Transcript Carrying out EFA

Carrying out EFA - stages
• Ensure that data are suitable
• Decide on the model - PAF or PCA
• Decide how many factors are required to
represent you data
• When using PAF estimate the communality
of each factor
• Factor extraction
• Rotate the factors ensuring that simple
structure has been reached
• Compute factor scores
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Stage1 - Data suitability
• Packages will happily churn out results on any data!
• Must be continuous, categorical = inappropriate
• Variables are normally distributed and outliers have
been appropriately dealt with
• Relationship between all variables appear to be
linear or at least not U-shaped or J shaped
• All variable are independent - thus variables cannot
be calculated from any other variables
– e.g., if item a was height and b was weight then it would
be inappropriate for c to be a height to weight ratio since
it would necessarily be correlated to both a and b
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Stage1 - Data suitability
• There are at least some correlations in the
matrix that are above .3
– If correlations are smaller than this then there
would seem to be no real relationship between
any of the items
• Must be at least 100 participants and more
participants than items
– Nunnally (1978) advocates 10 times as many
participants as items
– Barrett & Kline (1981) so long as no. of Ps >
than items then ratio is not as important
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Stage 2 - Model
• The PCA is the simplest - sufficient for any
analysis performed at an undergraduate level
– Carroll argues is nonsense since all items must
have unique variance
Stage 3 - deciding on No. of factors to extract
• Theory and past experience
• If increasing the number of factors does not
increase the simplicity then it is of little use result should always be plausible
• Large correlations (>.5) between factors
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should be considered as suspect
Identifying No. of Factors
• Different tests produce different results
– Kaiser-Guttman criterion - generated factors with
eigenvalues above 1 are removed as real factors
• Problem - is sensitive to the number of items. Increase in
items = increase in eigenvalue.
– Scree test - based on eigenvalues of an unrotated
PC solution - depends on the relative values of
eigenvalues and therefore should be independent of
item number
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Eigenvalue
Scree Plot
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2.5
2
1.5
1
0.5
0
2 Factors to be extracted
0
1
2
3
4
5
6
7
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No. of factor extracted
• Going from left to right draw the first straight line
that shows the data leveling of - elbow
• No. of factors = the number of factors above this
line
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• Stage 4 - Estimation of the communality
– PCA - assumed to be 100% and therefore no
estimation required
– With PAF no agreed way to do this
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Stage 5 - Factor extraction
I1
I2
F1
I3
I4
• Initially factors are placed arbitrarily
• Successive factors are placed
F2
– At right angles to each other
– In a position that explains a substantial amount of
variance of the items
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Stage 6 - Factor rotation
• Changes the position of the factors to ease
interpretation
• Each factor should have some large
loadings and some small ones - each factor
should only have substantial loadings on
only a few items - known as simple
structure
• Large number of mediocre loadings should
be avoided
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Table showing rotated and unrotated
solutions for 4 hypothetical items
Notice
Unrotated
Rotated
h2
F1
F2
F1
F2
Comprehension
.4
.3
.5
.00
.25
Spelling
.4
.5
.64
.00
.41
Addition
.4
-.4
.13
.55
.32
Subtraction
.5
-.3
.06
.58
.34
Eigenvalue
.73
.59
.68
.64
1.32
Variance of
squared
loadings
.002
.006
.038
.034
– Communality of each
variable remains the
same
– Eigenvalues do not
– Factors are
positioned such that
the variance of the
squared loadings is
as large as possible most stat packages
use VARIMAX
• MAXimises the
VARIance of the
(squared factor
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loadings)
Factor scores
• Score on anyone factor can be determined
by considering the responses to items that
load onto that factor
• Can take into account the factor loading
such that item with greater loading have a
higher weighting
• Factor scores are determined through the
sum of scores produced on all the relevant
items
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Hierarchical FA
• If factors are obliquely rotated the resultant
matrix of correlations can itself be factoranalysed – hierarchical analysis or secondorder analysis
• Can be difficult to conceptualise what such
a higher order factor might represent
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EFA vs CFA
• EFA seeks to determine the number and
nature of factors which underpin a set of
data
• CFA allows you to choose between
alternative hypotheses which purport to
represent your data
– Given a set of data you could determine which
factor theory of personality best represented the
data
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CFA
• Can be carried out by principle axis
factoring or principle components method
• Is the simplest form of Structural equation
modelling
• Packages that have been created specially to
perform EFA
– LISREL
– EQS
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Errors of Factor Analysis
• Interpreting the unrotated solution
• Applying rigid rules to the extraction of
factors – KG vs Scree method. Which
solution makes most sense?
• Replication is very important
• Factor validity is not attested to only by
item content (face validity). Must be
compared with some other measure
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