chapter 43 Research Methods Presentation Factor Analysis

Information about Factor analysis

1. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
FACTOR ANALYSIS, CLUSTER
ANALYSIS AND STRUCTURAL
EQUATION MODELLING
© LOUIS COHEN, LAWRENCE MANION AND
KEITH MORRISON

2. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
STRUCTURE OF THE CHAPTER
•Factor analysis
•What to look for in factor analysis output
•Cluster analysis
•A note on structural equation modelling
•A note on multilevel modelling

3. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
WHAT IS FACTOR ANALYSIS?
•A method of grouping together variables which have
something in common.
•The researcher can take a set of variables and reduce them
to a smaller number of underlying factors (latent variables)
which account for as many variables as possible.
•It detects structures and commonalities in the relationships
between variables. Researchers can identify where
different variables in fact are addressing the same
underlying concept.
•It detects latent (unobservable) factors.

4. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
TWO MAIN FORMS OF FACTOR ANALYSIS

5. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
SAFETY CHECKS FOR FACTOR ANALYSIS
•Sample size.
•Number of variables.
•Ratio of sample size to number of variables.
•Interval and ratio data.
•Sampling adequacy.
•Intercorrelations between variables.

6. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
SAFETY CHECKS FOR FACTOR ANALYSIS
•Intercorrelations between factors.
•Normal distributions.
•Linearity.
•Outliers.
•Selection bias/proper specification.
•Theoretical underpinning of factors.

7. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
STAGE ONE IN FACTOR ANALYSIS
1.Check that the data are suitable for factor analysis:
(a) Sample size (varies in the literature, from a minimum
of 30 to a minimum of 300); if the sample size is small then
the factor loadings should be high to be included).
(b) Number of variables.
(c) Ratio of sample size to number of variables (different
ratios given in literature, from 5:1 to 30:1).
(c) Strength of intercorrelations should be no less than 0.3.
(d) Bartlett’s test of sphericity should be statistically
significant ( < 0.05).
(e) Kaiser-Mayer-Olkin measure of sampling adequacy
should be 0.6 or higher (maximum is 1).

8. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
STAGE TWO IN FACTOR ANALYSIS
2.Decide which form of extraction method to use:
(a) Principal components analysis is widely used.
(b) Set the Kaiser criterion (the Eigenvalues to be
set at greater than 1); the Eigenvalue of a factor
indicates the amount of the total variance
explained by that factor – if it is less than 1.00
then it does not have any additional explanatory
value and should be ignored (SPSS does this
automatically).
(c) Unrotated factor solution to be set.
(d) Scree plot to be set.

9. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
SCREE PLOT IN SPSS
Scree Plot
Component Number
2321191715131197531
E
ig
e
n
v
a
lu
e
10
8
6
4
2
0

10. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
STAGE THREE IN FACTOR ANALYSIS
3.Conduct the factor rotation:
(a) Decide which of the two main approaches to
use:
i.Oblique (related variables): Direct Oblimin
ii.Orthogonal (unrelated variables): Varimax.
(b) People often use the varimax solution when it
should not be used, as it is sometimes easier to
use than other kinds.
(c) Check that the rotated solution is set.

11. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
ROTATION
Rotation keeps together those items
that are closely related and separates
them clearly from other items, i.e. it
includes and excludes (keeps together
a group of homogeneous items and
keeps them apart from other groups).

12. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
EXAMPLE OF FACTOR ANALYSIS USING SPSS
•Factor analysis for an oblique rotation.
•Direct Oblimin rotation.

13. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
Analyze  Dimension Reduction  Factor  Move the variables
to be included to the ‘Variables’ box 

14. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
Click on ‘Descriptives’  Click on ‘KMO and Bartlett’s test of
sphericity’  Click on Coefficients’  Click ‘Continue’ 

15. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
Click on ‘Extraction’  Click on ‘Principal components’  Click on
‘Correlation matrix’  Click on ‘Unrotated factor solution’  Click on ‘Scree
plot’  Click on ‘Based on Eigenvalue’  Click ‘Continue’ 

16. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
Click on ‘Rotation’  Click on ‘Direct Oblimin’ or ‘Varimax’ (depending
on whether the rotation is oblique or orthogonal)  Click ‘Continue’
 return to main screen and click ‘OK’

17. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
ANALYSIS OF THE EXAMPLE FROM SPSS
•SPSS produces many tables for factor analysis. Be
selective but fair to the data.

18. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
Check correlation coefficients (most should be over 0.3)
(selection only reproduced here, not the full table)
How much do
you feel that
working with
colleagues all
day is really a
strain for you?
How much
do you feel
emotionally
drained by
your work?
How much do
you worry that
your job is
hardening
you
emotionally?
How much
frustration
do you feel
in your
job?
Correlation
How much do you
feel that working
with colleagues all
day is really a
strain for you?
1.000 0.554 0.507 0.461
How much do you
feel emotionally
drained by your
work?
0.554 1.000 0.580 0.518
How much do you
worry that your
job is hardening
you emotionally?
0.507 0.580 1.000 0.646
How much
frustration do you
feel in your job?
0.461 0.518 0.646 1.000

19. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
SUITABILITY FOR FACTOR ANALYSIS
KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
0.845
Bartlett's Test of
Sphericity
Approx. Chi-Square 5460.475
df 36
Sig. 0.000
KMO > 0.6
Bartlett’s test Sig.:  < 0.05
The data are suitable for factor analysis

20. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
How much of the variance is explained by each
item (lower than 0.3 and the item is a poor fit)
Communalities
Initial Extraction
How hard do you feel you are working in your job? 1.000 .779
How much do you feel exhausted by the end of the
workday?
1.000 .818
How much do you feel that you cannot cope with your
job any longer?
1.000 .578
How much do you feel that you treat colleagues as
impersonal objects?
1.000 .578
How much do you feel that working with colleagues all
day is really a strain for you?
1.000 .602
How much do you feel emotionally drained by your
work?
1.000 .629
How tired do you feel in the morning, having to face
another school day?
1.000 .595
How much do you worry that your job is hardening you
emotionally?
1.000 .661
How much frustration do you feel in your job? 1.000 .595
Extraction Method: Principal Component Analysis.

21. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
Two factors found: factor one explains 45.985 per cent of total
variance; factor two explains 18.852 per cent of total variance.
Total Variance Explained
Compo
nent
Initial Eigenvalues
Extraction Sums of Squared
Loadings
Rotation
Sums of
Squared
Loadings
a
Total
% of
Variance
Cumulative
% Total
% of
VarianceCumulative %Total
1 4.13945.985 45.9854.13945.985 45.985 4.028
2 1.69718.851 64.8361.69718.851 64.836 1.991
3 .661 7.342 72.178
4 .542 6.023 78.202
5 .531 5.900 84.102
6 .451 5.006 89.107
7 .395 4.390 93.497
8 .323 3.593 97.090
9 .262 2.910100.000
Extraction Method: Principal Component Analysis.
a. When components are correlated, sums of squared loadings cannot be added to
obtain a total variance.

22. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
Pattern Matrix
a
Component
1 2
How hard do you feel you are working in your job? .005.882
How much do you feel exhausted by the end of the workday? .252.834
How much do you feel that you cannot cope with your job any longer? .691.234
How much do you feel that you treat colleagues as impersonal objects?.674-.459
How much do you feel that working with colleagues all day is really a
strain for you?
.782-.158
How much do you feel emotionally drained by your work? .774.096
How tired do you feel in the morning, having to face another school day?.697.247
How much do you worry that your job is hardening you emotionally?.814-.008
How much frustration do you feel in your job? .752.097
Extraction Method: Principal Component Analysis.
Rotation Method: Oblimin with Kaiser Normalization.
a. Rotation converged in 6 iterations.
Decide the cut-off points and
which variables to include.

23. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
WHICH VARIABLES TO INCLUDE IN A FACTOR
For each variable:
1.Include the highest scoring variables.
2.Omit the low scoring variables.
3.Look for where there is a clear scoring distance between
those included and those excluded.
4.Review your selection to check that no lower scoring variables
have been excluded which are conceptually close to those
included.
5.Review your selection to check whether some higher scoring
variables should be excluded if they are not sufficiently
conceptually close to the others that have been included.
6.Review your final selection to see that they are conceptually
similar.
NB. Inclusion and exclusion are an art, not a science; there is no
simple formula, so you have to use your judgement.

24. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
WHAT TO REPORT
1.Method of factor analysis used (Principal components;
Direct Oblimin); KMO and Bartlett test of sphericity;
Eigenvalues greater than 1; scree test; rotated solution).
2.How many factors were extracted with Eigenvalues greater
than 1.
3.How many factors were included as a result of the scree
test.
4.Give a name/title to each of the factors.
5.Indicate how much of the total variance was explained by
each factor.
6.Report the cut-off point for the variables included in each
factor.
7.Indicate the factor loadings of each variable in the factor.
8.What the results tell us.

25. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
FIVE STAGES IN FACTOR ANALYSIS

26. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
CLUSTER ANALYSIS

27. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
TWO MAIN FORMS OF CLUSTER ANALYSIS
1. Hierarchical cluster analysis.
2. Non-hierarchical analysis.
•Researchers often use K-Means Cluster
(nonhierarchical cluster analysis), Hierarchical
‑
Cluster and Two-step Cluster, of which Hierarchical
Cluster analysis is the most widely used.
•Cluster analysis can work with interval, ratio,
ordinal and nominal data, using different statistics
for each kind of data.

28. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
DENDROGRAM IN CLUSTER ANALYSIS

29. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
INTERPRETING THE DENDROGRAM

30. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
ANALYSING AND REPORTING CLUSTERS
•What is the similarity criterion that combines
individual cases into a single cluster?
•How many cases (and who) are in each cluster?
•How similar are the cases within each cluster?
•What differentiates that cluster from another?
•What is the criterion or characteristic that combines
clusters?
•How similar/dissimilar are the clusters?
•At what level in the hierarchy is it most advisable to
cease combining clusters?

31. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
STRUCTURAL EQUATION MODELLING
•The name given to a group of techniques that
enable researchers to construct models of
putative causal relations, and to test those
models against data.
•It is designed to enable researchers to confirm,
modify and test their models of causal relations
between variables.
•It is based on multiple regression and factor
analysis.

32. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
STRUCTURAL EQUATION MODELLING
•It works with observed and unobserved variables,
not latent factors (as in factor analysis).
•It is a particular kind of multiple regression
analysis that enables the researcher to see the
relative weightings of observed independent
variables on each other and on a dependent
variable, to establish pathways of causation, and
to determine the direct and indirect effects of
independent variables on a dependent variable.

33. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
A CAUSAL MODEL (USING AMOS WITH SPSS)
Part-time work
Level of motivation
for academic study
Class of degree
Socio-economic status
e1
e2
e3

34. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
THE CAUSAL MODEL WITH CALCULATIONS ADDED
Part-time work
Level of motivation
for academic study
Class of degree
Socio-economic status
.18
e1
e2
e3
.04
.52
-.21
-1.45 1.37
-.01

35. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
INTERPRETING THE CAUSAL MODEL
–Socio-economic’ status exerts a direct powerful influence
on class of degree (.18), which is higher than the direct
influence of either ‘part-time work’ (–.01) or ‘level of
motivation for academic study’ (.04).
–‘Socio-economic status’ exerts a powerful direct influence
on ‘level of motivation for academic study’ (.52), which is
higher than the influence of ‘socio-economic status’ on
‘class of degree’ (.18).
–‘Socio-economic status’ exerts a powerful direct and
negative influence on ‘part-time work’ (–.21), i.e. the
higher the socio-economic status, the lower the amount
of part-time work undertaken.

36. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
INTERPRETING THE CAUSAL MODEL
–‘Part-time work’ exerts a powerful direct influence on ‘level
of motivation for academic study’ (1.37), and this is higher
than the influence of ‘socio-economic status’ on ‘level of
motivation for academic study’ (.52).
–‘Level of motivation for academic study’ exerts a powerful
negative direct influence on ‘part-time work’ (–1.45), i.e. the
higher the level of motivation for academic study, the lower
the amount of part-time work undertaken.
–‘Level of motivation for academic study’ exerts a slightly more
powerful influence on ‘class of degree’ (.04) than does ‘part-
time work’ (–.01).
–‘Part-time work’ exerts a negative influence on the class of
degree (–.01), i.e. the more one works part-time, the lower
the class of degree obtained.

37. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
A STRUCTURAL EQUATION MODEL
(USING AMOS IN SPSS)
Ovals = factors
Rectangles = variables for
each factor
E = Error factor

38. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
A NOTE ON MULTILEVEL MODELLING

39. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors
A NOTE ON MULTILEVEL MODELLING
•Data are ‘nested’, e.g. individual-level data are nested
within group, class, school, regional etc. levels.
•A dependent variable is affected by independent
variables at different levels, i.e. data are hierarchical.
•Multilevel modelling uses regression analysis and
multilevel regression.
•Multilevel modelling enables the researcher to
calculate the relative impact on a dependent variable
of one or more independent variables at each level of
the hierarchy, and thereby to identify factors at each
level of the hierarchy that are associated with the
impact of that level.