Factor Analysis

1. 1
Factor Analysis
Factor analysis attempts to bring inter-correlated
variables together under more general, underlying
variables.
More specifically, the goal of factor analysis is to
reduce “the dimensionality of the original space and
to give an interpretation to the new space, spanned
by a reduced number of new dimensions which are
supposed to underlie the old ones” (Rietveld & Van
Hout 1993:254).
Rietveld, T. & Van Hout, R. (1993). Statistical Techniques for the Study of Language and
Language Behaviour. Berlin – New York: Mouton de Gruyter.
Friday, November 11, 2016 05:15 PM

2. 2
Factor Analysis
Or to explain the variance in the observed variables
in terms of underlying latent factors” (Habing
2003).
Thus, factor analysis offers not only the possibility
of gaining a clear view of the data, but also the
possibility of using the output in subsequent
analyses (Field 2000; Rietveld & Van Hout 1993).
Field, A. (2000). Discovering Statistics using SPSS for Windows. London – Thousand Oaks
– New Delhi: Sage publications.
Rietveld, T. & Van Hout, R. (1993). Statistical Techniques for the Study of Language and
Language Behaviour. Berlin – New York: Mouton de Gruyter.
Friday, November 11, 2016 05:15 PM

3. 3
Factor Analysis
The starting point of factor analysis is a correlation
matrix, in which the inter-correlations between the
studied variables are presented. The dimensionality
of this matrix can be reduced by “looking for
variables that correlate highly with a group of other
variables, but correlate very badly with variables
outside of that group” (Field 2000: 424). These
variables with high inter-correlations could well
measure one underlying variable, which is called a
‘factor’.
Field, A. (2000). Discovering Statistics using SPSS for Windows. London – Thousand Oaks
– New Delhi: Sage publications.
Friday, November 11, 2016 05:15 PM

4. 4
Factor Analysis
Factor analysis is a method of dimension reduction.
It does this by seeking underlying unobservable
(latent) variables that are reflected in the observed
variables (manifest variables).
Friday, November 11, 2016 05:15 PM

5. 5
Factor Analysis
There are many different methods that can be used
to conduct a factor analysis
There are many different types of rotations that
can be done after the initial extraction of factors.
You also need to determine the number of factors
that you want to extract.

6. 6
Factor Analysis
Given the number of factor analytic techniques and
options, it is not surprising that different analysts
could reach very different results analysing the
same data set.

7. 7
Factor Analysis
However, all analysts are looking for a simple
structure.
Simple structure is a pattern of results such that
each variable loads highly onto one and only one
factor.

8. 8
Factor Analysis
Factor analysis is a technique that requires a large
sample size.
Factor analysis is based on the correlation matrix of
the variables involved, and correlations usually need
a large sample size before they stabilize.

9. 9
Factor Analysis
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

10. 10
Factor Analysis
In this example I have included many options, while
you may not wish to use all of these options, I have
included them here to aid in the explanation of the
analysis.

11. 11
Factor Analysis
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.

12. 12
Factor Analysis
In this example we examine students assessment of
academic courses. We restrict attention to 12 variables.
Scored on a five point Likert scale.

13. 13
Factor Analysis
Analyze > Dimension Reduction > Factor

14. 14
Factor Analysis
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.

15. 15
Factor Analysis
Use the buttons at the side of the previous screen to set
the Descriptives. Employ the Continue button to return to
the main Factor Analysis screen.
Note the request for a determinant.

16. 16
Factor Analysis
Use the buttons at the side of the main screen to set the
Extraction. Employ the Continue button to return to the
main Factor Analysis screen.
Note the request for Principal axis factoring, 3 factors
and a scree plot.

17. 17
Factor Analysis
Use the buttons at the side of the main screen to set the
Rotation (Varimax). Employ the Continue button to return
to the main Factor Analysis screen.

18. 18
Factor Analysis
Varimax rotation tries to maximize the variance of each of
the factors, so the total amount of variance accounted for
is redistributed over the three extracted factors.

19. 19
Factor Analysis
Select the OK button to proceed with the analysis, or
Paste to preserve the syntax.
Syntax for varimax and 3 factors, alternatives promax and 2
factor
/variables item13 item14 item15 item16 item17 item18 item19 item20
item21 item22 item23 item24
/print initial det kmo repr extraction rotation fscore univaratiate
/format blank(.30)
/plot eigen rotation
/criteria factors(3)
/extraction paf
/rotation varimax
/method = correlation.

20. 20
Factor Analysis
The descriptive statistics
table is output because we
used the univariate option.
Mean - These are the means
of the variables used in the
factor analysis.
Are they meaningful for a
Likert scale!
Norman, G. (2010). Likert scales, levels of
measurement and the “laws” of statistics.
Advances in health sciences education, 15(5),
625-632.
Descriptive Statistics
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
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
Mean Std. Deviation Analysis N

21. 21
Factor Analysis
The descriptive statistics
table is output because we
used the univariate option.
Std. Deviation - These
are the standard
deviations of the variables
used in the factor
analysis.
Are they meaningful for a
Likert scale!
Descriptive Statistics
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
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
Mean Std. Deviation Analysis N

22. 22
Factor Analysis
The descriptive statistics
table is output because we
used the univariate option.
Analysis N - This is the
number of cases used in
the factor analysis.
Note N is 1365.
Descriptive Statistics
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
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
Mean Std. Deviation Analysis N

23. 23
Factor Analysis
The correlation matrix is included in the output
because we used the determinant option.
All we want to see in this table is that the
determinant is not 0.
If the determinant is 0, then there will be
computational problems with the factor analysis, and
SPSS may issue a warning message or be unable to
complete the factor analysis.
Correlation Matrixa
Determinant = .002a.

24. 24
Factor Analysis
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
.934
8676.712
66
.000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity

25. 25
Factor Analysis
Bartlett's Test of Sphericity (see the ANOVA
slides) - 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 (indicates a
lack of correlation). You want to reject this null
hypothesis.
KMO and Bartlett's Test
.934
8676.712
66
.000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity

26. 26
Factor Analysis
Taken together, these tests provide a minimum
standard, which should be passed before a factor
analysis (or a principal components analysis) should be
conducted.
KMO and Bartlett's Test
.934
8676.712
66
.000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity

27. 27
Factor Analysis
Communalities - This
is the proportion of
each variable's
variance that can be
explained by the
factors (e.g., the
underlying latent
continua).
Communalities
.564 .676
.551 .619
.538 .592
.447 .468
.585 .623
.572 .679
.456 .576
.326 .369
.516 .549
.397 .444
.662 .791
.526 .632
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
Initial Extraction
Extraction Method: Principal Axis Factoring.

28. 28
Factor Analysis
Initial - With principal
factor axis factoring, the
initial values on the diagonal
of the correlation matrix
are determined by the
squared multiple correlation
of the variable with the
other variables. For
example, if you regressed
items 14 through 24 on item
13, the squared multiple
correlation coefficient
would be 0.564.
Communalities
.564 .676
.551 .619
.538 .592
.447 .468
.585 .623
.572 .679
.456 .576
.326 .369
.516 .549
.397 .444
.662 .791
.526 .632
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
Initial Extraction
Extraction Method: Principal Axis Factoring.

29. 29
Factor Analysis
Extraction - The values in
this column indicate the
proportion of each
variable's variance that can
be explained by the retained
factors. 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.)
Communalities
.564 .676
.551 .619
.538 .592
.447 .468
.585 .623
.572 .679
.456 .576
.326 .369
.516 .549
.397 .444
.662 .791
.526 .632
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
Initial Extraction
Extraction Method: Principal Axis Factoring.

30. 30
Factor Analysis
Factor - The initial number of factors is the same as
the number of variables used in the factor analysis.
However, not all 12 factors will be retained. In this
example, only the first three factors will be retained
(as we requested).
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.

31. 31
Factor Analysis
Initial Eigenvalues - Eigenvalues are the variances of the
factors. Because we conducted our factor 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
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.

32. 32
Factor Analysis
Initial Eigenvalues - Total - This column contains the
eigenvalues. The first factor will always account for the
most variance (and hence have the highest eigenvalue), and
the next factor will account for as much of the left over
variance as it can, and so on. Hence, each successive factor
will account for less and less variance.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.

33. 33
Factor Analysis
Initial Eigenvalues - % of Variance - This column contains
the percent of total variance accounted for by each factor
(6.249/12 = .52 or 52%).
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.

34. 34
Factor Analysis
Initial Eigenvalues - Cumulative % - This column contains
the cumulative percentage of variance accounted for by
the current and all preceding factors. For example, the
third row shows a value of 68.313. This means that the
first three factors together account for 68.313% of the
total variance.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.

35. 35
Factor Analysis
Extraction Sums of Squared Loadings - The number of
rows in this panel of the table correspond to the number
of factors retained. The values are based on the common
variance (of the retained factors). The values in this panel
of the table will always be lower than the values in the left
panel of the table, because they are based on the common
variance, which is always smaller than the total variance.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.

36. 36
Factor Analysis
Rotation Sums of Squared Loadings - The values in this
panel of the table represent the distribution of the
variance after the varimax rotation. Varimax rotation
tries to maximize the variance of each of the factors, so
the total amount of variance accounted for is
redistributed over the three extracted factors.
Note the more even split.
Total Variance Explained
6.249 52.076 52.076 5.851 48.759 48.759 2.950 24.583 24.583
1.229 10.246 62.322 .806 6.719 55.478 2.655 22.127 46.710
.719 5.992 68.313 .360 3.000 58.478 1.412 11.769 58.478
.613 5.109 73.423
.561 4.676 78.099
.503 4.192 82.291
.471 3.927 86.218
.389 3.240 89.458
.368 3.066 92.524
.328 2.735 95.259
.317 2.645 97.904
.252 2.096 100.000
Factor
1
2
3
4
5
6
7
8
9
10
11
12
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Extraction Method: Principal Axis Factoring.

37. 37
Factor Analysis
The scree plot graphs the eigenvalue (variance) against the
factor number. You can see these values in the first two
columns of the variance explained table.

38. 38
Factor Analysis
From the third factor on, you can see that the line is almost
flat, meaning the each successive factor is accounting for
smaller and smaller amounts of the total variance.
You need to locate this,
so called, elbow!
In other words, when
the drop ceases and the
curve makes an elbow
toward a less steep
decline.

39. 39
Factor Analysis
Factor Matrix - This table
contains the unrotated
factor loadings, which are
the correlations between the
variable and the factor.
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.
Factor Matrixa
.713 -.398
.703 -.339
.721
.648
.783
.740 .345
.616 .415
.550
.732
.613
.819 -.345
.695 -.386
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
1 2 3
Factor
Extraction Method: Principal Axis Factoring.
3 factors extracted. 7 iterations required.a.

40. 40
Factor Analysis
Factor - The columns under
this heading are the
unrotated factors that have
been extracted. As you can
see by the footnote provided
by SPSS, three factors were
extracted (the three
factors that we requested).
Factor Matrixa
.713 -.398
.703 -.339
.721
.648
.783
.740 .345
.616 .415
.550
.732
.613
.819 -.345
.695 -.386
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
1 2 3
Factor
Extraction Method: Principal Axis Factoring.
3 factors extracted. 7 iterations required.a.

41. 41
Factor Analysis
The plot shows the
items (variables) in
the rotated factor
space.
While this picture
may not be
particularly
helpful, when you
get this graph in
the SPSS output,
you can
interactively
rotate it.

42. 42
Factor Analysis
Rotation may help
you to see how the
items (variables)
are organized in
the common factor
space.

43. 43
Factor Analysis
Another run of the factor analysis program is conducted
with a promax rotation. It is included to show how
different the rotated solutions can be, and to better
illustrate what is meant by simple structure.
As you will see with an oblique rotation, such as a promax
rotation, the factors are permitted to be correlated with
one another. With an orthogonal rotation, such as the
varimax shown above, the factors are not permitted to
be correlated (they are orthogonal to one another).
Oblique rotations, such as promax, produce both factor
pattern and factor structure matrices. For orthogonal
rotations, such as varimax and equimax, the factor
structure and the factor pattern matrices are the same.

44. 44
Factor Analysis
Use the buttons at the bottom of the screen to set the
alternate Rotation, employ the Continue button to return to
the main Factor Analysis screen.

45. 45
Factor Analysis
The resulting plot with a “simple” structure is shown.

46. 46
Factor Analysis
For a recent review see Factor Analysis at 100. Historical
Developments and Future Directions. By Robert Cudeck,
and Robert C. MacCallum (Eds.). Lawrence Earlbaum
Associates, Mahwah, NJ, 2007, xiii+381 pp., ISBN:978-0-
8058-5347-6 (hardcover), and, ISBN 978-0-8058-6212-6
(paperback).

47. 47
Factor Analysis
Summary
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.
A Beginner’s Guide to Factor Analysis: Focusing on
Exploratory Factor Analysis
An Gie Yong and Sean Pearce
Tutorials in Quantitative Methods for Psychology 2013
9(2) 79-94

48. 48
Factor Analysis
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 matrix X implemented by each method
produces a set of fewer homogenous variables – the true scores –
which contain most of the model’s information.

49. 49
Factor Analysis
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.
For Principal Components, see next weeks lecture.

50. 50
Factor Analysis
Overview of the steps in a
factor analysis. From: Rietveld
& Van Hout (1993: 291).
Rietveld, T. & Van Hout, R. (1993).
Statistical Techniques for the Study of
Language and
Language Behaviour. Berlin – New York:
Mouton de Gruyter.

51. 51
Factor Analysis
After having obtained the correlation matrix, it is time to decide which
type of analysis to use: factor analysis or principal component analysis. The
main difference between these types of analysis lies in the way the
communalities are used. In principal component analysis it is assumed that
the communalities are initially 1. In other words, principal component
analysis assumes that the total variance of the variables can be accounted
for by means of its components (or factors), and hence that there is no
error variance. On the other hand, factor analysis does assume error
variance. This is reflected in the fact that in factor analysis the
communalities have to estimated, which makes factor analysis more
complicated than principal component analysis, but also more conservative.
For further details see "Factor Analysis" Kootstra 2004

52. 52
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.