```Mplus VERSION 8.8
MUTHEN & MUTHEN
04/19/2022  10:22 PM

INPUT INSTRUCTIONS

title:	this is an example of a threshold
structure CFA for categorical factor indicators

montecarlo:
names = u1a u1b u1c u2a u2b u2c;
generate = u1a-u2c(1);
categorical = u1a-u2c;
nobs = 500;
nreps = 1;
save = ex5.10.dat;

model population:

f1 by u1a-u1c@1;
f2 by u2a-u2c@1;
[u1a\$1*-.5 u1b\$1*-.5 u1c\$1*-.5] (1);
[u2a\$1*.5 u2b\$1*.5 u2c\$1*.5] (2);

u1a-u2c*.5;

f1-f2*.5; f1 with f2*.2;

! note that u* variances should be 1
! in order for the Delta parameterization
! to give estimates in the correct metric

model:

f1 by u1a-u1c@1;
f2 by u2a-u2c@1;
[u1a\$1*-.5 u1b\$1*-.5 u1c\$1*-.5] (1);
[u2a\$1*.5 u2b\$1*.5 u2c\$1*.5] (2);

f1-f2*.5; f1 with f2*.2;

this is an example of a threshold
structure CFA for categorical factor indicators

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         500

Number of replications
Requested                                                    1
Completed                                                    1
Value of seed                                                    0

Number of dependent variables                                    6
Number of independent variables                                  0
Number of continuous latent variables                            2

Observed dependent variables

Binary and ordered categorical (ordinal)
U1A         U1B         U1C         U2A         U2B         U2C

Continuous latent variables
F1          F2

Estimator                                                    WLSMV
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20
Parameterization                                             DELTA

SAMPLE STATISTICS FOR THE FIRST REPLICATION

SAMPLE STATISTICS

Means/Intercepts/Thresholds
U1A\$1         U1B\$1         U1C\$1         U2A\$1         U2B\$1
________      ________      ________      ________      ________
-0.479        -0.577        -0.490         0.577         0.536

Means/Intercepts/Thresholds
U2C\$1
________
0.507

Covariances/Correlations/Residual Correlations
U1A           U1B           U1C           U2A           U2B
________      ________      ________      ________      ________
U1A
U1B            0.504
U1C            0.464         0.513
U2A            0.215         0.085         0.154
U2B            0.196         0.082         0.204         0.470
U2C            0.303         0.211         0.110         0.514         0.579

Covariances/Correlations/Residual Correlations
U2C
________

MODEL FIT INFORMATION

Number of Free Parameters                        5

Chi-Square Test of Model Fit

Degrees of freedom                      16

Mean                                15.119
Std Dev                              0.000
Number of successful computations        1

Proportions                   Percentiles
Expected    Observed         Expected       Observed
0.990       1.000            5.812         15.119
0.980       1.000            6.614         15.119
0.950       1.000            7.962         15.119
0.900       1.000            9.312         15.119
0.800       1.000           11.152         15.119
0.700       1.000           12.624         15.119
0.500       0.000           15.338         15.119
0.300       0.000           18.418         15.119
0.200       0.000           20.465         15.119
0.100       0.000           23.542         15.119
0.050       0.000           26.296         15.119
0.020       0.000           29.633         15.119
0.010       0.000           32.000         15.119

RMSEA (Root Mean Square Error Of Approximation)

Mean                                 0.000
Std Dev                              0.000
Number of successful computations        1

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       1.000

CFI/TLI

CFI

Mean                                 1.000
Std Dev                              0.000
Number of successful computations        1

Cumulative Distribution Function
Value    Function Value
0.990       0.000
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                                 1.000
Std Dev                              0.000
Number of successful computations        1

Cumulative Distribution Function
Value    Function Value
0.990       0.000
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                                 0.044
Std Dev                              0.000
Number of successful computations        1

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       0.000
0.010       0.000

MODEL RESULTS

ESTIMATES              S. E.     M. S. E.  95%  % Sig
Population   Average   Std. Dev.   Average             Cover Coeff

F1       BY
U1A                 1.000     1.0000     0.0000     0.0000     0.0000 1.000 0.000
U1B                 1.000     1.0000     0.0000     0.0000     0.0000 1.000 0.000
U1C                 1.000     1.0000     0.0000     0.0000     0.0000 1.000 0.000

F2       BY
U2A                 1.000     1.0000     0.0000     0.0000     0.0000 1.000 0.000
U2B                 1.000     1.0000     0.0000     0.0000     0.0000 1.000 0.000
U2C                 1.000     1.0000     0.0000     0.0000     0.0000 1.000 0.000

F1       WITH
F2                  0.200     0.1760     0.0000     0.0400     0.0006 1.000 1.000

Thresholds
U1A\$1              -0.500    -0.5146     0.0000     0.0433     0.0002 1.000 1.000
U1B\$1              -0.500    -0.5146     0.0000     0.0433     0.0002 1.000 1.000
U1C\$1              -0.500    -0.5146     0.0000     0.0433     0.0002 1.000 1.000
U2A\$1               0.500     0.5397     0.0000     0.0441     0.0016 1.000 1.000
U2B\$1               0.500     0.5397     0.0000     0.0441     0.0016 1.000 1.000
U2C\$1               0.500     0.5397     0.0000     0.0441     0.0016 1.000 1.000

Variances
F1                  0.500     0.4940     0.0000     0.0431     0.0000 1.000 1.000
F2                  0.500     0.5257     0.0000     0.0435     0.0007 1.000 1.000

QUALITY OF NUMERICAL RESULTS

Average Condition Number for the Information Matrix      0.504E+00
(ratio of smallest to largest eigenvalue)

TECHNICAL OUTPUT

PARAMETER SPECIFICATION

TAU
U1A\$1         U1B\$1         U1C\$1         U2A\$1         U2B\$1
________      ________      ________      ________      ________
1             1             1             2             2

TAU
U2C\$1
________
2

NU
U1A           U1B           U1C           U2A           U2B
________      ________      ________      ________      ________
0             0             0             0             0

NU
U2C
________
0

LAMBDA
F1            F2
________      ________
U1A                0             0
U1B                0             0
U1C                0             0
U2A                0             0
U2B                0             0
U2C                0             0

ALPHA
F1            F2
________      ________
0             0

BETA
F1            F2
________      ________
F1                 0             0
F2                 0             0

PSI
F1            F2
________      ________
F1                 3
F2                 4             5

STARTING VALUES

TAU
U1A\$1         U1B\$1         U1C\$1         U2A\$1         U2B\$1
________      ________      ________      ________      ________
-0.500        -0.500        -0.500         0.500         0.500

TAU
U2C\$1
________
0.500

NU
U1A           U1B           U1C           U2A           U2B
________      ________      ________      ________      ________
0.000         0.000         0.000         0.000         0.000

NU
U2C
________
0.000

LAMBDA
F1            F2
________      ________
U1A            1.000         0.000
U1B            1.000         0.000
U1C            1.000         0.000
U2A            0.000         1.000
U2B            0.000         1.000
U2C            0.000         1.000

ALPHA
F1            F2
________      ________
0.000         0.000

BETA
F1            F2
________      ________
F1             0.000         0.000
F2             0.000         0.000

PSI
F1            F2
________      ________
F1             0.500
F2             0.200         0.500

POPULATION VALUES

TAU
U1A\$1         U1B\$1         U1C\$1         U2A\$1         U2B\$1
________      ________      ________      ________      ________
-0.500        -0.500        -0.500         0.500         0.500

TAU
U2C\$1
________
0.500

NU
U1A           U1B           U1C           U2A           U2B
________      ________      ________      ________      ________
0.000         0.000         0.000         0.000         0.000

NU
U2C
________
0.000

LAMBDA
F1            F2
________      ________
U1A            1.000         0.000
U1B            1.000         0.000
U1C            1.000         0.000
U2A            0.000         1.000
U2B            0.000         1.000
U2C            0.000         1.000

THETA
U1A           U1B           U1C           U2A           U2B
________      ________      ________      ________      ________
U1A            0.500
U1B            0.000         0.500
U1C            0.000         0.000         0.500
U2A            0.000         0.000         0.000         0.500
U2B            0.000         0.000         0.000         0.000         0.500
U2C            0.000         0.000         0.000         0.000         0.000

THETA
U2C
________
U2C            0.500

ALPHA
F1            F2
________      ________
0.000         0.000

BETA
F1            F2
________      ________
F1             0.000         0.000
F2             0.000         0.000

PSI
F1            F2
________      ________
F1             0.500
F2             0.200         0.500

SAVEDATA INFORMATION

Order of variables

U1A
U1B
U1C
U2A
U2B
U2C

Save file
ex5.10.dat

Save file format           Free
Save file record length    10000

Beginning Time:  22:22:57
Ending Time:  22:22:57
Elapsed Time:  00:00:00

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```