### Mplus Textbook Examples Exploratory and Confirmatory Factor Analysis:  Understanding Concepts and Applications by Bruce Thompson

The data are shown on pages 163-167.

• Please note that a subset of the data is used for some examples.
• Some of the output has been omitted to save space.
• The values for the chi-squared tests are slightly different to those shown in the text due to differences in the algorithms in the programs used.
• Relevant results are bolded.

#### Chapter 11:  Some confirmatory Factor Analysis Interpretation Principles

Page 136 Table 11.1

Model 1  A text file (of the faculty data) with the data ready for use in Mplus can be downloaded here.

title: page 136 Model 1
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;
model: f1 by per1 - per4;
f2 by per8 per5 per6 per7;
f1 with f2 @0;
output:  standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            2

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Continuous latent variables
F1          F2

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                             48.757
Degrees of Freedom                    20
P-Value                           0.0003

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.959
TLI                                0.943

Loglikelihood

H0 Value                       -1231.770
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters             16
Akaike (AIC)                    2495.540
Bayesian (BIC)                  2537.223
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.120
90 Percent C.I.                    0.077  0.163
Probability RMSEA <= .05           0.006

SRMR (Standardized Root Mean Square Residual)

Value                              0.238

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

F1       BY
PER1               1.000    0.000      0.000    1.291    0.950
PER2               1.052    0.073     14.353    1.358    0.882
PER3               0.803    0.093      8.589    1.036    0.686
PER4               1.018    0.072     14.148    1.314    0.877

F2       BY
PER8               1.000    0.000      0.000    1.703    0.793
PER5               1.134    0.101     11.245    1.931    0.934
PER6               1.083    0.100     10.800    1.843    0.908
PER7               1.026    0.091     11.221    1.746    0.933

F1       WITH
F2                 0.000    0.000      0.000    0.000    0.000

Variances
F1                 1.667    0.266      6.264    1.000    1.000
F2                 2.899    0.614      4.718    1.000    1.000

Residual Variances
PER1               0.181    0.062      2.923    0.181    0.098
PER2               0.529    0.099      5.331    0.529    0.223
PER3               1.206    0.180      6.684    1.206    0.529
PER4               0.521    0.096      5.434    0.521    0.232
PER5               0.544    0.123      4.436    0.544    0.127
PER6               0.724    0.137      5.298    0.724    0.176
PER7               0.456    0.101      4.496    0.456    0.130
PER8               1.706    0.264      6.456    1.706    0.371

Model 2

title: page 136 Model 2
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;
model: f1 by per2 per1 per3 per4;
f2 by per6 per5 per7 per8;
f1 with f2 @0;
output:  standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            2

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Continuous latent variables
F1          F2

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                             48.757
Degrees of Freedom                    20
P-Value                           0.0003

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.959
TLI                                0.943

Loglikelihood

H0 Value                       -1231.770
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters             16
Akaike (AIC)                    2495.540
Bayesian (BIC)                  2537.223
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.120
90 Percent C.I.                    0.077  0.163
Probability RMSEA <= .05           0.006

SRMR (Standardized Root Mean Square Residual)

Value                              0.238

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

F1       BY
PER2               1.000    0.000      0.000    1.358    0.882
PER1               0.950    0.066     14.353    1.291    0.950
PER3               0.763    0.094      8.130    1.036    0.686
PER4               0.968    0.078     12.440    1.314    0.877

F2       BY
PER6               1.000    0.000      0.000    1.843    0.908
PER5               1.048    0.066     15.780    1.931    0.934
PER7               0.948    0.060     15.718    1.746    0.933
PER8               0.924    0.086     10.800    1.703    0.793

F1       WITH
F2                 0.000    0.000      0.000    0.000    0.000

Variances
F1                 1.845    0.334      5.529    1.000    1.000
F2                 3.397    0.581      5.848    1.000    1.000

Residual Variances
PER1               0.181    0.062      2.923    0.181    0.098
PER2               0.529    0.099      5.331    0.529    0.223
PER3               1.206    0.180      6.684    1.206    0.529
PER4               0.521    0.096      5.434    0.521    0.232
PER5               0.544    0.123      4.436    0.544    0.127
PER6               0.724    0.137      5.298    0.724    0.176
PER7               0.456    0.101      4.496    0.456    0.130
PER8               1.706    0.264      6.456    1.706    0.371

Model 3

title: page 136 Model 3
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;
model: f1 by per1* per2 - per4;
f2 by per5* per6 - per8;
f1 with f2 @0;
f1@1 f2@1;
output:  standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            2

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Continuous latent variables
F1          F2

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                             48.757
Degrees of Freedom                    20
P-Value                           0.0003

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.959
TLI                                0.943

Loglikelihood

H0 Value                       -1231.770
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters             16
Akaike (AIC)                    2495.540
Bayesian (BIC)                  2537.223
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.120
90 Percent C.I.                    0.077  0.163
Probability RMSEA <= .05           0.006

SRMR (Standardized Root Mean Square Residual)

Value                              0.238

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

F1       BY
PER1               1.291    0.103     12.529    1.291    0.950
PER2               1.358    0.123     11.058    1.358    0.882
PER3               1.036    0.135      7.674    1.036    0.686
PER4               1.314    0.120     10.957    1.314    0.877

F2       BY
PER5               1.931    0.157     12.287    1.931    0.934
PER6               1.843    0.158     11.696    1.843    0.908
PER7               1.746    0.143     12.254    1.746    0.933
PER8               1.703    0.180      9.436    1.703    0.793

F1       WITH
F2                 0.000    0.000      0.000    0.000    0.000

Variances
F1                 1.000    0.000      0.000    1.000    1.000
F2                 1.000    0.000      0.000    1.000    1.000

Residual Variances
PER1               0.181    0.062      2.923    0.181    0.098
PER2               0.529    0.099      5.331    0.529    0.223
PER3               1.206    0.180      6.684    1.206    0.529
PER4               0.521    0.096      5.434    0.521    0.232
PER5               0.544    0.123      4.436    0.544    0.127
PER6               0.724    0.137      5.298    0.724    0.176
PER7               0.456    0.101      4.496    0.456    0.130
PER8               1.706    0.264      6.456    1.706    0.371

Page 144 Table 11.4

Model 1:  Independence

title: page 144 Model - Independence
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;

*** WARNING in Model command
All variables are uncorrelated with all other variables in the model.
Check that this is what is intended.
1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            0

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.000
TLI                                0.000

Loglikelihood

H0 Value                       -1573.308
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters              8
Akaike (AIC)                    3162.616
Bayesian (BIC)                  3183.457
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.501
90 Percent C.I.                    0.470  0.533
Probability RMSEA <= .05           0.000

SRMR (Standardized Root Mean Square Residual)

Value                              0.500

MODEL RESULTS

Estimates     S.E.  Est./S.E.

Variances
PER1               1.847    0.261      7.071
PER2               2.374    0.336      7.071
PER3               2.280    0.322      7.071
PER4               2.248    0.318      7.071
PER5               4.274    0.604      7.071
PER6               4.122    0.583      7.071
PER7               3.506    0.496      7.071
PER8               4.605    0.651      7.071

Model 2:  One factor

title: page 144 Model - one factor
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;
model: f1 by per1 - per8;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            1

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Continuous latent variables
F1

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                            288.554
Degrees of Freedom                    20
P-Value                           0.0000

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.618
TLI                                0.466

Loglikelihood

H0 Value                       -1351.669
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters             16
Akaike (AIC)                    2735.337
Bayesian (BIC)                  2777.020
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.366
90 Percent C.I.                    0.330  0.404
Probability RMSEA <= .05           0.000

SRMR (Standardized Root Mean Square Residual)

Value                              0.230

MODEL RESULTS

Estimates     S.E.  Est./S.E.

F1       BY
PER1               1.000    0.000      0.000
PER2               0.920    0.303      3.039
PER3               1.051    0.311      3.376
PER4               1.048    0.310      3.385
PER5               3.098    0.641      4.832
PER6               2.979    0.620      4.803
PER7               2.836    0.585      4.845
PER8               2.817    0.608      4.635

Variances
F1                 0.378    0.162      2.335

Residual Variances
PER1               1.469    0.211      6.977
PER2               2.053    0.293      7.014
PER3               1.862    0.266      6.989
PER4               1.833    0.262      6.988
PER5               0.641    0.132      4.867
PER6               0.763    0.142      5.380
PER7               0.462    0.102      4.522
PER8               1.601    0.251      6.372

Model 3:  Uncorrelated factors (#3)

title: page 144 Model - two factors, uncorrelated
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;
model: f1 by per1 - per4;
f2 by per8 per5 per6 per7;
f1 with f2 @0;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            2

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Continuous latent variables
F1          F2

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                             48.757
Degrees of Freedom                    20
P-Value                           0.0003

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.959
TLI                                0.943

Loglikelihood

H0 Value                       -1231.770
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters             16
Akaike (AIC)                    2495.540
Bayesian (BIC)                  2537.223
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.120
90 Percent C.I.                    0.077  0.163
Probability RMSEA <= .05           0.006

SRMR (Standardized Root Mean Square Residual)

Value                              0.238

MODEL RESULTS

Estimates     S.E.  Est./S.E.

F1       BY
PER1               1.000    0.000      0.000
PER2               1.052    0.073     14.353
PER3               0.803    0.093      8.589
PER4               1.018    0.072     14.148

F2       BY
PER8               1.000    0.000      0.000
PER5               1.134    0.101     11.245
PER6               1.083    0.100     10.800
PER7               1.026    0.091     11.221

F1       WITH
F2                 0.000    0.000      0.000

Variances
F1                 1.667    0.266      6.264
F2                 2.899    0.614      4.718

Residual Variances
PER1               0.181    0.062      2.923
PER2               0.529    0.099      5.331
PER3               1.206    0.180      6.684
PER4               0.521    0.096      5.434
PER5               0.544    0.123      4.436
PER6               0.724    0.137      5.298
PER7               0.456    0.101      4.496
PER8               1.706    0.264      6.456

Model 4:  Correlated factors (#4)

title: page 144 Model - two factors, correlated
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;
model: f1 by per1 - per4;
f2 by per8 per5 per6 per7;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            2

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Continuous latent variables
F1          F2

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                             31.682
Degrees of Freedom                    19
P-Value                           0.0339

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.982
TLI                                0.973

Loglikelihood

H0 Value                       -1223.232
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters             17
Akaike (AIC)                    2480.465
Bayesian (BIC)                  2524.753
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.082
90 Percent C.I.                    0.023  0.130
Probability RMSEA <= .05           0.146

SRMR (Standardized Root Mean Square Residual)

Value                              0.059

MODEL RESULTS

Estimates     S.E.  Est./S.E.

F1       BY
PER1               1.000    0.000      0.000
PER2               1.048    0.073     14.320
PER3               0.806    0.093      8.678
PER4               1.018    0.071     14.241

F2       BY
PER8               1.000    0.000      0.000
PER5               1.125    0.099     11.339
PER6               1.076    0.099     10.913
PER7               1.021    0.090     11.373

F2       WITH
F1                 0.926    0.261      3.544

Variances
F1                 1.669    0.266      6.279
F2                 2.932    0.616      4.758

Residual Variances
PER1               0.178    0.061      2.948
PER2               0.540    0.100      5.422
PER3               1.195    0.179      6.683
PER4               0.519    0.095      5.459
PER5               0.564    0.124      4.557
PER6               0.728    0.137      5.318
PER7               0.450    0.100      4.478
PER8               1.674    0.260      6.438

Page 148 Table 11.5

title: page 148
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per12 ;
model: f1 by per4 per1 - per3;
f2 by per5 - per8;
f3 by per9 - per12;
f4 by f1* f2 f3; ! this statement makes it
! a second order model and has f1, f2 and f3
! predicting f4.;
f4@1;
output: standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                   12
Number of independent variables                                  0
Number of continuous latent variables                            4

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8        PER9        PER10       PER11       PER12

Continuous latent variables
F1          F2          F3          F4

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                             69.640
Degrees of Freedom                    51
P-Value                           0.0424

Chi-Square Test of Model Fit for the Baseline Model

Value                            889.592
Degrees of Freedom                    66
P-Value                           0.0000

CFI/TLI

CFI                                0.977
TLI                                0.971

Loglikelihood

H0 Value                       -1949.113
H1 Value                       -1914.293

Information Criteria

Number of Free Parameters             27
Akaike (AIC)                    3952.225
Bayesian (BIC)                  4022.565
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.060
90 Percent C.I.                    0.012  0.094
Probability RMSEA <= .05           0.300

SRMR (Standardized Root Mean Square Residual)

Value                              0.062

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

F1       BY
PER4               1.000    0.000      0.000    1.323    0.882
PER1               0.969    0.068     14.355    1.283    0.944
PER2               1.027    0.081     12.600    1.358    0.882
PER3               0.790    0.096      8.237    1.045    0.692

F2       BY
PER5               1.000    0.000      0.000    1.928    0.933
PER6               0.956    0.061     15.758    1.844    0.908
PER7               0.905    0.053     17.041    1.745    0.932
PER8               0.888    0.078     11.358    1.712    0.798

F3       BY
PER9               1.000    0.000      0.000    1.070    0.577
PER10              1.296    0.264      4.907    1.387    0.716
PER11              0.927    0.197      4.709    0.991    0.663
PER12              0.740    0.196      3.772    0.792    0.480

F4       BY
F1                 0.920    0.155      5.939    0.695    0.695
F2                 1.160    0.214      5.424    0.602    0.602
F3                 1.059    0.212      4.993    0.990    0.990

Variances
F4                 1.000    0.000      0.000    1.000    1.000

Residual Variances
PER1               0.202    0.060      3.376    0.202    0.109
PER2               0.529    0.098      5.404    0.529    0.223
PER3               1.187    0.178      6.670    1.187    0.521
PER4               0.497    0.092      5.388    0.497    0.221
PER5               0.555    0.122      4.543    0.555    0.130
PER6               0.722    0.136      5.319    0.722    0.175
PER7               0.460    0.101      4.573    0.460    0.131
PER8               1.675    0.260      6.445    1.675    0.364
PER9               2.298    0.374      6.152    2.298    0.668
PER10              1.831    0.360      5.080    1.831    0.488
PER11              1.256    0.224      5.605    1.256    0.561
PER12              2.091    0.320      6.523    2.091    0.769
F1                 0.904    0.232      3.890    0.516    0.516
F2                 2.372    0.470      5.046    0.638    0.638
F3                 0.022    0.225      0.100    0.020    0.020

R-SQUARE

Observed
Variable  R-Square

PER1         0.891
PER2         0.777
PER3         0.479
PER4         0.779
PER5         0.870
PER6         0.825
PER7         0.869
PER8         0.636
PER9         0.332
PER10        0.512
PER11        0.439
PER12        0.231

Latent
Variable  R-Square

F1           0.484
F2           0.362
F3           0.980

Page 150 Table 11.6

title: page 150
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per8;
model: f1 by per1 - per4;
f2 by per5 - per8;
f4 by f1* (1)
f2 (1); ! this statement makes it
! a second order model and has f1 and f2
! predicting f4.  The star indicates that the
! coefficient should be estimated.  The (1) labels
! the coefficient so that it can be constrained to
! be equal for both f1 and f2.  The syntax will
! run without the star and you will get the correct
! number of df, but the standardized coefficients
! won't be correct b/c the unstandardized coefficient
! was set to one (by default) instead of estimated.
! The df don't change b/c it doesn't matter if the
! coefficient for f1 is set to one and then f2 is
! constrained to equal it, or if the coefficient for
! f1 is estimated and then f2 is constrained to
! equal it.;
f4@1;
! the statement above sets the variance of the
! second order factor (named f4) to equal 1.;
output: standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                    8
Number of independent variables                                  0
Number of continuous latent variables                            3

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8

Continuous latent variables
F1          F2          F4

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                             31.682
Degrees of Freedom                    19
P-Value                           0.0339

Chi-Square Test of Model Fit for the Baseline Model

Value                            731.833
Degrees of Freedom                    28
P-Value                           0.0000

CFI/TLI

CFI                                0.982
TLI                                0.973

Loglikelihood

H0 Value                       -1223.232
H1 Value                       -1207.392

Information Criteria

Number of Free Parameters             17
Akaike (AIC)                    2480.465
Bayesian (BIC)                  2524.753
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.082
90 Percent C.I.                    0.023  0.130
Probability RMSEA <= .05           0.146

SRMR (Standardized Root Mean Square Residual)

Value                              0.059

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

F1       BY
PER1               1.000    0.000      0.000    1.292    0.950
PER2               1.048    0.073     14.320    1.354    0.879
PER3               0.806    0.093      8.678    1.041    0.690
PER4               1.018    0.071     14.241    1.315    0.877

F2       BY
PER5               1.000    0.000      0.000    1.926    0.932
PER6               0.956    0.061     15.656    1.842    0.907
PER7               0.908    0.053     17.039    1.748    0.934
PER8               0.889    0.078     11.339    1.712    0.798

F4       BY
F1                 1.021    0.140      7.291    0.790    0.790
F2                 1.021    0.140      7.291    0.530    0.530

Variances
F4                 1.000    0.000      0.000    1.000    1.000

Residual Variances
PER1               0.178    0.061      2.948    0.178    0.097
PER2               0.540    0.100      5.422    0.540    0.228
PER3               1.195    0.179      6.683    1.195    0.524
PER4               0.519    0.095      5.459    0.519    0.231
PER5               0.564    0.124      4.557    0.564    0.132
PER6               0.728    0.137      5.318    0.728    0.177
PER7               0.450    0.100      4.478    0.450    0.128
PER8               1.674    0.260      6.438    1.674    0.363
F1                 0.627    0.272      2.305    0.376    0.376
F2                 2.668    0.519      5.142    0.719    0.719

R-SQUARE

Observed
Variable  R-Square

PER1         0.903
PER2         0.772
PER3         0.476
PER4         0.769
PER5         0.868
PER6         0.823
PER7         0.872
PER8         0.637

Latent
Variable  R-Square

F1           0.624
F2           0.281

Page 156 Table 12.1  data set

title: page 156
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson.txt";
variable:  names are id type per1 - per12;
usevar per1-per12 type ;
grouping is type (2 = gs 3 = fac);
! the values in () are necessary;
model: f1 by per1* per2 - per4;
f2 by per5* per6 - per8;
f3 by per9* per10 - per12;
! the stars tell Mplus to estimate the parameter,
! by default, the parameter of the first variable
! listed is set to one.  We don't need this b/c
! we have set the variances of the factors to
! equal one (see below);
f1@1 f2@1 f3@1; ! setting the variance of the
! factors to equal one so that all of the
! parameters are estimated;
model fac: f1 by per1* per2 - per4;
f2 by per5* per6 - per8;
f3 by per9* per10 - per12;
! You need this second model statement (with a name)
! because, by default, Mplus will constrain the
! parameter estimates to be the same in both groups.
! This is not what you want.  However, the correlations
! and error variances are not constrained to be the same
! by default.
output: standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 2
Number of observations
Group GS                                                    100
Group FAC                                                   100

Number of dependent variables                                   12
Number of independent variables                                  0
Number of continuous latent variables                            3

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8        PER9        PER10       PER11       PER12

Continuous latent variables
F1          F2          F3

Variables with special functions

Grouping variable     TYPE

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                            136.390
Degrees of Freedom                   102
P-Value                           0.0130

Chi-Square Test of Model Fit for the Baseline Model

Value                           1793.727
Degrees of Freedom                   132
P-Value                           0.0000

CFI/TLI

CFI                                0.979
TLI                                0.973

Loglikelihood

H0 Value                       -3988.643
H1 Value                       -3920.448

Information Criteria

Number of Free Parameters             54
Akaike (AIC)                    8085.285
Bayesian (BIC)                  8263.394
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.058
90 Percent C.I.                    0.028  0.082

SRMR (Standardized Root Mean Square Residual)

Value                              0.057

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

Group GS

F1       BY
PER1               1.608    0.129     12.481    1.608    0.945
PER2               1.559    0.140     11.171    1.559    0.886
PER3               1.556    0.153     10.159    1.556    0.835
PER4               1.506    0.152      9.914    1.506    0.822

F2       BY
PER5               1.730    0.150     11.496    1.730    0.907
PER6               1.439    0.166      8.648    1.439    0.753
PER7               1.647    0.148     11.150    1.647    0.890
PER8               1.734    0.172     10.068    1.734    0.834

F3       BY
PER9               1.519    0.159      9.541    1.519    0.827
PER10              1.586    0.195      8.145    1.586    0.739
PER11              1.499    0.161      9.304    1.499    0.813
PER12              1.122    0.176      6.383    1.122    0.613

F2       WITH
F1                 0.569    0.075      7.590    0.569    0.569

F3       WITH
F1                 0.642    0.071      9.066    0.642    0.642
F2                 0.567    0.081      7.013    0.567    0.567

Variances
F1                 1.000    0.000      0.000    1.000    1.000
F2                 1.000    0.000      0.000    1.000    1.000
F3                 1.000    0.000      0.000    1.000    1.000

Residual Variances
PER1               0.308    0.090      3.428    0.308    0.106
PER2               0.666    0.124      5.384    0.666    0.215
PER3               1.052    0.174      6.059    1.052    0.303
PER4               1.090    0.177      6.162    1.090    0.325
PER5               0.649    0.151      4.308    0.649    0.178
PER6               1.580    0.250      6.317    1.580    0.433
PER7               0.713    0.150      4.767    0.713    0.208
PER8               1.312    0.229      5.724    1.312    0.304
PER9               1.066    0.228      4.681    1.066    0.316
PER10              2.094    0.363      5.764    2.094    0.454
PER11              1.155    0.235      4.917    1.155    0.340
PER12              2.097    0.326      6.430    2.097    0.625

Group FAC

F1       BY
PER1               1.283    0.103     12.420    1.283    0.944
PER2               1.358    0.123     11.072    1.358    0.882
PER3               1.045    0.135      7.761    1.045    0.692
PER4               1.323    0.119     11.089    1.323    0.882

F2       BY
PER5               1.928    0.157     12.261    1.928    0.933
PER6               1.844    0.158     11.706    1.844    0.908
PER7               1.745    0.143     12.244    1.745    0.932
PER8               1.712    0.180      9.513    1.712    0.798

F3       BY
PER9               1.070    0.190      5.629    1.070    0.577
PER10              1.386    0.190      7.296    1.386    0.716
PER11              0.991    0.149      6.649    0.991    0.662
PER12              0.792    0.174      4.563    0.792    0.480

F2       WITH
F1                 0.418    0.088      4.746    0.418    0.418

F3       WITH
F1                 0.689    0.076      9.008    0.689    0.689
F2                 0.596    0.087      6.874    0.596    0.596

Variances
F1                 1.000    0.000      0.000    1.000    1.000
F2                 1.000    0.000      0.000    1.000    1.000
F3                 1.000    0.000      0.000    1.000    1.000

Residual Variances
PER1               0.202    0.060      3.377    0.202    0.109
PER2               0.529    0.098      5.404    0.529    0.223
PER3               1.187    0.178      6.670    1.187    0.521
PER4               0.497    0.092      5.388    0.497    0.221
PER5               0.555    0.122      4.543    0.555    0.130
PER6               0.722    0.136      5.319    0.722    0.175
PER7               0.460    0.101      4.573    0.460    0.131
PER8               1.675    0.260      6.445    1.675    0.364
PER9               2.298    0.373      6.152    2.298    0.668
PER10              1.832    0.361      5.082    1.832    0.488
PER11              1.256    0.224      5.605    1.256    0.561
PER12              2.090    0.320      6.523    2.090    0.769

R-SQUARE

Group GS

Observed
Variable  R-Square

PER1         0.894
PER2         0.785
PER3         0.697
PER4         0.675
PER5         0.822
PER6         0.567
PER7         0.792
PER8         0.696
PER9         0.684
PER10        0.546
PER11        0.660
PER12        0.375

Group FAC

Observed
Variable  R-Square

PER1         0.891
PER2         0.777
PER3         0.479
PER4         0.779
PER5         0.870
PER6         0.825
PER7         0.869
PER8         0.636
PER9         0.332
PER10        0.512
PER11        0.439
PER12        0.231

Page 158 Table 12.2

title: page 158
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson.txt";
variable:  names are id type per1 - per12;
usevar per1-per12 type ;
grouping is type (2 = gs 3 = fac);
! the values in () are necessary;
model: f1 by per1 - per4;
f2 by per5 - per8;
f3 by per10 per9 per11 per12;
! per10 is listed first because it needs to be
! constrained to equal 1, and the first variable
! listed is constrained to equal 1 by default.
output: standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 2
Number of observations
Group GS                                                    100
Group FAC                                                   100

Number of dependent variables                                   12
Number of independent variables                                  0
Number of continuous latent variables                            3

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8        PER9        PER10       PER11       PER12

Continuous latent variables
F1          F2          F3

Variables with special functions

Grouping variable     TYPE

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                            145.926
Degrees of Freedom                   111
P-Value                           0.0147

Chi-Square Test of Model Fit for the Baseline Model

Value                           1793.727
Degrees of Freedom                   132
P-Value                           0.0000

CFI/TLI

CFI                                0.979
TLI                                0.975

Loglikelihood

H0 Value                       -3993.411
H1 Value                       -3920.448

Information Criteria

Number of Free Parameters             45
Akaike (AIC)                    8076.821
Bayesian (BIC)                  8225.246
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.056
90 Percent C.I.                    0.026  0.080

SRMR (Standardized Root Mean Square Residual)

Value                              0.067

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

Group GS

F1       BY
PER1               1.000    0.000      0.000    1.593    0.944
PER2               1.009    0.049     20.403    1.606    0.893
PER3               0.911    0.060     15.294    1.451    0.812
PER4               0.989    0.052     18.862    1.575    0.834

F2       BY
PER5               1.000    0.000      0.000    1.741    0.907
PER6               0.923    0.050     18.405    1.606    0.789
PER7               0.923    0.043     21.412    1.607    0.884
PER8               0.943    0.058     16.176    1.642    0.816

F3       BY
PER10              1.000    0.000      0.000    1.684    0.763
PER9               0.896    0.096      9.360    1.509    0.825
PER11              0.860    0.091      9.444    1.448    0.798
PER12              0.658    0.092      7.128    1.108    0.607

F2       WITH
F1                 1.575    0.340      4.629    0.568    0.568

F3       WITH
F1                 1.719    0.367      4.689    0.641    0.641
F2                 1.657    0.389      4.257    0.565    0.565

NOTE:  The estimates 1.575, 1.719 and 1.657 are covariances.  The values given in the StdYX column are correlations (which are standardized covariances).  The values in the Estimates (covariances) and StdYX (correlations) columns are often the same because we often constrain the variances of the factors to equal 1.  In this case, we did not, so we get both the covariance and the correlation estimates.

Variances
F1                 2.537    0.395      6.418    1.000    1.000
F2                 3.030    0.489      6.192    1.000    1.000
F3                 2.837    0.613      4.625    1.000    1.000

Residual Variances
PER1               0.313    0.087      3.613    0.313    0.110
PER2               0.655    0.124      5.297    0.655    0.202
PER3               1.088    0.174      6.248    1.088    0.341
PER4               1.083    0.178      6.099    1.083    0.304
PER5               0.650    0.147      4.411    0.650    0.177
PER6               1.566    0.254      6.161    1.566    0.378
PER7               0.718    0.144      5.003    0.718    0.218
PER8               1.352    0.227      5.945    1.352    0.334
PER9               1.070    0.224      4.781    1.070    0.320
PER10              2.037    0.364      5.599    2.037    0.418
PER11              1.195    0.230      5.196    1.195    0.363
PER12              2.101    0.325      6.469    2.101    0.631

Group FAC

F1       BY
PER1               1.000    0.000      0.000    1.297    0.947
PER2               1.009    0.049     20.403    1.308    0.870
PER3               0.911    0.060     15.294    1.182    0.735
PER4               0.989    0.052     18.862    1.282    0.874

F2       BY
PER5               1.000    0.000      0.000    1.919    0.932
PER6               0.923    0.050     18.405    1.770    0.898
PER7               0.923    0.043     21.412    1.771    0.935
PER8               0.943    0.058     16.176    1.809    0.814

F3       BY
PER10              1.000    0.000      0.000    1.230    0.658
PER9               0.896    0.096      9.360    1.102    0.589
PER11              0.860    0.091      9.444    1.058    0.690
PER12              0.658    0.092      7.128    0.809    0.488

F2       WITH
F1                 1.050    0.285      3.685    0.422    0.422

F3       WITH
F1                 1.115    0.236      4.725    0.699    0.699
F2                 1.423    0.334      4.257    0.603    0.603

Variances
F1                 1.682    0.261      6.438    1.000    1.000
F2                 3.681    0.582      6.324    1.000    1.000
F3                 1.513    0.360      4.200    1.000    1.000

Residual Variances
PER1               0.194    0.058      3.318    0.194    0.103
PER2               0.547    0.097      5.647    0.547    0.242
PER3               1.186    0.180      6.574    1.186    0.459
PER4               0.509    0.091      5.580    0.509    0.236
PER5               0.558    0.121      4.617    0.558    0.132
PER6               0.752    0.135      5.548    0.752    0.194
PER7               0.450    0.100      4.479    0.450    0.125
PER8               1.668    0.261      6.380    1.668    0.337
PER9               2.286    0.368      6.218    2.286    0.653
PER10              1.984    0.342      5.795    1.984    0.567
PER11              1.232    0.223      5.531    1.232    0.524
PER12              2.090    0.318      6.568    2.090    0.762

R-SQUARE

Group GS

Observed
Variable  R-Square

PER1         0.890
PER2         0.798
PER3         0.659
PER4         0.696
PER5         0.823
PER6         0.622
PER7         0.782
PER8         0.666
PER9         0.680
PER10        0.582
PER11        0.637
PER12        0.369

Group FAC

Observed
Variable  R-Square

PER1         0.897
PER2         0.758
PER3         0.541
PER4         0.764
PER5         0.868
PER6         0.806
PER7         0.875
PER8         0.663
PER9         0.347
PER10        0.433
PER11        0.476
PER12        0.238

Page 161 Table 12.3

title: page 158
of Exploratory and Confirmatory Factor Analysis;
data: file is "D:\thompson_fac.txt";
variable:  names are id type per1 - per12;
usevar per1-per12;
model: f1 by per1@1.61
per2@1.60
per3@1.56
per4@1.51;
f2 by per5@1.73
per6@1.44
per7@1.65
per8@1.73;
f3 by per9@1.52
per10@1.59
per11@1.50
per12@1.12;
f1@1 f2@1 f3@1;
output: standardized;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         100

Number of dependent variables                                   12
Number of independent variables                                  0
Number of continuous latent variables                            3

Observed dependent variables

Continuous
PER1        PER2        PER3        PER4        PER5        PER6
PER7        PER8        PER9        PER10       PER11       PER12

Continuous latent variables
F1          F2          F3

Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value                            107.174
Degrees of Freedom                    63
P-Value                           0.0004

Chi-Square Test of Model Fit for the Baseline Model

Value                            889.592
Degrees of Freedom                    66
P-Value                           0.0000

CFI/TLI

CFI                                0.946
TLI                                0.944

Loglikelihood

H0 Value                       -1967.880
H1 Value                       -1914.293

Information Criteria

Number of Free Parameters             15
Akaike (AIC)                    3965.759
Bayesian (BIC)                  4004.837
(n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

Estimate                           0.084
90 Percent C.I.                    0.056  0.110
Probability RMSEA <= .05           0.027

SRMR (Standardized Root Mean Square Residual)

Value                              0.262

MODEL RESULTS

Estimates     S.E.  Est./S.E.    Std     StdYX

F1       BY
PER1               1.610    0.000      0.000    1.610    0.966
PER2               1.600    0.000      0.000    1.600    0.908
PER3               1.560    0.000      0.000    1.560    0.816
PER4               1.510    0.000      0.000    1.510    0.901

F2       BY
PER5               1.730    0.000      0.000    1.730    0.917
PER6               1.440    0.000      0.000    1.440    0.837
PER7               1.650    0.000      0.000    1.650    0.929
PER8               1.730    0.000      0.000    1.730    0.801

F3       BY
PER9               1.520    0.000      0.000    1.520    0.709
PER10              1.590    0.000      0.000    1.590    0.754
PER11              1.500    0.000      0.000    1.500    0.808
PER12              1.120    0.000      0.000    1.120    0.613

F2       WITH
F1                 0.461    0.077      6.005    0.461    0.461

F3       WITH
F1                 0.785    0.048     16.463    0.785    0.785
F2                 0.575    0.075      7.646    0.575    0.575

Variances
F1                 1.000    0.000      0.000    1.000    1.000
F2                 1.000    0.000      0.000    1.000    1.000
F3                 1.000    0.000      0.000    1.000    1.000

Residual Variances
PER1               0.185    0.057      3.245    0.185    0.067
PER2               0.547    0.096      5.701    0.547    0.176
PER3               1.218    0.187      6.525    1.218    0.334
PER4               0.527    0.091      5.817    0.527    0.188
PER5               0.569    0.121      4.722    0.569    0.160
PER6               0.888    0.146      6.073    0.888    0.300
PER7               0.430    0.100      4.280    0.430    0.136
PER8               1.674    0.265      6.313    1.674    0.359
PER9               2.292    0.375      6.120    2.292    0.498
PER10              1.916    0.329      5.816    1.916    0.431
PER11              1.194    0.227      5.255    1.194    0.347
PER12              2.086    0.321      6.510    2.086    0.625

R-SQUARE

Observed
Variable  R-Square

PER1         0.933
PER2         0.824
PER3         0.666
PER4         0.812
PER5         0.840
PER6         0.700
PER7         0.864
PER8         0.641
PER9         0.502
PER10        0.569
PER11        0.653
PER12        0.375

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