FACTOR ANALYSIS &
SPSS
• First, let’s check the reliability of the scale
• Go to
• Analyze,
• Scale and
• Reliability analysis
• Select the items and transfer them into the box on the right
(under ‘Items)
• Click on Statistics and select the analysis that you need:
•
•
•
•
•
Item
Scale
Scale if item deleted
Correlations
Means, etc
• Then click continue, and then OK
• You will see the output and the analysis you asked for
• Look at Alpha
Reliability Statistics
Cronbach's Alpha
Cronbach's Alpha Based
N of Items
on Standardized Items
,722
,728
30
Acepted to be reliable
• Alpha is over .70 so it is reliable enough.
• No need to delete an item
• But let’s see if deleting an item will make the alpha much
higher
Item-Total Statistics
Scale Mean if
Scale Variance
Corrected Item-
Squared
Cronbach's
Item Deleted
if Item Deleted
Total
Multiple
Alpha if Item
Correlation
Correlation
Deleted
Q1
88,6117
155,671
-,358
,550
,751
Q2
88,0583
156,448
-,341
,625
,756
Q3
87,1553
141,250
,197
,572
,718
Q4
87,1748
141,989
,175
,509
,719
Q5
87,6893
134,471
,331
,426
,709
Q6
88,3786
129,159
,483
,594
,697
Q7
87,9417
133,957
,413
,462
,704
Q8
88,5049
155,978
-,365
,581
,752
Q9
87,5146
145,174
-,011
,365
,732
Q10
87,2136
137,777
,333
,558
,711
Q11
88,4175
130,148
,519
,543
,696
Q12
87,8835
131,320
,439
,612
,701
Q13
87,5825
136,677
,340
,529
,710
Q14
88,0097
136,853
,283
,523
,713
Q15
88,0388
134,077
,372
,586
,706
Q16
87,8350
148,159
-,107
,638
,737
Q17
87,5825
132,775
,507
,592
,700
Q18
89,0000
128,647
,563
,607
,693
Q19
87,8544
132,126
,433
,633
,702
Q20
87,6796
135,926
,365
,625
,708
Q21
88,7184
130,302
,540
,620
,696
Q22
88,2816
137,028
,294
,567
,712
Q23
88,1068
150,469
-,187
,442
,741
Q24
88,6796
146,867
-,065
,577
,735
Q25
87,8641
129,432
,583
,686
,693
Q26
87,9029
134,814
,401
,654
,706
Q27
88,5437
131,937
,416
,543
,703
Q28
88,2621
127,156
,584
,651
,690
Q29
88,3592
127,958
,580
,584
,691
Q30
88,6893
150,903
-,188
,560
,745
Factor Analysis
• When do we need factor analysis?
•
•
•
•
•
•
•
•
•
•
•
to explore a content area,
to structure a domain,
to map unknown concepts,
to classify or reduce data,
to show causal relationships,
to screen or transform data,
to define relationships,
to test hypotheses,
to formulate theories,
to control variables,
to make inferences.
Two Aproaches
• 1. Exploratory factor analysis (EFA)
• It is used to uncover the underlying structure of a relatively
large set of variables.
• The goal is to identify the underlying relationships between
measured variables.
• It is commonly used by researchers when developing a scale
and serves to identify a set of latent constructs underlying a
battery of measured variables.
• It should be used when the researcher has no a priori
hypothesis about factors or patterns of measured variables.
• 2. Confirmatory factor analysis (CFA)
• It is used to test whether measures of a construct are
consistent with a researcher's understanding of the nature of
that construct (or factor).
• The objective of confirmatory factor analysis is to test whether
the data fit a hypothesized measurement model.
• This hypothesized model is based on theory and/or previous
analytic research.
• In CFA, the researcher first develops a hypothesis about what
factors s/he believes are underlying the measures s/he has
used and may impose constraints on the model based on
these a priori hypotheses
Using factor analysis in SPSS
(EFA)
• Step 1.
• Go to Analyze
• Select Reduction (Dimension Reduction)
• Select Factor
• Step 2.
• Select the variables and transfer them to Variables box
•
•
•
•
Step 3.
Select Descriptives
Click on the statistics that you need
E.g.
• coefficients
• Significance levels
• KMO and Barlett’s
• Then click Continue
• Step 4. Click Extraction
• Select Scree Plot
• Make sure
• Method is Principal Component
• Correlation matrix is checked
• Eigenvalue is greater than 1
• Click Continue
• Step 5. Click Rotation
• Select Varimax
• Click Continue
• Step 6. Click Options
• Select Sorted by Size
• Make sure Exclude cases listwise is selected
• Click continue
• Step 7: Click OK
• You will see the analysis results as Output document
• A) Correlation matrix
Analysing correlation matrix
• If a variable has no relationship with any other variable, it
should be taken out.
• If a variable has a correlation of .9 or above (perfect
correlation) with another variable, you should consider taking
it out.
B) KMO & Barlett’s Test
KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy.
Approx. Chi-Square
Bartlett's Test of Sphericity
,804
1404,825
df
435
Sig.
,000
KMO shows the suitability of your data for factor analysis.
0,93 shows that this data is perfect.
Above 0,8 very good
0,7-0,8 good,
0,5-0,7 medium,
below 0,5, you should collect more data
Bartlett shows the significance.
C) Commonalities
This shows the common variances.
We understand to what extent the variance after
factor extraction is common.
E.g. For Question 1, 66% of the variance is
common. Some info is missing.
The present factors cannot explain all the variance
Communalities
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
Q10
Q11
Q12
Q13
Q14
Q15
Q16
Q17
Q18
Q19
Q20
Q21
Q22
Q23
Q24
Q25
Q26
Q27
Q28
Q29
Q30
Initial Extraction
1,000
,665
1,000
,729
1,000
,801
1,000
,804
1,000
,538
1,000
,707
1,000
,567
1,000
,635
1,000
,498
1,000
,753
1,000
,619
1,000
,731
1,000
,753
1,000
,673
1,000
,717
1,000
,730
1,000
,664
1,000
,760
1,000
,743
1,000
,784
1,000
,691
1,000
,717
1,000
,786
1,000
,702
1,000
,764
1,000
,772
1,000
,689
1,000
,677
1,000
,626
1,000
,647
D) Total Variance Explained
Total Variance Explained
Component
Total
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
,840
,751
,713
,683
,643
,619
,557
,526
,455
,446
,429
,349
,332
,307
,270
,265
,219
,186
,175
,153
,138
Initial Eigenvalues
Extraction Sums of Squared Loadings
Rotation Sums of Squared Loadings
% of Variance Cumulative %
Total
Total
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
2,799
2,503
2,376
2,276
2,145
2,065
1,856
1,754
1,518
1,486
1,430
1,165
1,108
1,024
,901
,882
,731
,621
,582
,509
,459
Extraction Method: Principal Component Analysis.
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
72,610
75,114
77,490
79,766
81,910
83,975
85,830
87,585
89,102
90,589
92,019
93,184
94,291
95,315
96,216
97,098
97,829
98,450
99,032
99,541
100,000
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
% of Variance Cumulative %
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
3,132
2,895
2,814
2,699
2,544
2,218
1,896
1,384
1,360
% of Variance Cumulative %
10,440
9,649
9,380
8,997
8,480
7,394
6,321
4,614
4,535
10,440
20,089
29,470
38,467
46,947
54,341
60,662
65,276
69,811
Eigenvalues before and after rotation.
There are 9 factors with eigenvalues bigger than
1.
The first factor covers 29% of the variance.
Rotation equals the importance of the factors
Factor 1’s contribution reduces to 10% from
29%.
The 9 factors explain about 70% of the total
variance
D) Total Variance Explained
Total Variance Explained
Component
Total
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
,840
,751
,713
,683
,643
,619
,557
,526
,455
,446
,429
,349
,332
,307
,270
,265
,219
,186
,175
,153
,138
Initial Eigenvalues
Extraction Sums of Squared Loadings
Rotation Sums of Squared Loadings
% of Variance Cumulative %
Total
Total
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
2,799
2,503
2,376
2,276
2,145
2,065
1,856
1,754
1,518
1,486
1,430
1,165
1,108
1,024
,901
,882
,731
,621
,582
,509
,459
Extraction Method: Principal Component Analysis.
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
72,610
75,114
77,490
79,766
81,910
83,975
85,830
87,585
89,102
90,589
92,019
93,184
94,291
95,315
96,216
97,098
97,829
98,450
99,032
99,541
100,000
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
% of Variance Cumulative %
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
3,132
2,895
2,814
2,699
2,544
2,218
1,896
1,384
1,360
% of Variance Cumulative %
10,440
9,649
9,380
8,997
8,480
7,394
6,321
4,614
4,535
10,440
20,089
29,470
38,467
46,947
54,341
60,662
65,276
69,811
Eigenvalues before and after rotation.
There are 9 factors with eigenvalues bigger than
1.
The first factor covers 29% of the variance.
Rotation equals the importance of the factors
Factor 1’s contribution reduces to 10% from
29%.
The 9 factors explain about 70% of the total
variance
D) Total Variance Explained
Total Variance Explained
Component
Total
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
,840
,751
,713
,683
,643
,619
,557
,526
,455
,446
,429
,349
,332
,307
,270
,265
,219
,186
,175
,153
,138
Initial Eigenvalues
Extraction Sums of Squared Loadings
Rotation Sums of Squared Loadings
% of Variance Cumulative %
Total
Total
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
2,799
2,503
2,376
2,276
2,145
2,065
1,856
1,754
1,518
1,486
1,430
1,165
1,108
1,024
,901
,882
,731
,621
,582
,509
,459
Extraction Method: Principal Component Analysis.
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
72,610
75,114
77,490
79,766
81,910
83,975
85,830
87,585
89,102
90,589
92,019
93,184
94,291
95,315
96,216
97,098
97,829
98,450
99,032
99,541
100,000
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
% of Variance Cumulative %
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
3,132
2,895
2,814
2,699
2,544
2,218
1,896
1,384
1,360
% of Variance Cumulative %
10,440
9,649
9,380
8,997
8,480
7,394
6,321
4,614
4,535
10,440
20,089
29,470
38,467
46,947
54,341
60,662
65,276
69,811
Eigenvalues before and after rotation.
There are 9 factors with eigenvalues bigger than
1.
The first factor covers 29% of the variance.
Rotation equals the importance of the factors
Factor 1’s contribution reduces to 10% from
29%.
The 9 factors explain about 70% of the total
variance
D) Total Variance Explained
Total Variance Explained
Component
Total
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
,840
,751
,713
,683
,643
,619
,557
,526
,455
,446
,429
,349
,332
,307
,270
,265
,219
,186
,175
,153
,138
Initial Eigenvalues
Extraction Sums of Squared Loadings
Rotation Sums of Squared Loadings
% of Variance Cumulative %
Total
Total
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
2,799
2,503
2,376
2,276
2,145
2,065
1,856
1,754
1,518
1,486
1,430
1,165
1,108
1,024
,901
,882
,731
,621
,582
,509
,459
Extraction Method: Principal Component Analysis.
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
72,610
75,114
77,490
79,766
81,910
83,975
85,830
87,585
89,102
90,589
92,019
93,184
94,291
95,315
96,216
97,098
97,829
98,450
99,032
99,541
100,000
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
% of Variance Cumulative %
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
3,132
2,895
2,814
2,699
2,544
2,218
1,896
1,384
1,360
% of Variance Cumulative %
10,440
9,649
9,380
8,997
8,480
7,394
6,321
4,614
4,535
10,440
20,089
29,470
38,467
46,947
54,341
60,662
65,276
69,811
Eigenvalues before and after rotation.
There are 9 factors with eigenvalues bigger than
1.
The first factor covers 29% of the variance.
Rotation equals the importance of the factors
Factor 1’s contribution reduces to 10% from
29%.
The 9 factors explain about 70% of the total
variance
D) Total Variance Explained
Total Variance Explained
Component
Total
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
,840
,751
,713
,683
,643
,619
,557
,526
,455
,446
,429
,349
,332
,307
,270
,265
,219
,186
,175
,153
,138
Initial Eigenvalues
Extraction Sums of Squared Loadings
Rotation Sums of Squared Loadings
% of Variance Cumulative %
Total
Total
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
2,799
2,503
2,376
2,276
2,145
2,065
1,856
1,754
1,518
1,486
1,430
1,165
1,108
1,024
,901
,882
,731
,621
,582
,509
,459
Extraction Method: Principal Component Analysis.
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
72,610
75,114
77,490
79,766
81,910
83,975
85,830
87,585
89,102
90,589
92,019
93,184
94,291
95,315
96,216
97,098
97,829
98,450
99,032
99,541
100,000
8,663
2,648
2,232
1,491
1,408
1,257
1,163
1,056
1,025
% of Variance Cumulative %
28,876
8,827
7,441
4,970
4,694
4,190
3,876
3,521
3,415
28,876
37,703
45,143
50,113
54,808
58,998
62,874
66,396
69,811
3,132
2,895
2,814
2,699
2,544
2,218
1,896
1,384
1,360
% of Variance Cumulative %
10,440
9,649
9,380
8,997
8,480
7,394
6,321
4,614
4,535
10,440
20,089
29,470
38,467
46,947
54,341
60,662
65,276
69,811
Eigenvalues before and after rotation.
There are 9 factors with eigenvalues bigger than
1.
The first factor covers 29% of the variance.
Rotation equals the importance of the factors
Factor 1’s contribution reduces to 10% from
29%.
The 9 factors explain about 70% of the total
variance
E) Component Matrix (factor
loads)
• Normally loadings bigger than 0,3 are accepted to be
important
• But the number of the sampling is also important
•
•
•
•
For 50 : 0,722
For 100: 0,512
For 200: 0,364
For 300: 0,298
• For our data (150 participants) we can accept 0,40.
• To analyze the loadings let’s look at the component matrix first
Component Matrixa
Component
1
2
3
4
5
6
7
8
9
Q25
,736
-,001
,145
-,100
,135
-,256
-,001
,325
-,048
Q28
,725
-,071
,218
,120
,072
-,173
-,135
,088
,152
Q17
,715
-,156
,109
,010
-,132
,038
,179
,254
-,042
Q2
-,642
,119
,166
,282
,216
-,201
,115
,304
-,047
Q26
,641
-,081
-,030
,150
,261
-,310
-,353
-,105
-,179
Q8
-,629
,180
,098
,113
,183
-,184
-,078
,067
,328
Q1
-,616
,102
,097
,284
,083
-,317
,232
,073
-,138
Q29
,616
,421
,030
,028
,058
-,113
,085
,051
-,203
Q20
,609
-,090
,000
-,383
,353
-,033
,145
,244
-,230
Q12
,597
-,045
,315
,014
-,317
,218
-,260
-,102
,217
Q19
,584
-,006
,081
-,113
,402
-,111
,155
-,148
,403
Q6
,569
,279
,159
,026
-,473
-,151
,072
-,114
-,121
Q13
,567
-,213
,034
,140
-,235
-,124
,286
,161
,432
Q5
,557
-,147
,041
-,150
-,005
,358
-,112
,196
-,048
Q11
,556
,334
,115
,210
-,274
-,184
-,145
-,058
-,086
Q21
,556
,426
,027
-,026
,285
,230
-,016
-,077
,243
Q30
-,514
,377
,360
-,153
,090
,178
,040
-,023
-,215
Q18
,503
,256
,487
,039
,110
-,208
-,321
-,191
,088
Q7
,485
,337
-,147
,186
-,115
,105
,050
,297
-,216
Q27
,482
,057
,475
-,430
,122
-,002
,095
,045
-,128
Q10
,476
-,298
,314
,209
-,141
,234
,468
,032
,026
Q15
,380
,614
-,265
,255
,021
,143
,092
-,174
-,036
Q22
,302
,564
-,281
,083
,367
,277
,089
,001
,056
Q3
,367
-,500
,138
,478
,294
,116
,155
-,195
-,077
Q9
-,256
,214
,577
-,026
-,161
,011
,000
-,107
-,129
Q16
-,483
,109
,567
,299
,075
-,075
-,009
,242
,062
Q24
-,404
,385
,449
-,002
,041
,237
,299
-,196
,056
Q14
,434
,405
-,447
,246
-,142
-,130
,120
,078
,057
Q4
,298
-,388
,114
,525
,201
,342
-,234
-,046
-,249
Q23
-,381
,230
,081
,088
-,066
,339
-,373
,519
,214
Before rotation, most
variables are related to the
first factor (the ones over
0,40)
You can also see this in the
scree plot
F) Scree Plot
• To see the common themes of the variables
under each factor, we should check the loadings
after rotation
• Let’s accept the ones loading above 0,40
Rotated Component Matrixa
Component
1
2
3
4
5
6
7
8
9
Q15
,813
,141
,108
-,093
,030
-,015
,014
,075
-,095
Q22
,746
-,094
,075
,116
,018
-,144
,035
,323
,082
Q14
,684
,141
,006
-,047
-,346
,190
-,144
-,047
-,071
Q7
,589
,151
,094
,265
-,117
,172
,061
-,235
,128
Q29
,550
,367
,073
,393
-,008
,085
-,010
,004
-,142
Q21
,528
,208
,276
,198
,071
,043
,046
,487
,086
Q18
,085
,770
,105
,182
,154
-,016
,089
,291
-,011
Q11
,391
,624
,133
,055
-,043
,175
-,002
-,140
-,066
Q26
,137
,555
,088
,291
-,421
-,143
,326
,117
-,186
Q28
,096
,541
,151
,317
-,241
,314
,202
,233
,018
Q6
,318
,526
,263
,100
,077
,339
-,146
-,240
-,225
Q2
-,111
-,177
-,749
-,093
,201
-,116
-,006
-,078
,233
Q1
-,093
-,145
-,718
-,196
,199
-,088
-,050
-,169
-,065
Q12
,011
,500
,564
,018
,049
,335
,138
,080
,147
Q16
-,231
,114
-,559
-,093
,450
,049
,091
-,024
,358
Q8
-,172
-,079
-,513
-,333
,151
-,189
-,210
,231
,263
Q5
,091
,056
,503
,391
-,110
,169
,214
,014
,185
Q20
,116
,004
,198
,805
-,190
,040
,055
,148
-,146
Q25
,153
,406
,068
,645
-,272
,261
,033
,107
-,004
Q27
-,056
,276
,254
,642
,281
,114
-,052
,158
-,114
Q24
,059
-,120
-,179
-,159
,776
,000
-,085
,133
,011
Q30
-,024
-,124
-,199
,011
,665
-,326
-,134
-,087
,131
Q9
-,163
,232
-,116
-,039
,615
-,028
-,062
-,136
,037
Q13
,053
,182
,125
,080
-,294
,757
-,011
,185
-,034
Q10
,026
-,002
,178
,207
,148
,700
,378
,005
-,153
Q17
,151
,201
,266
,454
-,179
,513
,158
-,057
-,029
Q4
,006
,109
,137
,018
-,106
,019
,864
-,075
,097
Q3
-,059
,037
-,005
,057
-,125
,274
,781
,169
-,250
Q19
,131
,198
,132
,281
-,148
,231
,050
,689
-,195
Q23
,005
-,097
-,082
-,139
,128
-,116
-,094
-,083
,840
• Now it’s time to decide the factors, their labels, and the items under each factor.
• After reviewing the literature, you have decided to group the items under these categories
• External attribution: Assessment measures students future and intelligence or
the quality of schooling
• Improvement: Assessment improves students’ learning and teachers’ teaching
• Irrelevance: Assessment is ignored or perceived negatively
• Affect: Assessment is enjoyable and benefits the class environment
• Now, look at the questionnaire items and factor loadings and decide the factors.
• At this step, we may need some qualitative decisions as well.
• See which items can be grouped under the lables
• (if you don’t have previously decided labels, you also need to decide the lables
of the factos)
• If there are some unrelated items, see whether it can fit in another factor.
References
• yunus.hacettepe.edu.tr/~tonta/courses/spring2008/bby208/
• Büyüköztürk, Ş. (2009). Sosyal Bilimler İçin Veri Analizi El
Kitabı, Ankara:Pegem Akademi
• http://www.hawaii.edu/powerkills/UFA.HTM