
fair
300
good
500
very good
1000
excellent
Comrey and Lee (1992) A First Course In Factor Analysis
3
Principal Components
In this example we have included many options,
while you may not wish to use all of these options,
we have included them here to aid in the
explanation of the analysis.
4
Principal Components
In this example we examine students assessment of
academic courses. We restrict attention to 12 variables.
Item 13
INSTRUCTOR WELL PREPARED
Item 14
INSTRUCTOR SCHOLARLY GRASP
Item 15
INSTRUCTOR CONFIDENCE
Item 16
INSTRUCTOR FOCUS LECTURES
Item 17
INSTRUCTOR USES CLEAR RELEVANT EXAMPLES
Item 18
INSTRUCTOR SENSITIVE TO STUDENTS
Item 19
INSTRUCTOR ALLOWS ME TO ASK QUESTIONS
Item 20
INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS
Item 21
INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING
Item 22
I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION
Item 23
COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS
Item 24
COMPARED TO OTHER COURSES THIS COURSE WAS
Scored on a five point Likert scale, seven is better.
5
Principal Components
In this example we examine students assessment of
academic courses. We restrict attention to 12 variables.
Scored on a five point Likert scale.
6
Principal Components
Analyze > Dimension Reduction > Factor
7
Principal Components
Select variables 13-24 that is “instructor well
prepared” to “compared to other courses this
course was”. By using the arrow button.
Use the buttons at the side of the screen to set additional options. 8
Principal Components
Use the buttons at the side of the screen to set the
Descriptives employ the Continue button to return to the
main Factor Analysis screen.
9
Principal Components
Use the buttons at the side of the screen to set the
Extraction employ the Continue button to return to the
main Factor Analysis screen.
Select the appropriate method and the eigen value
criteria, set at 1. It is essential to obtain a scree plot.
10
Principal Components
Select the OK button to proceed with the analysis, or Paste
to preserve the syntax.
Syntax
factor
/variables item13 item14 item15 item16 item17 item18 item19 item20 item21 item22 item23 item24
/print initial correlation det kmo repr extraction univariate
/format blank(.30)
/plot eigen
/extraction pc
/method = correlate.
After “/extraction” you can introduce a promax rotation
/rotation promax(4)
11
Principal Components
Descriptive Statistics
Mean
The descriptive statistics
table is output because we
used the univariate option.
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TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Std. Deviation
Analysis N
4.46
.729
1365
4.53
.700
1365
4.45
.732
1365
4.28
.829
1365
4.17
.895
1365
3.93
1.035
1365
4.08
.964
1365
3.78
.909
1365
3.77
.984
1365
3.61
1.116
1365
3.81
.957
1365
3.67
.926
1365
12
Principal Components
Descriptive Statistics
Mean
Mean - These are the
means of the variables
used in the factor analysis.
Are these appropriate for
a Likert scale?
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INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Std. Deviation
Analysis N
4.46
.729
1365
4.53
.700
1365
4.45
.732
1365
4.28
.829
1365
4.17
.895
1365
3.93
1.035
1365
4.08
.964
1365
3.78
.909
1365
3.77
.984
1365
3.61
1.116
1365
3.81
.957
1365
3.67
.926
1365
13
Principal Components
Descriptive Statistics
Mean
Std. Deviation - These are
the standard deviations of
the variables used in the
factor analysis.
Are these appropriate for
a Likert scale?
INSTRUC WELL
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GRASP
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CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Std. Deviation
Analysis N
4.46
.729
1365
4.53
.700
1365
4.45
.732
1365
4.28
.829
1365
4.17
.895
1365
3.93
1.035
1365
4.08
.964
1365
3.78
.909
1365
3.77
.984
1365
3.61
1.116
1365
3.81
.957
1365
3.67
.926
1365
14
Principal Components
Descriptive Statistics
Mean
Analysis N - This is the
number of cases used in
the factor analysis.
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Std. Deviation
Analysis N
4.46
.729
1365
4.53
.700
1365
4.45
.732
1365
4.28
.829
1365
4.17
.895
1365
3.93
1.035
1365
4.08
.964
1365
3.78
.909
1365
3.77
.984
1365
3.61
1.116
1365
3.81
.957
1365
3.67
.926
1365
15
Principal Components
The correlation matrix table was included in the
output because we included the correlation option.
This table gives the correlations between the original
variables (which were specified). 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 0.9), 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).
16
Principal Components
If the correlations are too low, say below 0.1, 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).
17
Principal Components
The correlation matrix is extremely large.
Correlation Matrixa
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TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
INSTRUCTO
R ALLOWS
ME TO ASK
QUESTIONS
INSTRUCTOR
AWARE OF
STUDENTS
UNDERSTAN
DING
I AM
SATISFIED
WITH
STUDENT
PERFORMAN
CE
EVALUATION
COMPARED
TO OTHER
INSTRUCTO
RS, THIS
INSTRUCTO
R IS
COMPARED
TO OTHER
COURSES
THIS
COURSE
WAS
INSTRUC
SCHOLARLY
GRASP
INSTRUC
TOR
CONFIDE
NCE
INSTRUCT
OR FOCUS
LECTURES
INSTRUCT
OR USES
CLEAR
RELEVANT
EXAMPLES
1.000
.661
.600
.566
.577
.409
.286
.304
.476
.333
.564
.454
.661
1.000
.635
.500
.552
.433
.320
.315
.449
.333
.565
.443
.600
.635
1.000
.505
.587
.457
.359
.356
.509
.369
.582
.435
.566
.500
.505
1.000
.586
.405
.335
.317
.452
.363
.459
.430
.577
.552
.587
.586
1.000
.555
.449
.417
.595
.450
.613
.521
.409
.433
.457
.405
.555
1.000
.627
.521
.554
.536
.569
.474
.286
.320
.359
.335
.449
.627
1.000
.446
.499
.484
.444
.374
.304
.315
.356
.317
.417
.521
.446
1.000
.425
.383
.410
.357
.476
.449
.509
.452
.595
.554
.499
.425
1.000
.507
.598
.500
.333
.333
.369
.363
.450
.536
.484
.383
.507
1.000
.493
.444
.564
.565
.582
.459
.613
.569
.444
.410
.598
.493
1.000
.705
.454
.443
.435
.430
.521
.474
.374
.357
.500
.444
.705
1.000
INSTRUC
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PREPARED
Correlation
INSTRUCTOR
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TO
STUDENTS
INSTRUCT
OR IS
ACCESSIB
LE TO
STUDENTS
OUTSIDE
CLASS
a. Determinant = .002
18
Principal Components
The correlation matrix is extremely large.
Correlation Matrixa
INSTRUCT
OR FOCUS
LECTURES
INSTRUCT
OR USES
CLEAR
RELEVANT
EXAMPLES
INSTRUCTOR
SENSITIVE
TO
STUDENTS
INSTRUCTO
R ALLOWS
ME TO ASK
QUESTIONS
INSTRUCT
OR IS
ACCESSIB
LE TO
STUDENTS
OUTSIDE
CLASS
.600
.566
.577
.409
.286
.304
1.000
.635
.500
.552
.433
.320
.315
.600
.635
1.000
.505
.587
.457
.359
.356
.566
.500
.505
1.000
.586
.405
.335
.317
.577
.552
.587
.586
1.000
.555
.449
.417
.409
.433
.457
.405
.555
1.000
.627
.521
.286
.320
.359
.335
.449
.627
1.000
.446
.304
.315
.356
.317
.417
.521
.446
1.000
INSTRUC
SCHOLARLY
GRASP
INSTRUC
TOR
CONFIDE
NCE
1.000
.661
.661
INSTRUC
WELL
PREPARED
Correlation
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
19
.476
.449
.509
.452
.595
.554
.499
.425
Principal Components
Kaiser-Meyer-Olkin Measure of Sampling Adequacy
This measure varies between 0 and 1, and values closer
to 1 are better. A value of 0.6 is a suggested minimum.
KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Bartlett' s Test of
Sphericity
Approx. Chi-Square
df
Sig .
.934
8676.712
66
.000
20
Principal Components
Bartlett's Test of Sphericity - This tests the null
hypothesis that the correlation matrix is an identity
matrix. An identity matrix is matrix in which all of the
diagonal elements are 1 and all off diagonal elements
are 0. You want to reject this null hypothesis.
KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Bartlett' s Test of
Sphericity
Approx. Chi-Square
df
Sig .
.934
8676.712
66
.000
21
Principal Components
Taken together, these tests provide a minimum
standard, which should be passed before a principal
components analysis (or a factor analysis) should be
conducted.
KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Bartlett' s Test of
Sphericity
Approx. Chi-Square
df
Sig .
.934
8676.712
66
.000
22
Principal Components
Communalities
Initial
Communalities - This is the
proportion of each variable's
variance that can be explained
by the principal components
(e.g. the underlying latent
continua).
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Extraction
1.000
.731
1.000
.690
1.000
.652
1.000
.549
1.000
.661
1.000
.704
1.000
.658
1.000
.494
1.000
.601
1.000
.557
1.000
.673
1.000
.509
Extraction Method: Principal Component Analysis.
23
Principal Components
Communalities
Initial
Initial - By definition, the
initial
value
of
the
communality in a principal
components analysis is 1.
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INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Extraction
1.000
.731
1.000
.690
1.000
.652
1.000
.549
1.000
.661
1.000
.704
1.000
.658
1.000
.494
1.000
.601
1.000
.557
1.000
.673
1.000
.509
Extraction Method: Principal Component Analysis.
24
Principal Components
Communalities
Initial
Extraction - The values in this
column indicate the proportion
of each variable's variance that
can be explained by the
principal components. Variables
with high values are well
represented in the common
factor space, while variables
with low values are not well
represented. (In this example,
we don't have any particularly
low values.)
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INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
Extraction
1.000
.731
1.000
.690
1.000
.652
1.000
.549
1.000
.661
1.000
.704
1.000
.658
1.000
.494
1.000
.601
1.000
.557
1.000
.673
1.000
.509
Extraction Method: Principal Component Analysis.
25
Principal Components
Component - There are as many components extracted during a
principal components analysis, as there are variables that are put
into it. In our example, we used 12 variables (item13 through
item24), so we have 12 components.
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
Total
6.249
1.229
.719
.613
.561
.503
.471
.389
.368
.328
.317
.252
Initial Eigenvalues
% of Variance
Cumulative %
52.076
52.076
10.246
62.322
5.992
68.313
5.109
73.423
4.676
78.099
4.192
82.291
3.927
86.218
3.240
89.458
3.066
92.524
2.735
95.259
2.645
97.904
2.096
100.000
Extraction Method: Principal Component Analysis.
Extraction Sums of Squared Loading s
Total
% of Variance
Cumulative %
6.249
52.076
52.076
1.229
10.246
62.322
26
Principal Components
Initial eigen values - eigen values are the variances of the principal
components. Because we conducted our principal components
analysis on the correlation matrix, the variables are standardized,
which means that the each variable has a variance of 1, and the
total variance is equal to the number of variables used in the
analysis, in this case, 12.
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
Total
6.249
1.229
.719
.613
.561
.503
.471
.389
.368
.328
.317
.252
Initial Eigenvalues
% of Variance
Cumulative %
52.076
52.076
10.246
62.322
5.992
68.313
5.109
73.423
4.676
78.099
4.192
82.291
3.927
86.218
3.240
89.458
3.066
92.524
2.735
95.259
2.645
97.904
2.096
100.000
Extraction Sums of Squared Loading s
Total
% of Variance
Cumulative %
6.249
52.076
52.076
1.229
10.246
62.322
Extraction Method: Principal Component Analysis.
27
Principal Components
Initial eigen values - Total - This column contains the eigen values.
The first component will always account for the most variance (and
hence have the highest eigen value), 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.
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
Total
6.249
1.229
.719
.613
.561
.503
.471
.389
.368
.328
.317
.252
Initial Eigenvalues
% of Variance
Cumulative %
52.076
52.076
10.246
62.322
5.992
68.313
5.109
73.423
4.676
78.099
4.192
82.291
3.927
86.218
3.240
89.458
3.066
92.524
2.735
95.259
2.645
97.904
2.096
100.000
Extraction Sums of Squared Loading s
Total
% of Variance
Cumulative %
6.249
52.076
52.076
1.229
10.246
62.322
Extraction Method: Principal Component Analysis.
28
Principal Components
Initial eigen values - % of Variance - This column
contains the percent of variance accounted for by each
principal component (6.249/12 = 0.52).
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
Total
6.249
1.229
.719
.613
.561
.503
.471
.389
.368
.328
.317
.252
Initial Eigenvalues
% of Variance
Cumulative %
52.076
52.076
10.246
62.322
5.992
68.313
5.109
73.423
4.676
78.099
4.192
82.291
3.927
86.218
3.240
89.458
3.066
92.524
2.735
95.259
2.645
97.904
2.096
100.000
Extraction Sums of Squared Loading s
Total
% of Variance
Cumulative %
6.249
52.076
52.076
1.229
10.246
62.322
Extraction Method: Principal Component Analysis.
29
Principal Components
Initial eigen values - Cumulative % - This column contains the
cumulative percentage of variance accounted for by the current
and all preceding principal components. For example, the second
row shows a value of 62.322. This means that the first two
components together account for 62.322% of the total variance.
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
Total
6.249
1.229
.719
.613
.561
.503
.471
.389
.368
.328
.317
.252
Initial Eigenvalues
% of Variance
Cumulative %
52.076
52.076
10.246
62.322
5.992
68.313
5.109
73.423
4.676
78.099
4.192
82.291
3.927
86.218
3.240
89.458
3.066
92.524
2.735
95.259
2.645
97.904
2.096
100.000
Extraction Sums of Squared Loading s
Total
% of Variance
Cumulative %
6.249
52.076
52.076
1.229
10.246
62.322
Extraction Method: Principal Component Analysis.
30
Principal Components
Extraction Sums of Squared Loadings - The three columns in this
half of the table exactly reproduce the values given on the same
row on the left side of the table. The number of rows reproduced
on the right side of the table is determined by the number of
principal components whose eigen values are 1 or greater.
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
Total
6.249
1.229
.719
.613
.561
.503
.471
.389
.368
.328
.317
.252
Initial Eigenvalues
% of Variance
Cumulative %
52.076
52.076
10.246
62.322
5.992
68.313
5.109
73.423
4.676
78.099
4.192
82.291
3.927
86.218
3.240
89.458
3.066
92.524
2.735
95.259
2.645
97.904
2.096
100.000
Extraction Sums of Squared Loading s
Total
% of Variance
Cumulative %
6.249
52.076
52.076
1.229
10.246
62.322
Totally agree
Extraction Method: Principal Component Analysis.
31
Principal Components
The scree plot graphs the eigen value against the
component number.
32
Principal Components
In general, we are interested in keeping only those
principal components whose eigen values are greater
than 1 (we set this value).
33
Principal Components
Component Matrixa
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. It is
usual to not report any
correlations that are less
than |.3|. As shown.
Component
1
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
2
.727
-.449
.724
-.408
.746
-.308
.685
.806
.755
.366
.641
.497
.593
.378
.763
.651
.364
.819
.714
Extraction Method: Principal Component Analysis.
a. 2 components extracted.
34
Principal Components
Component Matrixa
Component
Component - The columns under
this heading are the principal
components
that
have
been
extracted. As you can see by the
footnote provided by SPSS, two
components were extracted (the
two components that had an eigen
value greater than 1).
1
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
2
.727
-.449
.724
-.408
.746
-.308
.685
.806
.755
.366
.641
.497
.593
.378
.763
.651
.364
.819
.714
Extraction Method: Principal Component Analysis.
a. 2 components extracted.
35
Principal Components
Component Matrixa
You usually do not try to interpret
the components in 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 dimension reduction (as
opposed to factor analysis where
you are looking for underlying
latent continua).
Component
1
INSTRUC WELL
PREPARED
INSTRUC SCHOLARLY
GRASP
INSTRUCTOR
CONFIDENCE
INSTRUCTOR FOCUS
LECTURES
INSTRUCTOR USES
CLEAR RELEVANT
EXAMPLES
INSTRUCTOR SENSITIVE
TO STUDENTS
INSTRUCTOR ALLOWS
ME TO ASK QUESTIONS
INSTRUCTOR IS
ACCESSIBLE TO
STUDENTS OUTSIDE
CLASS
INSTRUCTOR AWARE OF
STUDENTS
UNDERSTANDING
I AM SATISFIED WITH
STUDENT
PERFORMANCE
EVALUATION
COMPARED TO OTHER
INSTRUCTORS, THIS
INSTRUCTOR IS
COMPARED TO OTHER
COURSES THIS
COURSE WAS
2
.727
-.449
.724
-.408
.746
-.308
.685
.806
.755
.366
.641
.497
.593
.378
.763
.651
.364
.819
.714
Extraction Method: Principal Component Analysis.
a. 2 components extracted.
36
Principal Components
For a component plot employ the Rotation option
37
Principal Components
Its always wise to plot your results. Note the clusters.
38
Principal Components
The advantages in adopting Factor Analysis as opposed to
Principal Components Analysis for component evaluation
and/or instrumental variable estimation purposes are
reported (Travaglini 2011). Under Factor Analysis, the
scores are in fact shown to produce more efficient slope
estimators when utilized as regressor’s and/or instruments.
Together with the factors they also exhibit a higher
degree of consistency even for large sample dimensions.
Finally under Factor Analysis, dimension reduction is
definitely more stringent, greatly facilitating the search
and identification of the common components of the
available dataset (Travaglini 2011).
39
Principal Components
Principal Components Analysis and Factor Analysis share
the search for a common structure characterized by few
common components, usually known as “scores” that
determine the observed variables contained in matrix X.
However, the two methods differ on the characterization
of the scores as well as on the technique adopted for
selecting their true number. In Principal Components
Analysis the scores are the orthogonalised principal
components obtained through rotation, while in Factor
Analysis the scores are latent variables determined by
unobserved factors and loadings which involve idiosyncratic
error terms. The dimension reduction of X implemented by
each method produces a set of fewer homogenous variables
– the true scores – which contain most of the model’s
40
information.
Principal Components
For a detailed discussion and a brief numerical derivation see Velicer
and Jackson (1990), who also give an extensive bibliography.
“Should one do a component analysis? The choice is not obvious,
because the two broad classes of procedures serve a similar purpose,
and share many important mathematical characteristics. Despite many
textbooks describing common factor analysis as the preferred
procedure, principal component analysis has been the most widely
applied.”
Velicer, W.F. and Jackson, D.N. 1990 “Component Analysis Versus
Common Factor Analysis: Some Issues In Selecting An Appropriate
Procedure” Multivariate Behavioral Research 25(1) 1-28.
41
Principal Components
After some mathematics!
“An examination of the algebraic representations of the two methods
of analysis has served to highlight the differences between them.
However, when the same number of components or factors are
extracted, the results from different types of component or factor
analysis procedures typically yield highly similar results. Discrepancies
are rarely, if ever, of any practical importance in subsequent
interpretations.”
Velicer, W.F. and Jackson, D.N. 1990 “Component Analysis Versus
Common Factor Analysis: Some Issues In Selecting An Appropriate
Procedure” Multivariate Behavioral Research 25(1) 1-28.
42
Principal Components
Summary
Principal Components is used to help understand the
covariance structure in the original variables and/or to
create a smaller number of variables using this structure.
Factor Analysis like principal components is used to
summarise the data covariance structure in a smaller
number of dimensions. The emphasis is the identification of
underlying “factors” that might explain the dimensions
associated with large data variability.
43
Similarities
Principal Components Analysis and Factor Analysis have
these assumptions in common:
Measurement scale is interval or ratio level.
Random sample - at least 5 observations per observed
variable and at least 100 observations.
Larger sample sizes recommended for more stable
estimates, 10-20 observations per observed variable.
44
Similarities
Principal Components Analysis and Factor Analysis have
these assumptions in common:
Over sample to compensate for missing values
Linear relationship between observed variables
Normal distribution for each observed variable
Each pair of observed variables has a bivariate normal
distribution
Are both variable reduction techniques. If communalities
45
are large, close to 1.00, results could be similar.
Similarities
Principal Components Analysis assumes the absence of
outliers in the data.
Factor Analysis assumes a multivariate normal distribution
when using Maximum Likelihood extraction method.
46
Differences
Principal Component Analysis
Exploratory Factor Analysis
Principal Components retained account for a
maximal amount of variance of observed
variables
Factors account for common variance in the
data
Analysis decomposes correlation matrix
Analysis decomposes adjusted correlation
matrix
Ones on the diagonals of the correlation
matrix
Diagonals of correlation matrix adjusted with
unique factors
Minimizes sum of squared perpendicular
distance to the component axis
Estimates factors which influence responses
on observed variables
Component scores are a linear combination of
the observed variables weighted by
eigenvectors
Observed variables are linear combinations of
the underlying and unique factors
47
SPSS Tips
Now you should go and try for yourself.
Each week our cluster (5.05) is booked for 2 hours after this session.
This will enable you to come and go as you please.
Obviously other timetabled sessions for this module take precedence.
48