Lecture 7:
Factor Analysis
Laura McAvinue
School of Psychology
Trinity College Dublin
The Relationship between Variables
• Previous lectures…
• Correlation
– Measure of strength of association between two variables
• Simple linear regression
– Describes the relationship between two variables by expressing
one variable as a function of the other, enabling us to predict one
variable on the basis of the other
The Relationship between Variables
• Multiple Regression
– Describes the relationship between several variables, expressing
one variable as a function of several others, enabling us to
predict this variable on the basis of the combination of the other
variables
• Factor Analysis
– Also a tool used to investigate the relationship between several
variables
– Investigates whether the pattern of correlations between a
number of variables can be explained by any underlying
dimensions, known as ‘factors’
Uses of Factor Analysis
Test / questionnaire construction
For example, you wish to design an anxiety questionnaire…
Create 50 items, which you think measure anxiety
Give your questionnaire to a large sample of people
Calculate correlations between the 50 items & run a factor
analysis on the correlation matrix
o If all 50 items are indeed measuring anxiety…
o
o
o
o
• All correlations will be high
• One underlying factor, ‘anxiety’
Verification of test / questionnaire structure
o Hospital Anxiety & Depression Scale
o Expect two factors, ‘anxiety’ & ‘depression’
Uses of Factor Analysis
Examining of the structure of a psychological
construct
What is ‘attention’?
A single ability? Several different abilities?
Some neuropsychological evidence for existence of different
neural pathways for ‘selective’ & ‘sustained’ attention
Administer tests measuring both aspects to large sample &
run factor analysis
One underlying factor? Two?
An example
Visual Imagery Ability
Two kinds of measure
 Self-report questionnaires v Objective tests
Is self-reported imagery related to imagery measured by
objective tests?
Do tests and questionnaires measure the same thing?
How does it work?
• Correlation Matrix
– Analyses the pattern of correlations between variables in the
correlation matrix
– Which variables tend to correlate highly together?
– If variables are highly correlated, likely that they represent the
same underlying dimension
• Factor analysis pinpoints the clusters of high correlations
between variables and for each cluster, it will assign a
factor
Correlation Matrix
Q1
•
•
•
Q2
Q3
Q4
Q5
Q1
1
Q2
.987
1
Q3
.801
.765
1
Q4
-.003
-.088
0
1
Q5
-.051
.044
.213
.968
1
Q6
-.190
-.111
0.102
.789
.864
Q6
1
Q1-3 correlate strongly with each other and hardly at all with 4-6
Q4-6 correlate strongly with each other and hardly at all with 1-3
Two factors!
Factor Analysis
• Two main things you want to know…
– How many factors underlie the correlations
between the variables?
– What do these factors represent?
• Which variables belong to which factors?
Steps of Factor Analysis
1. Suitability of the Dataset
2. Choosing the method of extraction
3. Choosing the number of factors to extract
4. Interpreting the factor solution
1. Suitability of Dataset
Selection of Variables
Sample Characteristics
Statistical Considerations
Selection of Variables
 Are the variables meaningful?
• Factor analysis can be run on any dataset
•
‘Garbage in, garbage out’ (Cooper, 2002)
 Psychometrics
• The field of measurement of psychological constructs
• Good measurement is crucial in Psychology
• Indicator approach
• Measurement is often indirect
• Can’t measure ‘depression’ directly, infer on the basis of an
indicator, such as questionnaire
 Based on some theoretical / conceptual framework, what
are these variables measuring?
Selection of Variables, Example
 Variables selected were measures of key aspects of
imagery ability, according to theory
 Questionnaires (Richardson, 1994)
 Vividness
 Control
 Preference
 Objective Tests (Kosslyn, 1999)





Generation
Inspection
Maintenance
Transformation
Visual STM
Sample Characteristics
 Size
 At least 100 participants
 Participant : Variable Ratio
 Estimates vary
 Minimum of 5 : 1, ideal of 10 : 1
 Characteristics
 Representative of the population of interest?
 Contains different subgroups?
Sample Characteristics, Example
 Size
 101 participants
 Participant : Variable Ratio
 101
:
9
 11.22
:
1
 Characteristics
 Interested in imagery ability of general adult population so took a
mixed sample of males and females, varying widely in age,
educational and employment backgrounds
Statistical Considerations
 Assumptions of factor analysis regarding data
 Continuous
 Normally distributed
 Linear relationships
 These properties affect the correlations between variables
 Independence of variables
 Variables should not be calculated from each other
 e.g. Item 4 = Item 1 + 2 + 3
Statistical Considerations
 Are there enough significant correlations (> .3)
between the variables to merit factor analysis?
 Bartlett Test of Sphericity
 Tests Ho that all correlations between variables = 0
 If p < .05, reject Ho and conclude there are significant
correlations between variables so factor analysis is possible
Statistical Considerations
 Are there enough significant correlations (> .3)
between the variables to merit factor analysis?
 Kaiser-Meyer-Olkin Measure of Sampling Adequacy
 Quantifies the degree of inter-correlations among variables
 Value from 0 – 1, 1 meaning that each variable is perfectly
predicted by the others
 Closer to 1 the better
 If KMO > .6, conclude there is a sufficient number of
correlations in the matrix to merit factor analysis
Statistical Considerations, Example
• All variables
•
•
•
•
Continuous
Normally Distributed
Linear relationships
Independent
• Enough correlations?
• Bartlett Test of Sphericity (χ2 = 114.56; df = 36; p
< .001)
• KMO = .734
2. Choosing the method of extraction
Two methods
Factor Analysis
Principal Components Analysis
Differ in how they analyse the variance in the
correlation matrix
Variable
Specific
Error
Variance
Variance
Variance unique to
the variable itself
Variance due to
measurement
error or some
random, unknown
source
Common
Variance
Variance that a
variable shares
with other
variables in a
matrix
When searching for the factors underlying the relationships between a set
of variables, we are interested in detecting and explaining the common
variance
Principal Components Analysis
Factor Analysis
•Ignores the distinction between
the different sources of variance
•Analyses total variance in the
correlation matrix, assuming the
components derived can explain
all variance
•Result: Any component extracted
will include a certain amount of
error & specific variance
V
•Separates specific & error
variance from common
variance
•Attempts to estimate
common variance and
identify the factors underlying
this
Which to choose?
•Different opinions
•Theoretically, factor analysis is more sophisticated but statistical
calculations are more complicated, often leading to impossible results
•Often, both techniques yield similar solutions
2. Choosing the method of extraction,
Example
Tried both
Chose Principal Components Analysis as Factor
Analysis proved impossible (estimated communalities >
1)
3. Choosing the number of factors to
extract
• Statistical Modelling
– You can create many solutions using different
numbers of factors
• An important decision
– Aim is to determine the smallest number of factors
that adequately explain the variance in the matrix
– Too few factors
• Second-order factors
– Too many factors
• Factors that explain little variance & may be meaningless
Criteria for determining Extraction
Theory / past experience
Latent Root Criterion
Scree Test
Percentage of Variance Explained by the
factors
Latent Root Criterion (Kaiser-Guttman)
• Eigenvalues
– Expression of the amount of variance in the matrix
that is explained by the factor
– Factors with eigenvalues > 1 are extracted
– Limitations
• Sensitive to the number of variables in the matrix
• More variables… eigenvalues inflated… overestimation of
number of underlying factors
Scree Test (Cattell, 1966)
• Scree Plot
– Based on the relative values of the eigenvalues
– Plot the eigenvalues of the factors
– Cut-off point
• The last component before the slope of the line becomes flat
(before the scree)
Elbow in
the graph
Take the components above the elbow
Percentage of Variance
• Percentage of variance explained by the factors
– Convention
– Components should explain at least 60% of the
variance in the matrix (Hair et al., 1995)
3. Choosing the number of factors to
extract, Example
Scree Plot
4.0
• Three
components with
eigenvalues > 1
3.5
3.0
2.5
2.0
1.5
• Explained 67.26%
of the variance
1.0
.5
0.0
1
2
3
Component Number
4
5
6
7
8
9
4. Interpreting the Factor Solution
• Factor Matrix
– Shows the loadings of each of the variables on the
factors that you extracted
– Loadings are the correlations between the variables
and the factors
– Loadings allow you to interpret the factors
• Sign indicates whether the variable has a positive or negative
correlation with the factor
• Size of loading indicates whether a variable makes a
significant contribution to a factor
– ≥ .3
Variables
Component 1 Component 2
Component 3
Vividness Qu
-.198
-.805
.061
Control Qu
.173
.751
.306
Preference Qu
.353
.577
-.549
Generate Test
-.444
.251
.543
Inspect Test
-.773
.051
-.051
Maintain
.734
-.003
.384
Transform
(P&P) Test
.759
-.155
.188
Transform
(Comp) Test
-.792
.179
.304
Visual STM Test
.792
-.102
.215
Component 1 –
Visual imagery tests
Component 2 –
Visual imagery questionnaires
Component 3 –
?
Factor Matrix
• Interpret the factors
• Communality of the variables
– Percentage of variance in each variable that can be
explained by the factors
• Eigenvalues of the factors
– Helps us work out the percentage of variance in the
correlation matrix that the factor explains
Component 1
Component 2
Component 3
Communality
Vividness Qu
-.198
-.805
.061
69%
Control Qu
.173
.751
.306
69%
Preference Qu
.353
.577
-.549
76%
Generate Test
-.444
.251
.543
55%
Inspect Test
-.773
.051
-.051
60%
Maintain
.734
-.003
.384
69%
Transform (P&P)
Test
.759
-.155
.188
64%
Transform
(Comp) Test
-.792
.179
.304
75%
Visual STM Test
.792
-.102
.215
69%
Eigenvalues
3.36
1.677
1.018
/
% Variance
37.3%
18.6%
11.3%
/
Variables
Communality of Variable 1 (Vividness Qu) = (-.198)2 + (-.805)2 + (.061)2 = .
69 or 69%
Eigenvalue of Comp 1 = ( [-.198]2 + [.173]2 + [.353]2 + [-.444]2 + [-.773]2
+[.734]2 + [.759]2 + [-.792]2 + [.792]2 ) = 3.36
3.36 / 9 = 37.3%
Factor Matrix
• Unrotated Solution
– Initial solution
– Can be difficult to interpret
– Factor axes are arbitrarily aligned with the variables
• Rotated Solution
– Easier to interpret
– Simple structure
– Maximises the number of high and low loadings on
each factor
Factor Analysis through Geometry
• It is possible to represent correlation matrices
geometrically
• Variables
– Represented by straight lines of equal length
– All start from the same point
– High correlation between variables, lines positioned
close together
– Low correlation between variables, lines positioned
further apart
– Correlation = Cosine of the angle between the lines
V1 & V3
90º angle
V1
V2
Cosine = 0
No relationship
30º
60º
V3
The smaller the angle, the bigger the
cosine and the bigger the correlation
V1 & V2
30º angle
Cosine = .867
r = .867
V2 & V3
60º angle
Cosine = .5
R = .5
F1
V1
V2 V3
Factor Analysis
Fits a factor to
each cluster of
variables
V4
V5
V6
Factor Loading
Cosine of the angle between each
factor and the variable
F2
Passes a factor
line through the
groups of
variables
Two Methods of fitting Factors
F1
F1
V1 V2 V3
V1
V2 V3
V4
V4
V5
V5
V6
V6
F2
Orthogonal Solution
Oblique Solution
Factors are at right
angles
Factors are not at right
angles
Uncorrelated
Correlated
F2
Two Step Process
F1
F1
V1 V2
V1
V3
V4
V4
V5
V6
Factors are fit arbitrarily
V2 V3
F2
V5
F2
V6
Factors are rotated to fit the clusters
of variables better
For example…
Unrotated Solution
Variables
Solution following Orthogonal Rotation
C1
C2
C3
Variables
C1
C2
C3
Vividness Qu
-.198
-.805
.061
Vividness Qu
-.029
-.831
.008
Control Qu
.173
.751
.306
Control Qu
.174
.744
.323
Preference Qu
.353
.577
-.549
Preference Qu
-.010
.679
-.547
Generate Test
-.444
.251
.543
Generate Test
-.197
.112
.709
Inspect Test
-.773
.051
-.051
Inspect Test
-.717
-.103
.279
Maintain Test
.734
-.003
.384
Maintain Test
.819
.116
.043
Transform
(P&P) Test
.759
-.155
.188
Transform
(P&P) Test
.779
-.013
-.166
Transform
(Comp) Test
-.792
.179
.304
Transform
(Comp) Test
-.599
-.01
.626
Visual STM Test
.792
-.102
.215
VisualSTM Test
.813
.045
-.147
Factor Rotation
• Changes the position of the factors so that the
solution is easier to interpret
• Achieves simple structure
– Factor matrix where variables have either high or low
loadings on factors rather than lots of moderate
loadings
Evaluating your Factor Solution
• Is the solution interpretable?
– Should you re-run and extract a bigger or smaller number of
factors?
• What percentage of variance is explained by the factors?
– >60%?
• Are all variables represented by the factors?
– If the communality of one variable is very low, suggests it is not
related to the other variables, should re-run and exclude
For example…
First Solution
Variables
Vividness Qu
Control Qu
Preference Qu
C1
-.029
.174
-.010
C2
-.831
.744
.679
Second Solution
C3
Variables
Component 1
Component 2
Vividness Qu
.013
-.829
Control Qu
-.023
.770
Preference Qu
.195
.648
Generate Test
-.493
.130
Inspect Test
-.760
-.146
Maintain Test
.711
.183
Transform (P&P) Test
.773
.042
Transform
Test
-.811
-.028
.792
.103
.008
.323
-.547
Generate Test
-.197
.112
.709
Inspect Test
-.717
-.103
.279
Maintain Test
.819
.116
.043
Transform
(P&P) Test
.779
-.013
-.166
Transform
(Comp) Test
-.599
-.01
.626
VisualSTM Test
.813
.045
-.147
(Comp)
Visual STM Test
C1 = Efficiency of objective visual imagery
Component 3 = ?
C2 = Self-reported imagery efficacy
References
• Cooper, C. (1998). Individual differences.
London: Arnold.
• Kline, P. (1994). An easy guide to factor
analysis. London: Routledge.