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Principal component and exploratory factor analysis
Hans Baumgartner
Penn State University
Principal component and exploratory factor analysis
x1 x2 x3 x4 x5 x6 x7 x8
What’s the structure underlying
28 distinct covariances between
8 observed variables?
Principal component and exploratory factor analysis
The principal component model
x1 x2 x3 x4 x5 x6 x7 x8
x1 xn. . . . . . . . . . . . . . .
Principal component and exploratory factor analysis
How many components to retain?
▪ Eigenvalue greater than one rule
(Kaiser’s rule)
▪ Scree test (Cattell)
▪ Parallel analysis (Horn)
▪ Minimum Average Partial (MAP) test
(Velicer)
▪ Theoretical considerations
Principal component and exploratory factor analysis
Factor rotation
▪ Orthogonal rotation
e.g., Varimax
▪ Oblique rotation
e.g., Promax
Principal component and exploratory factor analysis
SAS specification
DATA coupon;
INFILE 'd:\ipss\factor.dat';
INPUT id aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2;
ods graphics on;
TITLE 'PRINCIPAL COMPONENT ANALYSIS, TWO COMPONENTS'; run;
PROC FACTOR NFACTORS=2 ROTATE=PROMAX PLOTS=(scree preloadings loadings);
VAR aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2;
run;
ods graphics off;
Principal component and exploratory factor analysis
The FACTOR Procedure
Initial Factor Method: Principal Components
Prior Communality Estimates: ONE
Eigenvalues of the Correlation Matrix: Total = 8 Average = 1
Eigenvalue Difference Proportion Cumulative
1 5.56628614 4.99193128 0.6958 0.6958
2 0.57435486 0.05222193 0.0718 0.7676
3 0.52213293 0.14851325 0.0653 0.8328
4 0.37361969 0.06185008 0.0467 0.8795
5 0.31176961 0.05897629 0.0390 0.9185
6 0.25279331 0.03126256 0.0316 0.9501
7 0.22153076 0.04401806 0.0277 0.9778
8 0.17751270 0.0222 1.0000
2 factors will be retained by the NFACTOR criterion.
Principal component and exploratory factor analysis
Principal component and exploratory factor analysis
Parallel Analysis:
Principal Components
Specifications for this Run:Ncases 250Nvars 8Ndatsets 100Percent 95
Random Data Eigenvalues
Root Means Prcntyle
1.000000 1.263893 1.3444632.000000 1.163970 1.2300473.000000 1.088389 1.1332354.000000 1.026321 1.0601315.000000 0.967248 1.0070766.000000 0.904000 0.9466257.000000 0.834617 0.8784278.000000 0.751562 0.807438
Principal component and exploratory factor analysis
Velicer's Minimum Average Partial (MAP) Test:
Eigenvalues
5.56600.57820.52100.37270.30760.25700.22050.1772
Average Partial Correlations
squared power4
0.0000 0.4280 0.18921.0000 0.0440 0.00412.0000 0.0662 0.01503.0000 0.1058 0.03974.0000 0.1692 0.09125.0000 0.2582 0.12246.0000 0.4869 0.36537.0000 1.0000 1.0000
The smallest average squared partial correlation is0.0440
The smallest average 4rth power partial correlation is0.0041
The Number of Components According to the Original (1976) MAP Test is1.0000
Principal component and exploratory factor analysis
Factor PatternFactor1 Factor2
aa1t1 0.81547 0.35121aa2t1 0.83341 0.34182aa3t1 0.77536 0.01958aa4t1 0.84339 0.23736aa1t2 0.86051 -0.03867aa2t2 0.86274 -0.23905aa3t2 0.82149 -0.37671aa4t2 0.85709 -0.27731
Variance Explained by Each Factor
Factor1 Factor2
5.5662861 0.5743549
Final Communality Estimates: Total = 6.140641
aa1t1 aa2t1 aa3t1 aa4t10.78833603 0.81141733 0.60156808 0.76764683
aa1t2 aa2t2 aa3t2 aa4t20.74197084 0.80145580 0.81674966 0.81149643
Principal component and exploratory factor analysis
The FACTOR Procedure
Prerotation Method: Varimax
Rotated Factor Pattern
Factor1 Factor2
aa1t1 0.33675 0.82154
aa2t1 0.35614 0.82739
aa3t1 0.54017 0.55658
aa4t1 0.43637 0.75976
aa1t2 0.64176 0.57455
aa2t2 0.78357 0.43298
aa3t2 0.85044 0.30577
aa4t2 0.80631 0.40169
Principal component and exploratory factor analysis
The FACTOR ProcedureRotation Method: Promax (power = 3)
Inter-Factor Correlations
Factor1 Factor2
Factor1 1.00000 0.67589Factor2 0.67589 1.00000
Rotated Factor Pattern (Standardized Regression Coefficients)
Factor1 Factor2
aa1t1 0.02338 0.87192aa2t1 0.04504 0.86973aa3t1 0.41062 0.43660aa4t1 0.17970 0.74463aa1t2 0.53035 0.40913aa2t2 0.77916 0.16017aa3t2 0.92612 -0.03361aa4t2 0.82327 0.10941
Principal component and exploratory factor analysis
The exploratory factor model
d1 d2 d3 d4 d5 d6 d7 d8
x1 x2 x3 x4 x5 x6 x7 x8
x1 xn
q11d q22d q33d q44d q55d q66d q77d q88d
11
. . . . . . . . . . . . . . .
Principal component and exploratory factor analysis
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7271
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4241
3231
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Two-factor model
Principal component and exploratory factor analysis
SAS specification
TITLE 'PRINCIPAL FACTOR ANALYSIS, TWO FACTORS'; run;PROC FACTOR NFACTORS=2 ROTATE=PROMAX PRIORS=SMC;VAR aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2;
run;
TITLE 'EXPLORATORY MAXIMUM LIKELIHOOD FACTOR ANALYSIS, TWO FACTORS'; run;PROC FACTOR METHOD=ML NFACTORS=2 ROTATE=PROMAX SE COVER=.40 ALPHA=.1;VAR aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2;
run;
Principal component and exploratory factor analysis
The FACTOR ProcedureInitial Factor Method: Principal Factors
Prior Communality Estimates: SMC
aa1t1 aa2t1 aa3t1 aa4t1
0.61896228 0.66565385 0.55622099 0.66929517
aa1t2 aa2t2 aa3t2 aa4t2
0.69849794 0.70624991 0.64787954 0.70324282
Eigenvalues of the Reduced Correlation Matrix:Total = 5.26600251 Average = 0.65825031
Eigenvalue Difference Proportion Cumulative
1 5.22802782 4.98990454 0.9928 0.99282 0.23812328 0.10148438 0.0452 1.03803 0.13663890 0.10807443 0.0259 1.06404 0.02856447 0.05139663 0.0054 1.06945 -.02283216 0.05684943 -0.0043 1.06506 -.07968159 0.02020619 -0.0151 1.04997 -.09988779 0.06306265 -0.0190 1.03098 -.16295043 -0.0309 1.0000
Principal component and exploratory factor analysis
Parallel Analysis:
Principal Axis / Common Factor Analysis
Specifications for this Run:Ncases 250Nvars 8Ndatsets 100Percent 95
Random Data Eigenvalues
Root Means Prcntyle
1.000000 0.297504 0.3896522.000000 0.193068 0.2688263.000000 0.113040 0.1589434.000000 0.048229 0.0872085.000000 -0.011975 0.0373076.000000 -0.073090 -0.0345227.000000 -0.137988 -0.1000638.000000 -0.213994 -0.163157
Principal component and exploratory factor analysis
Principal component and exploratory factor analysis
The FACTOR Procedure
Rotation Method: Promax (power = 3)
Inter-Factor Correlations
Factor1 Factor2
Factor1 1.00000 0.74679Factor2 0.74679 1.00000
Rotated Factor Pattern (Standardized Regression Coefficients)
Factor1 Factor2
aa1t1 0.15925 0.68215aa2t1 0.12325 0.74482aa3t1 0.35117 0.43689aa4t1 0.21434 0.66378aa1t2 0.51948 0.37873aa2t2 0.71140 0.18860aa3t2 0.72635 0.12017aa4t2 0.74590 0.14729
Principal component and exploratory factor analysis
Principal component and exploratory factor analysis
EXPLORATORY MAXIMUM LIKELIHOOD FACTOR ANALYSIS, TWO FACTORS
Significance Tests Based on 250 Observations
Pr >Test DF Chi-Square ChiSq
H0: No common factors 28 1534.0584
Principal component and exploratory factor analysis
The FACTOR Procedure
Rotation Method: Promax (power = 3)
Inter-Factor CorrelationsWith 90% confidence limits
Estimate/StdErr/LowerCL/UpperCL
Factor1 Factor2
Factor1 1.00000 0.728480.00000 0.01570. 0.70161. 0.75328
Factor2 0.72848 1.000000.01570 0.000000.70161 .0.75328 .
Principal component and exploratory factor analysis
Rotated Factor Pattern (Standardized Regression Coefficients)With 90% confidence limits; Cover |*| = 0.4?
Estimate/StdErr/LowerCL/UpperCL/Coverage Display
Factor1 Factor2
aa1t1 0.19857 0.646280.08017 0.077720.06389 0.500040.32616 0.75668
0[]* 0*[]
aa2t1 0.06195 0.827830.05207 0.05846-0.02394 0.704230.14693 0.90272
[0]* 0*[]