This document provides an overview of various multidimensional measurement and factor analysis techniques, including elementary linkage analysis, factor analysis, cluster analysis, multidimensional scaling, structural equation modeling, and multilevel modeling. It discusses the key stages and considerations for conducting factor analysis and interpreting the results, and provides examples of interpreting outputs from SPSS.

1. MULTI-DIMENSIONAL
MEASUREMENT AND FACTOR
ANALYSIS
© LOUIS COHEN, LAWRENCE
MANION & KEITH MORRISON

2. STRUCTURE OF THE CHAPTER
• Elementary linkage analysis
• Factor analysis
• What to look for in factor analysis output
• Cluster analysis
• Examples of studies using multidimensional
scaling and cluster analysis
• Multi-dimensional data: some words on
notation
• A note on structural equation modelling
• A note on multilevel modelling

3. ELEMENTARY LINKAGE ANALYSIS
• A way of exploring the relationship between
personal constructs, of assessing the
dimensionality of the judgements that are
made.
• It seeks to identify and define the clusterings
of certain variables within a set of variables.
• Like factor analysis, elementary linkage
analysis searches for interrelated groups of
correlation coefficients. The objective of the
search is to identify ‘types’.

4. WHAT IS FACTOR ANALYSIS?
• A method of grouping together variables which
have something in common.
• It enables the researcher to 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.

5. WHAT IS FACTOR ANALYSIS?
• Factor analysis can take two main forms:
– Exploratory factor analysis: the use of factor
analysis (principal components analysis in
particular) to explore previously unknown
groupings of variables, to seek underlying
patterns, clusterings and groups.
– Confirmatory factor analysis is more stringent,
testing a found set of factors against a
hypothesized model of groupings and
relationships.

6. 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 factors 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 .3.;
(d)Bartlett’s test of sphericity should be statistically
significant (ρ<.05);
(e) Kaiser-Mayer-Olkin measure of sampling
adequacy should be .6 or higher (maximum is 1).

7. 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.

8. SCREE PLOT IN SPSS
Scree Plot
Component Number
2321191715131197531
Eigenvalue
10
8
6
4
2
0

9. 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.

10. 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).

11. EXAMPLE OF FACTOR ANALYSIS
USING SPSS
• Factor analysis for an oblique rotation.
• Direct Oblimin rotation.

12. Analyze → Dimension Reduction → Factor → Move the
variables to be included to the ‘Variables’ box →

13. Click on ‘Descriptives’ → Click on ‘KMO and Bartlett’s test
of sphericity’ → Click on Coefficients’ → Click ‘Continue’ →

14. 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’ →

15. 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’

16. ANALYSIS OF THE EXAMPLE
FROM SPSS
• SPSS produces many tables for factor
analysis. Be selective but fair to the data.

17. Check correlation coefficients (most should be over .
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 .554 .507 .461
How much do you
feel emotionally
drained by your
work?
.554 1.000 .580 .518
How much do you
worry that your job
is hardening you
emotionally?
.507 .580 1.000 .646
How much
frustration do you
feel in your job?
.461 .518 .646 1.000

18. SUITABILITY FOR FACTOR ANALYSIS
KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
.845
Bartlett's Test of
Sphericity
Approx. Chi-Square 5460.475
df 36
Sig. .000
KMO >.6
Bartlett’s test Sig.: ρ<.05
∴The data are suitable for factor analysis

19. How much of the variance is explained by each
item (lower than .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.

20. 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
Loadingsa
Total
% of
Variance
Cumulative
% Total
% of
Variance Cumulative % Total
1 4.139 45.985 45.985 4.139 45.985 45.985 4.028
2 1.697 18.851 64.836 1.697 18.851 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 2.910 100.000
Extraction Method: Principal Component Analysis.
a. When components are correlated, sums of squared loadings cannot be added to
obtain a total variance.

21. Pattern Matrixa
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.

22. 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 other that have
been included;
6. Review your final selection to see that they are
conceptually similar.
N. B. Inclusion and exclusion are an art, not a
science; there is no simple formula, so you have
to use your judgement.

23. 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.

24. CLUSTER ANALYSIS
• Factor analysis and elementary linkage
analysis enable the researcher to group
together factors and variables, but cluster
analysis enables the researcher to group
together similar and homogeneous sub-
samples of people.
• SPSS creates a dendrogram of clusters of
people into groups.

25. DENDROGRAM IN CLUSTER ANALYSIS

26. INTERPRETING THE DENDROGRAM
• There are two main clusters:
– Cluster One: Persons 19, 20, 2, 13, 15, 9,
11, 18, 14, 16, 1, 10, 12, 5, 17
– Cluster Two: Persons 7, 8, 4, 3, 6
• If one wishes to have smaller clusters then
three clusters can be found:
– Cluster One: Persons 19, 20, 2, 13, 15, 9,
11, 18
– Cluster Two: Persons 14, 16, 1, 10, 12, 5,
17
– Cluster Three: Persons 7, 8, 4, 3, 6

27. 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.

28. 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.

29. 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

30. 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

31. 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 lesser is the
amount of part-time work undertaken;

32. 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 is the level of motivation for
academic study, the lesser is 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 is the class of degree obtained.

33. A STRUCTURAL EQUATION
MODEL (USING AMOS IN SPSS)
Ovals = factors
Rectangles = variables
for each factor
E = Error factor

34. A NOTE ON MULTILEVEL MODELLING
• Data and variables exist at individual and group
levels, e.g.:
– between students over all groups
– between groups
– between students within groups
– individual
– group
– class
– school
– local
– regional
– national
– international

35. A NOTE ON MULTILEVEL MODELLING
• Data are ‘nested’, i.e. 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.