Mplus VERSION 8.8
MUTHEN & MUTHEN
04/19/2022 11:06 PM
INPUT INSTRUCTIONS
TITLE: this is an example of a Monte Carlo
simulation study for an exploratory factor
analysis with continuous factor indicators
MONTECARLO:
NAMES ARE y1-y10;
NOBSERVATIONS = 500;
NREPS = 500;
MODEL POPULATION:
f1 BY y1-y7*.5;
f2 BY y4-y5*.25 y6-y10*.8;
f1-f2@1;
f1 WITH f2*.5;
y1-y10*.36;
MODEL:
f1 BY y1-y7*.5 y8-y10*0 (*1);
f2 BY y1-y3*.0 y4-y5*.25 y6-y10*.8 (*1);
f1 WITH f2*.5;
y1-y10*.36;
OUTPUT: TECH9;
*** WARNING in MODEL command
Values given in the MODEL command for EFA factors are used only as
population values in computing coverage. EFA factors cannot be given
starting values.
1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS
this is an example of a Monte Carlo
simulation study for an exploratory factor
analysis with continuous factor indicators
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 500
Number of replications
Requested 500
Completed 500
Value of seed 0
Number of dependent variables 10
Number of independent variables 0
Number of continuous latent variables 2
Observed dependent variables
Continuous
Y1 Y2 Y3 Y4 Y5 Y6
Y7 Y8 Y9 Y10
Continuous latent variables
EFA factors
*1: F1 F2
Estimator ML
Rotation GEOMIN
Row standardization CORRELATION
Type of rotation OBLIQUE
Epsilon value Varies
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Optimization Specifications for the Exploratory Factor Analysis
Rotation Algorithm
Number of random starts 30
Maximum number of iterations 10000
Derivative convergence criterion 0.100D-04
SAMPLE STATISTICS FOR THE FIRST REPLICATION
SAMPLE STATISTICS
Means
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
-0.057 0.067 0.023 -0.033 0.021
Means
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
-0.033 0.010 0.047 -0.013 -0.001
Covariances
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.604
Y2 0.286 0.613
Y3 0.251 0.266 0.572
Y4 0.327 0.344 0.342 0.848
Y5 0.327 0.389 0.354 0.511 0.860
Y6 0.459 0.455 0.474 0.759 0.799
Y7 0.483 0.495 0.451 0.787 0.771
Y8 0.223 0.199 0.241 0.452 0.489
Y9 0.181 0.211 0.251 0.427 0.446
Y10 0.173 0.156 0.213 0.397 0.378
Covariances
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
Y6 1.643
Y7 1.322 1.715
Y8 0.856 0.888 1.104
Y9 0.848 0.832 0.662 0.987
Y10 0.786 0.751 0.586 0.593 0.943
Correlations
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 1.000
Y2 0.471 1.000
Y3 0.427 0.449 1.000
Y4 0.457 0.477 0.491 1.000
Y5 0.454 0.535 0.505 0.598 1.000
Y6 0.461 0.454 0.489 0.643 0.673
Y7 0.475 0.483 0.456 0.652 0.635
Y8 0.273 0.242 0.303 0.467 0.502
Y9 0.234 0.271 0.334 0.466 0.484
Y10 0.229 0.205 0.290 0.444 0.420
Correlations
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
Y6 1.000
Y7 0.787 1.000
Y8 0.636 0.645 1.000
Y9 0.666 0.640 0.634 1.000
Y10 0.632 0.590 0.575 0.614 1.000
MODEL FIT INFORMATION
Number of Free Parameters 39
Loglikelihood
H0 Value
Mean -5475.366
Std Dev 48.491
Number of successful computations 500
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.994 -5588.171 -5586.155
0.980 0.982 -5574.953 -5567.199
0.950 0.960 -5555.129 -5553.799
0.900 0.894 -5537.512 -5539.458
0.800 0.786 -5516.176 -5519.058
0.700 0.690 -5500.795 -5501.706
0.500 0.492 -5475.366 -5476.087
0.300 0.298 -5449.937 -5450.482
0.200 0.198 -5434.556 -5435.323
0.100 0.104 -5413.219 -5412.413
0.050 0.054 -5395.603 -5394.122
0.020 0.016 -5375.779 -5377.226
0.010 0.008 -5362.561 -5363.708
H1 Value
Mean -5462.274
Std Dev 48.668
Number of successful computations 500
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.994 -5575.491 -5573.248
0.980 0.984 -5562.224 -5557.150
0.950 0.956 -5542.328 -5541.496
0.900 0.886 -5524.647 -5527.580
0.800 0.788 -5503.233 -5506.013
0.700 0.696 -5487.796 -5488.916
0.500 0.500 -5462.274 -5462.526
0.300 0.308 -5436.752 -5436.250
0.200 0.200 -5421.315 -5421.664
0.100 0.106 -5399.901 -5397.939
0.050 0.052 -5382.219 -5380.525
0.020 0.020 -5362.324 -5362.533
0.010 0.010 -5349.057 -5353.842
Information Criteria
Akaike (AIC)
Mean 11028.732
Std Dev 96.983
Number of successful computations 500
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.992 10803.121 10804.444
0.980 0.984 10829.559 10831.682
0.950 0.946 10869.205 10865.296
0.900 0.896 10904.439 10901.446
0.800 0.802 10947.111 10945.795
0.700 0.702 10977.874 10978.741
0.500 0.508 11028.732 11030.063
0.300 0.310 11079.590 11081.081
0.200 0.214 11110.353 11116.113
0.100 0.106 11153.025 11156.274
0.050 0.040 11188.259 11185.271
0.020 0.018 11227.905 11208.758
0.010 0.006 11254.343 11246.986
Bayesian (BIC)
Mean 11193.102
Std Dev 96.983
Number of successful computations 500
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.992 10967.491 10968.814
0.980 0.984 10993.928 10996.052
0.950 0.946 11033.575 11029.666
0.900 0.896 11068.809 11065.815
0.800 0.802 11111.481 11110.165
0.700 0.702 11142.244 11143.111
0.500 0.508 11193.102 11194.433
0.300 0.310 11243.959 11245.451
0.200 0.214 11274.722 11280.483
0.100 0.106 11317.395 11320.644
0.050 0.040 11352.628 11349.641
0.020 0.018 11392.275 11373.128
0.010 0.006 11418.712 11411.356
Sample-Size Adjusted BIC (n* = (n + 2) / 24)
Mean 11069.313
Std Dev 96.983
Number of successful computations 500
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.992 10843.702 10845.025
0.980 0.984 10870.140 10872.264
0.950 0.946 10909.786 10905.877
0.900 0.896 10945.020 10942.027
0.800 0.802 10987.693 10986.376
0.700 0.702 11018.456 11019.322
0.500 0.508 11069.313 11070.644
0.300 0.310 11120.171 11121.662
0.200 0.214 11150.934 11156.694
0.100 0.106 11193.606 11196.856
0.050 0.040 11228.840 11225.853
0.020 0.018 11268.486 11249.339
0.010 0.006 11294.924 11287.567
Chi-Square Test of Model Fit
Degrees of freedom 26
Mean 26.184
Std Dev 7.621
Number of successful computations 500
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.982 12.198 10.892
0.980 0.980 13.409 13.069
0.950 0.940 15.379 14.829
0.900 0.892 17.292 16.854
0.800 0.806 19.820 19.854
0.700 0.698 21.792 21.763
0.500 0.514 25.336 25.506
0.300 0.318 29.246 29.899
0.200 0.202 31.795 31.824
0.100 0.110 35.563 36.299
0.050 0.056 38.885 39.490
0.020 0.032 42.856 43.812
0.010 0.010 45.642 45.552
RMSEA (Root Mean Square Error Of Approximation)
Mean 0.010
Std Dev 0.012
Number of successful computations 500
Cumulative Distribution Function
Value Function Value
0.990 1.000
0.980 1.000
0.950 1.000
0.900 1.000
0.800 1.000
0.700 1.000
0.500 1.000
0.300 1.000
0.200 1.000
0.100 1.000
0.050 1.000
0.020 0.768
0.010 0.600
CFI/TLI
CFI
Mean 0.999
Std Dev 0.002
Number of successful computations 500
Cumulative Distribution Function
Value Function Value
0.990 0.004
0.980 0.000
0.950 0.000
0.900 0.000
0.800 0.000
0.700 0.000
0.500 0.000
0.300 0.000
0.200 0.000
0.100 0.000
0.050 0.000
0.020 0.000
0.010 0.000
TLI
Mean 0.998
Std Dev 0.003
Number of successful computations 500
Cumulative Distribution Function
Value Function Value
0.990 0.034
0.980 0.000
0.950 0.000
0.900 0.000
0.800 0.000
0.700 0.000
0.500 0.000
0.300 0.000
0.200 0.000
0.100 0.000
0.050 0.000
0.020 0.000
0.010 0.000
SRMR (Standardized Root Mean Square Residual)
Mean 0.012
Std Dev 0.002
Number of successful computations 500
Cumulative Distribution Function
Value Function Value
0.990 1.000
0.980 1.000
0.950 1.000
0.900 1.000
0.800 1.000
0.700 1.000
0.500 1.000
0.300 1.000
0.200 1.000
0.100 1.000
0.050 1.000
0.020 1.000
0.010 0.172
MODEL RESULTS
ESTIMATES S. E. M. S. E. 95% % Sig
Population Average Std. Dev. Average Cover Coeff
F1 BY
Y1 0.500 0.5062 0.0436 0.0437 0.0019 0.948 1.000
Y2 0.500 0.5056 0.0403 0.0437 0.0016 0.968 1.000
Y3 0.500 0.5082 0.0427 0.0437 0.0019 0.952 1.000
Y4 0.500 0.5060 0.0466 0.0516 0.0022 0.964 1.000
Y5 0.500 0.5088 0.0472 0.0516 0.0023 0.964 1.000
Y6 0.500 0.5055 0.0498 0.0615 0.0025 0.984 1.000
Y7 0.500 0.5058 0.0550 0.0618 0.0030 0.960 1.000
Y8 0.000 -0.0086 0.0367 0.0377 0.0014 0.982 0.018
Y9 0.000 -0.0052 0.0368 0.0370 0.0014 0.988 0.012
Y10 0.000 -0.0050 0.0358 0.0391 0.0013 0.990 0.010
F2 BY
Y1 0.000 -0.0131 0.0354 0.0392 0.0014 0.980 0.020
Y2 0.000 -0.0132 0.0355 0.0390 0.0014 0.990 0.010
Y3 0.000 -0.0145 0.0373 0.0392 0.0016 0.980 0.020
Y4 0.250 0.2359 0.0516 0.0572 0.0029 0.964 0.964
Y5 0.250 0.2327 0.0490 0.0573 0.0027 0.966 0.966
Y6 0.800 0.7824 0.0558 0.0654 0.0034 0.974 1.000
Y7 0.800 0.7838 0.0567 0.0658 0.0035 0.974 1.000
Y8 0.800 0.8029 0.0451 0.0468 0.0020 0.964 1.000
Y9 0.800 0.8008 0.0459 0.0466 0.0021 0.956 1.000
Y10 0.800 0.8009 0.0472 0.0472 0.0022 0.950 1.000
F1 WITH
F2 0.500 0.5295 0.0481 0.0596 0.0032 0.952 0.998
Intercepts
Y1 0.000 -0.0024 0.0359 0.0349 0.0013 0.944 0.056
Y2 0.000 -0.0003 0.0357 0.0349 0.0013 0.936 0.064
Y3 0.000 0.0006 0.0336 0.0349 0.0011 0.970 0.030
Y4 0.000 0.0004 0.0408 0.0400 0.0017 0.946 0.054
Y5 0.000 0.0000 0.0411 0.0399 0.0017 0.958 0.042
Y6 0.000 -0.0002 0.0554 0.0574 0.0031 0.954 0.046
Y7 0.000 0.0021 0.0574 0.0575 0.0033 0.950 0.050
Y8 0.000 0.0013 0.0424 0.0446 0.0018 0.958 0.042
Y9 0.000 0.0006 0.0439 0.0446 0.0019 0.940 0.060
Y10 0.000 0.0010 0.0442 0.0447 0.0019 0.950 0.050
Variances
F1 1.000 1.0000 0.0000 0.0000 0.0000 1.000 0.000
F2 1.000 1.0000 0.0000 0.0000 0.0000 1.000 0.000
Residual Variances
Y1 0.360 0.3583 0.0269 0.0276 0.0007 0.954 1.000
Y2 0.360 0.3584 0.0292 0.0276 0.0009 0.918 1.000
Y3 0.360 0.3566 0.0276 0.0276 0.0008 0.936 1.000
Y4 0.360 0.3594 0.0274 0.0266 0.0008 0.950 1.000
Y5 0.360 0.3573 0.0272 0.0266 0.0007 0.930 1.000
Y6 0.360 0.3562 0.0302 0.0298 0.0009 0.934 1.000
Y7 0.360 0.3586 0.0293 0.0299 0.0009 0.956 1.000
Y8 0.360 0.3550 0.0292 0.0300 0.0009 0.956 1.000
Y9 0.360 0.3567 0.0292 0.0300 0.0009 0.938 1.000
Y10 0.360 0.3587 0.0304 0.0302 0.0009 0.946 1.000
QUALITY OF NUMERICAL RESULTS
Average Condition Number for the Information Matrix 0.392E-01
(ratio of smallest to largest eigenvalue)
TECHNICAL OUTPUT
TECHNICAL 1 OUTPUT FOR THE UNROTATED SOLUTION
PARAMETER SPECIFICATION
NU
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
1 2 3 4 5
NU
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
6 7 8 9 10
LAMBDA
F1 F2
________ ________
Y1 11 0
Y2 12 13
Y3 14 15
Y4 16 17
Y5 18 19
Y6 20 21
Y7 22 23
Y8 24 25
Y9 26 27
Y10 28 29
THETA
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 30
Y2 0 31
Y3 0 0 32
Y4 0 0 0 33
Y5 0 0 0 0 34
Y6 0 0 0 0 0
Y7 0 0 0 0 0
Y8 0 0 0 0 0
Y9 0 0 0 0 0
Y10 0 0 0 0 0
THETA
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
Y6 35
Y7 0 36
Y8 0 0 37
Y9 0 0 0 38
Y10 0 0 0 0 39
ALPHA
F1 F2
________ ________
0 0
BETA
F1 F2
________ ________
F1 0 0
F2 0 0
PSI
F1 F2
________ ________
F1 0
F2 0 0
STARTING VALUES
NU
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
NU
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
LAMBDA
F1 F2
________ ________
Y1 0.510 0.000
Y2 0.563 0.042
Y3 0.480 -0.065
Y4 0.649 -0.260
Y5 0.683 -0.251
Y6 0.889 -0.723
Y7 0.912 -0.693
Y8 0.425 -0.695
Y9 0.395 -0.693
Y10 0.334 -0.657
THETA
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.344
Y2 0.000 0.295
Y3 0.000 0.000 0.337
Y4 0.000 0.000 0.000 0.359
Y5 0.000 0.000 0.000 0.000 0.330
Y6 0.000 0.000 0.000 0.000 0.000
Y7 0.000 0.000 0.000 0.000 0.000
Y8 0.000 0.000 0.000 0.000 0.000
Y9 0.000 0.000 0.000 0.000 0.000
Y10 0.000 0.000 0.000 0.000 0.000
THETA
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
Y6 0.331
Y7 0.000 0.402
Y8 0.000 0.000 0.441
Y9 0.000 0.000 0.000 0.350
Y10 0.000 0.000 0.000 0.000 0.399
ALPHA
F1 F2
________ ________
0.000 0.000
BETA
F1 F2
________ ________
F1 0.000 0.000
F2 0.000 0.000
PSI
F1 F2
________ ________
F1 1.000
F2 0.000 1.000
TECHNICAL 1 OUTPUT FOR THE ROTATED SOLUTION
PARAMETER SPECIFICATION
NU
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
1 2 3 4 5
NU
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
6 7 8 9 10
LAMBDA
F1 F2
________ ________
Y1 11 12
Y2 13 14
Y3 15 16
Y4 17 18
Y5 19 20
Y6 21 22
Y7 23 24
Y8 25 26
Y9 27 28
Y10 29 30
THETA
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 31
Y2 0 32
Y3 0 0 33
Y4 0 0 0 34
Y5 0 0 0 0 35
Y6 0 0 0 0 0
Y7 0 0 0 0 0
Y8 0 0 0 0 0
Y9 0 0 0 0 0
Y10 0 0 0 0 0
THETA
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
Y6 36
Y7 0 37
Y8 0 0 38
Y9 0 0 0 39
Y10 0 0 0 0 40
ALPHA
F1 F2
________ ________
0 0
BETA
F1 F2
________ ________
F1 0 0
F2 0 0
PSI
F1 F2
________ ________
F1 0
F2 41 0
STARTING VALUES
NU
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
NU
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
LAMBDA
F1 F2
________ ________
Y1 0.000 0.000
Y2 0.000 0.000
Y3 0.000 0.000
Y4 0.000 0.000
Y5 0.000 0.000
Y6 0.000 0.000
Y7 0.000 0.000
Y8 0.000 0.000
Y9 0.000 0.000
Y10 0.000 0.000
THETA
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.360
Y2 0.000 0.360
Y3 0.000 0.000 0.360
Y4 0.000 0.000 0.000 0.360
Y5 0.000 0.000 0.000 0.000 0.360
Y6 0.000 0.000 0.000 0.000 0.000
Y7 0.000 0.000 0.000 0.000 0.000
Y8 0.000 0.000 0.000 0.000 0.000
Y9 0.000 0.000 0.000 0.000 0.000
Y10 0.000 0.000 0.000 0.000 0.000
THETA
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
Y6 0.360
Y7 0.000 0.360
Y8 0.000 0.000 0.360
Y9 0.000 0.000 0.000 0.360
Y10 0.000 0.000 0.000 0.000 0.360
ALPHA
F1 F2
________ ________
0.000 0.000
BETA
F1 F2
________ ________
F1 0.000 0.000
F2 0.000 0.000
PSI
F1 F2
________ ________
F1 1.000
F2 0.000 1.000
POPULATION VALUES
NU
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
NU
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
0.000 0.000 0.000 0.000 0.000
LAMBDA
F1 F2
________ ________
Y1 0.500 0.000
Y2 0.500 0.000
Y3 0.500 0.000
Y4 0.500 0.250
Y5 0.500 0.250
Y6 0.500 0.800
Y7 0.500 0.800
Y8 0.000 0.800
Y9 0.000 0.800
Y10 0.000 0.800
THETA
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.360
Y2 0.000 0.360
Y3 0.000 0.000 0.360
Y4 0.000 0.000 0.000 0.360
Y5 0.000 0.000 0.000 0.000 0.360
Y6 0.000 0.000 0.000 0.000 0.000
Y7 0.000 0.000 0.000 0.000 0.000
Y8 0.000 0.000 0.000 0.000 0.000
Y9 0.000 0.000 0.000 0.000 0.000
Y10 0.000 0.000 0.000 0.000 0.000
THETA
Y6 Y7 Y8 Y9 Y10
________ ________ ________ ________ ________
Y6 0.360
Y7 0.000 0.360
Y8 0.000 0.000 0.360
Y9 0.000 0.000 0.000 0.360
Y10 0.000 0.000 0.000 0.000 0.360
ALPHA
F1 F2
________ ________
0.000 0.000
BETA
F1 F2
________ ________
F1 0.000 0.000
F2 0.000 0.000
PSI
F1 F2
________ ________
F1 1.000
F2 0.500 1.000
TECHNICAL 9 OUTPUT
Error messages for each replication (if any)
Beginning Time: 23:06:54
Ending Time: 23:07:00
Elapsed Time: 00:00:06
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