Transcript Lecture 8: Factor Analysis Some practice!

Lecture 8:
Factor Analysis
Some practice!
Laura McAvinue
School of Psychology
Trinity College Dublin
Steps of Factor Analysis
•
•
•
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Suitability of dataset
Choose method of extraction
Choose the number of factors to extract
Interpret the factor solution
• Three steps to analysis
– Examine the dataset
– Preliminary analysis
– Main analysis
Have a look at the dataset
• Software / Kevin Thomas / Factor
Analysis (23 items)
• Includes data from a questionnaire
Worksheet Qus & Ans
1.
How many questions are in the questionnaire?
23
2.
3.
4.
5.
How many people have completed the questionnaire?
2571
What do the questions refer to?
See Variable View / Labels
What do you think this questionnaire is measuring?
SPSS anxiety
What will factor analysis of this dataset tell us?
How many factors underlie the correlations between
the items and the nature of these factors
Have a look at the correlation matrix
•
Analyse, correlate, bivariate
– Enter all 23 variables
6.
7.
8.
Do you notice something about the
statistical significance of the correlations?
All are highly statistically significant
Why is this so?
Huge sample size
Are all variables highly correlated with each
other or is there a mixture of high and low
correlations?
Mixture
Suitability of Dataset
• Selection of variables
• Sample characteristics
– Sample Size = 2571
– Participant : Variable Ratio
• 2571 : 23 / 112 : 1
– Characteristics?
• Statistical considerations
– Continuous?
• Yes (Likert scale, 1 – 5)
– Normally distributed, linear relationships, independence?
• Let’s assume so…
– Enough correlations?
• Return to this in a moment…
Steps of Factor Analysis
• Method of Extraction
– Principal Components Analysis
• Number of factors / components to extract
• Preliminary Analysis
– Check Bartlett Test of Sphericity & KMO
– Check the number of factors / components to extract
• Main Analysis
– Interpret the factor solution
Preliminary Analysis
• Analyse / data reduction / factor
– Select all variables and enter into box
– Descriptives
• Initial solution
• Tick KMO & Bartlett’s Test of Sphericity
– Extraction
• Method is principal Components
• Analyse correlation matrix
• Display unrotated factor solution & scree plot
– Options
• exclude cases pairwise
Output from Preliminary Analysis
• KMO & Bartlett’s Test of Sphericity
• Communalities
– Percentage of variance in each variable that’s explained by the
components
• Total variance explained
– The variance in the correlation matrix that is explained by the
components
• Scree plot
– Display of relative eigenvalues
• Component matrix
– Loadings of variables on components
Worksheet
9.
Is there a sufficient number of significant correlations
(>.3) to merit factor analysis?
Yes, KMO = .930 & Bartlett (X2 = 19334, df = 253, p <
.001)
10. How many components have eigenvalues > 1?
4
11. What does the scree plot suggest regarding the
extraction of components?
Difficult to judge, a little dip on the 5th component, so
perhaps best to take the 1st four
Scree Plot
8
6
4
2
0
1
3
5
7
9
11
13
15
17
19
21
23
Component Num ber
•
Here we see that the Scree plot is somewhat ambiguous. The curve is
difficult to interpret as it drops away after three variables, but there is
another drop after the fourth factor. Given our large sample size, we can
probably trust the Kaiser and accept that it is alright to retain four factors.
Worksheet
12. How many components have you decided to extract?
4
13. What percentage of variance in the correlation matrix
is explained by these components?
50%
Time to run the main analysis, using principal
components method and extracting four components
Recall that Factor Analysis is a 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
Need to decide the kind of rotation
F1
F1
V1 V2 V3
V1
V2 V3
V4
V4
V5
V5
V6
V6
F2
Orthogonal Rotation
Oblique Rotation
Varimax Method
Direct Oblimin
F2
Run the main factor analysis
• Analyse / data reduction / factor
– Enter all variables
– Descriptives
• Initial solution
• (not KMO or Bartletts)
– Extraction
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•
•
Method: principal components
Analyse correlation matrix
Display unrotated factor solution
(not scree plot)
Extract eigenvalues over 1
Run the main factor analysis
– Rotation
• Varimax
• Display rotated solution
– Options
• Exclude cases pairwise
• Sort coefficient display by size
• Suppress absolute values less than .4
The output
o Communalities
o Percentage of variance in each variable that’s explained by the
components
o Total variance explained
o The variance in the correlation matrix that is explained by the
components
o Component matrix
o Loadings of variables on components prior to rotation
o Rotated component matrix
o Loadings of variables on components following rotation
Worksheet Qus & Ans
‘Total Variance
Explained’
14. How many components
have we extracted?
4
15. What percentage of
variance in the
correlation matrix is
explained by each
component before and
after rotation?
Before Rotation
After Rotation
C1
31.7%
16.2%
C2
7.6%
14.5%
C3
5.7%
11.1%
C4
5.3%
8.5%
50.3%
50.3%
All four
Worksheet Qus & Ans
16. What has rotation done to the total percentage of
variance explained by all four components?
Nothing!
17. What has rotation done to the amount of variance
explained by each component?
Redistributed the variance, making it more even
Worksheet Qus & Ans
‘Component Matrix’
18. Examine the loadings of variables on components in the
component matrix and try to figure out what each
component represents.
C1:
C2:
C3:
C4:
SPSS, computer & stats anxiety
Maths & friends
?
Maths & friends
Worksheet Qus & Ans
‘Rotated Component Matrix’
19. Examine the loadings of variables on components in the
rotated component matrix and try to figure out what
each component represents.
C1:
C2:
C3:
C4:
Computer anxiety
Statistics anxiety
Mathematical ability
Anxiety over friends’ opinions
Worksheet Qus & Ans
20. Which matrix was easier to interpret?
The rotated Component Matrix
‘Communalities’
21. Which variable has the highest communality following
extraction?
‘I have never been good at mathematics’ .739
22. What does this tell us?
73.9% of the variance in this item is explained by the
four extracted components
Worksheet Qus & Ans
23. Which variable has the lowest communality following
extraction?
‘Computers are only useful for playing games’ .335
24. What does this tell us?
33.5% of the variance in this item is explained by the
four extracted components
Worksheet Qus & Ans
Are you happy with your factor solution?
25. Is the solution interpretable?
Yes!
26. What percentage of variance in the correlation matrix
is explained by the four components?
50%
Cronbach’s Alpha
• Questionnaire
– Each item on a test or questionnaire should measure
the same construct
– Each item on a subscale should measure the same
construct
• Cronbach’s Alpha
– Reliability index
– Internal consistency
– The average correlation among a set of items
Cronbach’s Alpha
– If a test is perfectly reliable, perfectly internally
consistent, the average correlation among items would
be 1
– The more error there is in the test, the lower the
average correlation
– Nunally (1978): alpha > .7
• internally consistent, reliable test with items that all measure
the same thing
Reliability of Scale
• Analyse, Scale, Reliability analysis
• Model; Alpha
• Statistics: Descriptives for,
The entire scale
Factor 1 comprises items 6, 7, 10, 13, 14, 15, 18:
Factor 2 comprises items1, 3, 4, 5, 12, 16, 20, 21:
Factor 3 comprises items 8, 17, 11
Factor 4 comprises items 2, 9, 19, 22, 23
Alpha =
Alpha =
Alpha =
Alpha =
Alpha =
The scale was developed to assess SPSS anxiety. Given the results
of the FA, would you be happy to use this scale as a measure of
SPSS anxiety?
Reliability of Scale
• Analyse, Scale, Reliability analysis
• Model; Alpha
• Statistics: Descriptives for,
The entire scale
Factor 1 comprises items 6, 7, 10, 13, 14, 15, 18:
Factor 2 comprises items 1, 3, 4, 5, 12, 16, 20, 21:
Factor 3 comprises items 8, 17, 11
Factor 4 comprises items 2, 9, 19, 22, 23
Alpha = .754
Alpha = .823
Alpha = .605
Alpha = .819
Alpha = .570
The scale was developed to assess SPSS anxiety. Given the results
of the FA, would you be happy to use this scale as a measure of
SPSS anxiety?
Example 2
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12-item scale measuring personality
Do these 12 items measure the same construct or do
different factors underlie the questionnaire?
Software / Kevin Thomas / Factor analysis dataset
1. Is there a sufficient number of correlations in the
matrix to merit factor analysis?
Yes!
KMO = .822
Bartlett Test of Sphericity : X2 = 738.89, df = 66,
p<.001
Example 2
2.
How many
components should
you extract?
Number of components
with eigenvalues > 1
3
% of variance explained
by these components?
74%
What does the scree plot
suggest?
Example 2
3.
What are the components measuring?
C1 - Sociability
C2 - Depression
C3 - Paranoia
4. Are the subscales reliable?
Scale
Alpha
Entire
Questionnaire
.823
C1
.917
C2
.842
C3
.946
5. Is the questionnaire
a valid and reliable
measure of
personality?
How many factors constitute personality?
•Example of use of FA for theoretical understanding
•Different theorists say there are different number of core personality
components
•Eysenck: 3 main components to personality (EPQ)
• Psychoticism
• Neuroticism
• Extraversion
•Big 5 model (NEO)
• Openness
• Conscientiousness
• Extraversion
• Agreeableness
• Neuroticism
•Perform FA on N=400 people comparing the EPQ with NEO
Examine the factor matrix…
Scale
NEO C
NEO A
NEO E
NEO O
NEO N
EPQ P
EPQ E
EPQ N
Eigenvalue
Percentage variance
Cum. % variance
Factor matrix table
Factor 1
Factor 2
-.009
-.760
.064
-.468
.902
-.193
.613
.175
-.203
.064
.221
.890
.873
.148
-.191
-.005
2.35
1.77
29.4
22.1
29.4
51.5
Factor 3
-.022
-.412
-.251
-.171
.916
.018
-.062
.930
1.42
17.8
69.3
h2
.593
.355
.883
.398
.841
.819
.780
.874
Example 3
What are the factors measuring?
Factor 1:
Extraversion
Factor 2:
Psychoticism
Factor 3:
Neuroticism
Based on the communalities…
Variable worst accounted for:
Variable best accounted for:
NEO Agreeableness
NEO Extraversion
Would you accept such a model of personality?