Prof. Andy Field. Slide . 2. Aims. Explore factor . a. nalysis and . p. rincipal . c. omponent . a. nalysis (PCA). What . Are . factors. ?. Representing . factors. Graphs and Equations. Extracting factors. ID: 634730
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Presentations text content in Exploratory Factor Analysis
Slide1
Exploratory Factor Analysis
Prof. Andy Field
Slide2
Slide
2
Aims
Explore factor
a
nalysis and
p
rincipal
c
omponent
a
nalysis (PCA)
What
Are
factors
?
Representing
factors
Graphs and Equations
Extracting factors
Methods and Criteria
Interpreting
factor
s
tructures
Factor
Rotation
Reliability
Cronbach’s alpha
Slide3
Slide
3
When and Why?
To test for clusters of variables or measures.
To see whether different measures are tapping aspects of a common dimension.
E.g. Anal-Retentiveness, Number of friends, and social skills might be aspects of the common dimension of ‘statistical ability’
Slide4
R-Matrix
In
f
actor analysis and
PCA
we look to reduce the R-matrix into smaller set of
correlated or uncorrelated
dimensions.
Slide
4
Slide5
Slide
5
Factors and components
Factor analysis attempts to achieve parsimony by explaining the maximum amount of
common variance
in a correlation matrix using the smallest number of explanatory constructs.
T
hese ‘explanatory constructs’
are
called
factors.
PCA tries to explain the maximum amount of total variance
in a correlation matrix.
It does this by transforming the original variables into a set of linear
components
.
Slide6
Slide
6
Graphical Representation
Slide7
Slide
7
Mathematical
Representation, PCA
Slide8
Mathematical Representation Continued
The factors in
f
actor analysis are not represented in the same way as components.
Variables = Variable Means + (Loadings × Common Factor) + Unique Factor
Slide9
Slide
9
Factor Loadings
Both factor analysis and PCA are linear models in which loadings are used as weights.
These loadings can be expressed as a matrix
This matrix is called the factor matrix or component matrix (if doing PCA).
The assumption of factor analysis (but not PCA) is that these algebraic factors represent real-world dimensions.
Slide10
Slide
10
The SAQ
Slide11
Slide
11
Initial Considerations
The quality of analysis depends upon the quality of the data (GI
GO).
Test variables should correlate quite well
r
>
.
3.
Avoid Multicollinearity:
several variables highly correlated,
r >
.80, tolerance
>
.20.
Avoid Singularity:
some variables perfectly correlated,
r
=
1, tolerance = 0.
Screen the correlation
matrix eliminate
any variables that obviously cause concern
.
Conduct multicollinearity analysis (as in multiple regression, but with case number as the dependent variable).
Slide12
Slide
12
Further Considerations
Determinant:
Indicator of multicollinearity
should be greater than 0.00001.
Kaiser-Meyer-Olkin (KMO):
Measures sampling adequacy
should be greater than 0.5.
Bartlett’s Test of Sphericity:
Tests whether the R-matrix is an identity matrix
should be significant at
p <
.05.
Anti-Image Matrix:
Measures of sampling adequacy on diagonal,
Off-diagonal elements should be small.
Reproduced:
Correlation matrix after rotation
most residuals should be < |0.05|
Slide13
Slide
13
Finding Factors: Communality
Common Variance:
Variance that a variable shares with other variables.
Unique Variance
:
Variance that is unique to a particular variable
.
The proportion of common variance in a variable is called the communality
.
Communality = 1, All variance shared.Communality = 0, No variance shared.
0 < Communality < 1 = Some variance shared.
Slide14
Slide
14
Variance of
Variable 1
Variance of
Variable 2
Variance of
Variable 3
Variance of
Variable 4
Communality = 1
Communality
= 0
Slide15
Slide
15
Finding
Factors
We find factors by calculating the amount of common variance
Circularity
Principal Components Analysis:
Assume all variance is shared
All Communalities = 1
Factor Analysis
Estimate Communality
Use Squared Multiple Correlation (SMC)
Slide16
Slide
16
Slide17
Slide
17
Factor Extraction
Kaiser’s
extraction
Kaiser (1960): retain factors with
eigenvalues
> 1.
Scree
plot
Cattell (1966): use ‘point of inflexion’ of the scree plot.
Which
rule?
Use
Kaiser’s
extraction whenless than 30 variables, communalities after extraction > 0.7.
sample size > 250 and mean communality
≥
0.6.
Scree plot is good if sample size is > 200
.
Parallel
analysis
Supported by SPSS macros written in SPSS syntax (O’Connor, 2000)
Slide18
Slide
18
Slide19
Slide
19
Scree Plots
Slide20
Slide
20
Rotation
To aid interpretation it is possible to maximise the loading of a variable on one factor while minimising its loading on all other factors
This is known as Factor Rotation
There are two types:
Orthogonal
(factors are uncorrelated)
Oblique
(factors intercorrelate)
Slide21
Slide
21
Orthogonal
Oblique
Slide22
Slide
22
Before Rotation
Slide23
Orthogonal
Rotation (varimax)
Slide
23
Slide24
Oblique Rotation
Slide
24
Slide25
Slide
25
Reliability
Test-Retest Method
What about practice effects/mood states?
Alternate Form Method
Expensive and Impractical
Split-Half Method
Splits the questionnaire into two random halves, calculates scores and correlates them.
Cronbach’s
alpha
Splits the questionnaire into all possible halves, calculates the scores, correlates them and averages the correlation for all splits (well, sort of …).
Ranges from 0 (no reliability) to 1 (complete reliability)
Slide26
Cronbach’s Alpha
Slide
26
Slide27
Slide
27
Interpreting Cronbach’s Alpha
Kline (1999)
Reliable if
>
.7
Depends on the number of
items
More questions = bigger
is *not* a measure of unidimensionality
Treat subscales
separately
Remember to reverse score reverse phrased items!
If not, is reduced and can even be negative
Slide28
Slide
28
Reliability for Fear of Computers Subscale
Slide29
Slide
29
Reliability for Fear of Statistics Subscale
Slide30
Slide
30
Reliability for Fear of Maths Subscale
Slide31
Slide
31
Reliability for the Peer Evaluation Subscale
Slide32
Slide
32
The End?
Describe Factor Structure/Reliability
What items should be retained?
What items did you eliminate and why?
Application
Where will your questionnaire be used?
How does it fit in with psychological theory?
Slide33
Slide
33
Conclusion
PCA and FA to reduce a larger set of measured variables to a smaller set of underlying dimensions
In PCA, components summarise information from set of variables
Download this presentation
DownloadNote - The PPT/PDF document "Exploratory Factor Analysis" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Presentations text content in Exploratory Factor Analysis
Exploratory Factor Analysis
Prof. Andy Field
Slide2Slide
2
Aims
Explore factor
a
nalysis and
p
rincipal
c
omponent
a
nalysis (PCA)
What
Are
factors
?
Representing
factors
Graphs and Equations
Extracting factors
Methods and Criteria
Interpreting
factor
s
tructures
Factor
Rotation
Reliability
Cronbach’s alpha
Slide3Slide
3
When and Why?
To test for clusters of variables or measures.
To see whether different measures are tapping aspects of a common dimension.
E.g. Anal-Retentiveness, Number of friends, and social skills might be aspects of the common dimension of ‘statistical ability’
Slide4R-Matrix
In
f
actor analysis and
PCA
we look to reduce the R-matrix into smaller set of
correlated or uncorrelated
dimensions.
Slide
4
Slide5Slide
5
Factors and components
Factor analysis attempts to achieve parsimony by explaining the maximum amount of
common variance
in a correlation matrix using the smallest number of explanatory constructs.
T
hese ‘explanatory constructs’
are
called
factors.
PCA tries to explain the maximum amount of total variance
in a correlation matrix.
It does this by transforming the original variables into a set of linear
components
.
Slide6Slide
6
Graphical Representation
Slide7Slide
7
Mathematical
Representation, PCA
Slide8Mathematical Representation Continued
The factors in
f
actor analysis are not represented in the same way as components.
Variables = Variable Means + (Loadings × Common Factor) + Unique Factor
Slide9Slide
9
Factor Loadings
Both factor analysis and PCA are linear models in which loadings are used as weights.
These loadings can be expressed as a matrix
This matrix is called the factor matrix or component matrix (if doing PCA).
The assumption of factor analysis (but not PCA) is that these algebraic factors represent real-world dimensions.
Slide10Slide
10
The SAQ
Slide11Slide
11
Initial Considerations
The quality of analysis depends upon the quality of the data (GI
GO).
Test variables should correlate quite well
r
>
.
3.
Avoid Multicollinearity:
several variables highly correlated,
r >
.80, tolerance
>
.20.
Avoid Singularity:
some variables perfectly correlated,
r
=
1, tolerance = 0.
Screen the correlation
matrix eliminate
any variables that obviously cause concern
.
Conduct multicollinearity analysis (as in multiple regression, but with case number as the dependent variable).
Slide12Slide
12
Further Considerations
Determinant:
Indicator of multicollinearity
should be greater than 0.00001.
Kaiser-Meyer-Olkin (KMO):
Measures sampling adequacy
should be greater than 0.5.
Bartlett’s Test of Sphericity:
Tests whether the R-matrix is an identity matrix
should be significant at
p <
.05.
Anti-Image Matrix:
Measures of sampling adequacy on diagonal,
Off-diagonal elements should be small.
Reproduced:
Correlation matrix after rotation
most residuals should be < |0.05|
Slide13Slide
13
Finding Factors: Communality
Common Variance:
Variance that a variable shares with other variables.
Unique Variance
:
Variance that is unique to a particular variable
.
The proportion of common variance in a variable is called the communality
.
Communality = 1, All variance shared.Communality = 0, No variance shared.
0 < Communality < 1 = Some variance shared.
Slide14Slide
14
Variance of
Variable 1
Variance of
Variable 2
Variance of
Variable 3
Variance of
Variable 4
Communality = 1
Communality
= 0
Slide15Slide
15
Finding
Factors
We find factors by calculating the amount of common variance
Circularity
Principal Components Analysis:
Assume all variance is shared
All Communalities = 1
Factor Analysis
Estimate Communality
Use Squared Multiple Correlation (SMC)
Slide16Slide
16
Slide17Slide
17
Factor Extraction
Kaiser’s
extraction
Kaiser (1960): retain factors with
eigenvalues
> 1.
Scree
plot
Cattell (1966): use ‘point of inflexion’ of the scree plot.
Which
rule?
Use
Kaiser’s
extraction whenless than 30 variables, communalities after extraction > 0.7.
sample size > 250 and mean communality
≥
0.6.
Scree plot is good if sample size is > 200
.
Parallel
analysis
Supported by SPSS macros written in SPSS syntax (O’Connor, 2000)
Slide18Slide
18
Slide19Slide
19
Scree Plots
Slide20Slide
20
Rotation
To aid interpretation it is possible to maximise the loading of a variable on one factor while minimising its loading on all other factors
This is known as Factor Rotation
There are two types:
Orthogonal
(factors are uncorrelated)
Oblique
(factors intercorrelate)
Slide21Slide
21
Orthogonal
Oblique
Slide22Slide
22
Before Rotation
Slide23Orthogonal
Rotation (varimax)
Slide
23
Slide24Oblique Rotation
Slide
24
Slide25Slide
25
Reliability
Test-Retest Method
What about practice effects/mood states?
Alternate Form Method
Expensive and Impractical
Split-Half Method
Splits the questionnaire into two random halves, calculates scores and correlates them.
Cronbach’s
alpha
Splits the questionnaire into all possible halves, calculates the scores, correlates them and averages the correlation for all splits (well, sort of …).
Ranges from 0 (no reliability) to 1 (complete reliability)
Slide26Cronbach’s Alpha
Slide
26
Slide27Slide
27
Interpreting Cronbach’s Alpha
Kline (1999)
Reliable if
>
.7
Depends on the number of
items
More questions = bigger
is *not* a measure of unidimensionality
Treat subscales
separately
Remember to reverse score reverse phrased items!
If not, is reduced and can even be negative
Slide28Slide
28
Reliability for Fear of Computers Subscale
Slide29Slide
29
Reliability for Fear of Statistics Subscale
Slide30Slide
30
Reliability for Fear of Maths Subscale
Slide31Slide
31
Reliability for the Peer Evaluation Subscale
Slide32Slide
32
The End?
Describe Factor Structure/Reliability
What items should be retained?
What items did you eliminate and why?
Application
Where will your questionnaire be used?
How does it fit in with psychological theory?
Slide33Slide
33
Conclusion
PCA and FA to reduce a larger set of measured variables to a smaller set of underlying dimensions
In PCA, components summarise information from set of variables
In FA, factors are underlying dimensions
How many factors to extract?
Rotation
Interpretation
Reliability analysis after PCA/FA