Multilevel Curve of Factors with categorical indicators

This is the second part of a lecture on multilevel approaches to modeling growth. This is a useful strategy when there are a lot of repeated observations.

This lecture builds on part 1 by treating the observed variables as categorical, rather than continuous.

I will run the following models:
1. Continuous variable CFA (model21, CFA Continuous)
2. Categorical variable CFA (model22, CFA Categorical)
3. Categorical variable CFA with covariates(model23, CFA Categorical)
4. Multiple indicator latent growth curve model (model24, MILGCM)
5. Multilevel curve of factors (model25, MLCOF)

Bottom Line Up Front

The multiple indicator latent growth curve model (MILGCM) and multilevel growth of factors (MLGOF) models for categorical variables are equivalent, with some constraints (no item level residual covariances in LILGCM; equal residual variances for time-specific common factors in LIGCM).

Parameter MILGCM MLCOF
Number of free parameters 32 32
Loglikelihood -29743.061 -29743.055
Correction factor 1.108 1.108
I with S -0.165 -0.165
Intercept(I) @0 @0
Intercept(S) 0.255 0.255
ResVar(I) 0.537 0.537
ResVar(S) 0.338 0.337
ResVar(F0), ResVar(FW) 0.433 0.434
Elapsed Time 00:06:39 00:10:49

Data

As in the previous example: this example uses data that are Centers for Epidemiologic Studies - Depression (CESD) scale responses collected from the New Haven Site of the EPESE (Established Populations for the Epidemiologic Study of the Elderly) study. In this lecture, the four-category responses of six CESD questions will be treated as categorical indicators of a common underlying trait.

These data are public and can be obtained from ICPSR.

Descriptives

kable(
   psych::describe(df, type = 3, skew = FALSE, ranges = TRUE, na.rm = TRUE)
   , format = "html", digits = 2)
vars n mean sd median min max range se
id 1 6165 1391.95 811.09 1393.00 1.00 2812.00 2811.00 10.33
sad 2 6119 1.47 0.75 1.00 1.00 4.00 3.00 0.01
blues 3 6109 1.35 0.74 1.00 1.00 4.00 3.00 0.01
depress 4 6130 1.52 0.82 1.00 1.00 4.00 3.00 0.01
happy 5 6071 1.70 1.00 1.00 1.00 4.00 3.00 0.01
enjoy 6 6063 1.54 0.94 1.00 1.00 4.00 3.00 0.01
hopeful 7 5693 1.94 1.19 1.00 1.00 4.00 3.00 0.02
t 8 6165 2.36 2.38 3.00 0.00 6.00 6.00 0.03
td 9 6165 0.24 0.24 0.30 0.00 0.60 0.60 0.00
sex 10 6165 1.60 0.49 2.00 1.00 2.00 1.00 0.01
age 11 6165 2.20 1.24 2.00 1.00 5.00 4.00 0.02
ymean 12 6165 1.59 0.64 1.50 1.00 4.00 3.00 0.01
yz 13 6165 0.03 1.01 -0.11 -0.91 3.87 4.77 0.01
female 14 6165 0.60 0.49 1.00 0.00 1.00 1.00 0.01
agec70 15 6165 0.30 0.62 0.20 -0.30 1.70 2.00 0.01

itemsummary

kable(itemsummary(df, c("sad","blues","depress","happy","enjoy","hopeful")))
Variable 1 2 3 4 NA
blues 4673 979 191 266 56
depress 3887 1642 248 353 35
enjoy 4319 574 789 381 102
happy 3828 691 1118 434 94
hopeful 3213 562 943 975 472
sad 3938 1724 204 253 46

Note: CES-D items are asked with a past week look-back, and response categories have the labels: rarely or none of the time (less than 1 day); some or a little of the time (1-2 days); occasionally or a moderate amount of time (3-4 days); and most or all of the time (5-7 days). Category labels are reversed for positive items (enjoy, happy, hopeful).

Continuous variable CFA

Use MplusAutomation to prepare data set for Mplus

cesditems <- df[df$t == 0, c("sad", "blues", "depress", "happy", "enjoy", "hopeful", "agec70","female")]
MplusAutomation::prepareMplusData(cesditems,"cesditems.dat")
TITLE: Your title goes here
DATA: FILE = "cesditems.dat";
VARIABLE: 
NAMES = sad blues depress happy enjoy hopeful agec70 female; 
MISSING=.;

Mplus ordinary CFA model, continuous items

TITLE:    Continuous item CFA of CESD at baseline
DATA:     FILE = cesditems.dat ;
VARIABLE: NAMES = sad blues depress happy enjoy hopeful; 
          MISSING = . ;
MODEL:    f by sad@1 blues-hopeful*;
OUTPUT:   STDYX  ;

Continuous item CFA of CESD at baseline

Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025  11:54 AM

INPUT INSTRUCTIONS

  TITLE:    Continuous item CFA of CESD at baseline
  DATA:     FILE = cesditems.dat ;
  VARIABLE: NAMES = sad blues depress happy enjoy hopeful;
            MISSING = . ;
  MODEL:    f by sad@1 blues-hopeful*;
  OUTPUT:   STDYX  ;



INPUT READING TERMINATED NORMALLY



Continuous item CFA of CESD at baseline

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        2728

Number of dependent variables                                    6
Number of independent variables                                  0
Number of continuous latent variables                            1

Observed dependent variables

  Continuous
   SAD         BLUES       DEPRESS     HAPPY       ENJOY       HOPEFUL

Continuous latent variables
   F


Estimator                                                       ML
Information matrix                                        OBSERVED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20
Maximum number of iterations for H1                           2000
Convergence criterion for H1                             0.100D-03

Input data file(s)
  cesditems.dat

Input data format  FREE


SUMMARY OF DATA

     Number of missing data patterns            22


COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     PROPORTION OF DATA PRESENT

           Covariance Coverage
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            0.993
 BLUES          0.987         0.993
 DEPRESS        0.991         0.989         0.996
 HAPPY          0.987         0.982         0.986         0.989
 ENJOY          0.986         0.981         0.986         0.981         0.988
 HOPEFUL        0.927         0.925         0.929         0.924         0.923

           Covariance Coverage
              HOPEFUL
              ________
 HOPEFUL        0.931



UNIVARIATE SAMPLE STATISTICS

     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     SAD                   1.458       1.812       1.000   65.26%       1.000      1.000      1.000
            2709.000       0.545       3.142       4.000    3.77%       1.000      2.000
     BLUES                 1.345       2.371       1.000   77.44%       1.000      1.000      1.000
            2708.000       0.552       5.035       4.000    4.36%       1.000      2.000
     DEPRESS               1.522       1.718       1.000   63.55%       1.000      1.000      1.000
            2716.000       0.686       2.339       4.000    6.04%       1.000      2.000
     HAPPY                 1.650       1.189       1.000   66.22%       1.000      1.000      1.000
            2697.000       1.001      -0.071       4.000    7.45%       1.000      3.000
     ENJOY                 1.465       1.735       1.000   75.91%       1.000      1.000      1.000
            2694.000       0.815       1.599       4.000    5.86%       1.000      2.000
     HOPEFUL               1.998       0.643       1.000   56.87%       1.000      1.000      1.000
            2539.000       1.543      -1.319       4.000   20.40%       2.000      4.000


THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       18

Loglikelihood

          H0 Value                      -18965.770
          H1 Value                      -18716.876

Information Criteria

          Akaike (AIC)                   37967.540
          Bayesian (BIC)                 38073.943
          Sample-Size Adjusted BIC       38016.752
            (n* = (n + 2) / 24)

Chi-Square Test of Model Fit

          Value                            497.788
          Degrees of Freedom                     9
          P-Value                           0.0000

RMSEA (Root Mean Square Error Of Approximation)

          Estimate                           0.141
          90 Percent C.I.                    0.131  0.152
          Probability RMSEA <= .05           0.000

CFI/TLI

          CFI                                0.889
          TLI                                0.815

Chi-Square Test of Model Fit for the Baseline Model

          Value                           4426.916
          Degrees of Freedom                    15
          P-Value                           0.0000

SRMR (Standardized Root Mean Square Residual)

          Value                              0.057



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                1.000      0.000    999.000    999.000
    BLUES              0.997      0.033     29.901      0.000
    DEPRESS            1.227      0.038     32.138      0.000
    HAPPY              1.193      0.046     25.777      0.000
    ENJOY              1.030      0.042     24.760      0.000
    HOPEFUL            0.882      0.055     15.941      0.000

 Intercepts
    SAD                1.460      0.014    102.922      0.000
    BLUES              1.346      0.014     94.356      0.000
    DEPRESS            1.523      0.016     95.907      0.000
    HAPPY              1.652      0.019     85.810      0.000
    ENJOY              1.468      0.017     84.452      0.000
    HOPEFUL            1.999      0.025     81.399      0.000

 Variances
    F                  0.261      0.014     18.521      0.000

 Residual Variances
    SAD                0.285      0.010     28.804      0.000
    BLUES              0.293      0.010     28.673      0.000
    DEPRESS            0.292      0.012     24.470      0.000
    HAPPY              0.631      0.021     30.492      0.000
    ENJOY              0.540      0.017     31.173      0.000
    HOPEFUL            1.339      0.039     34.265      0.000


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.143E-01
       (ratio of smallest to largest eigenvalue)


STANDARDIZED MODEL RESULTS


STDYX Standardization

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                0.692      0.013     53.469      0.000
    BLUES              0.686      0.013     51.723      0.000
    DEPRESS            0.758      0.012     63.900      0.000
    HAPPY              0.609      0.015     39.597      0.000
    ENJOY              0.582      0.016     36.453      0.000
    HOPEFUL            0.363      0.020     18.495      0.000

 Intercepts
    SAD                1.975      0.033     59.894      0.000
    BLUES              1.811      0.031     58.050      0.000
    DEPRESS            1.839      0.031     58.473      0.000
    HAPPY              1.650      0.030     55.831      0.000
    ENJOY              1.625      0.029     55.461      0.000
    HOPEFUL            1.610      0.030     53.678      0.000

 Variances
    F                  1.000      0.000    999.000    999.000

 Residual Variances
    SAD                0.522      0.018     29.143      0.000
    BLUES              0.530      0.018     29.161      0.000
    DEPRESS            0.426      0.018     23.718      0.000
    HAPPY              0.629      0.019     33.555      0.000
    ENJOY              0.661      0.019     35.504      0.000
    HOPEFUL            0.868      0.014     60.954      0.000


R-SQUARE

    Observed                                        Two-Tailed
    Variable        Estimate       S.E.  Est./S.E.    P-Value

    SAD                0.478      0.018     26.735      0.000
    BLUES              0.470      0.018     25.861      0.000
    DEPRESS            0.574      0.018     31.950      0.000
    HAPPY              0.371      0.019     19.798      0.000
    ENJOY              0.339      0.019     18.227      0.000
    HOPEFUL            0.132      0.014      9.248      0.000

     Beginning Time:  11:54:13
        Ending Time:  11:54:13
       Elapsed Time:  00:00:00



MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA  90066

Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com

Copyright (c) 1998-2024 Muthen & Muthen

Save output from Mplus Engine

file.copy("formplus.out", "model21.out", overwrite = TRUE)
file.copy("formplus.inp", "model21.inp", overwrite = TRUE)

Categorical item CFA of CESD at baseline

TITLE:    Categorical item CFA of CESD at baseline
DATA:     FILE = cesditems.dat ;
VARIABLE: NAMES = sad blues depress happy enjoy hopeful; 
          CATEGORICAL are all ;
          MISSING = . ;
ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
MODEL:    f by sad@1.24328 blues-hopeful*; 
          f*1;
OUTPUT:   TECH1 ; !SVALUES ;
Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025  11:54 AM

INPUT INSTRUCTIONS

  TITLE:    Categorical item CFA of CESD at baseline
  DATA:     FILE = cesditems.dat ;
  VARIABLE: NAMES = sad blues depress happy enjoy hopeful;
            CATEGORICAL are all ;
            MISSING = . ;
  ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
  MODEL:    f by sad@1.24328 blues-hopeful*;
            f*1;
  OUTPUT:   TECH1 ; !SVALUES ;



INPUT READING TERMINATED NORMALLY



Categorical item CFA of CESD at baseline

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        2728

Number of dependent variables                                    6
Number of independent variables                                  0
Number of continuous latent variables                            1

Observed dependent variables

  Binary and ordered categorical (ordinal)
   SAD         BLUES       DEPRESS     HAPPY       ENJOY       HOPEFUL

Continuous latent variables
   F


Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                 100
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-02
    Relative loglikelihood change                        0.100D-05
    Derivative                                           0.100D-02
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            10
  Minimum value for logit thresholds                           -10
  Minimum expected cell size for chi-square              0.100D-01
Maximum number of iterations for H1                           2000
Convergence criterion for H1                             0.100D-03
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            1
  Adaptive quadrature                                           ON
Link                                                        PROBIT
Cholesky                                                        ON

Input data file(s)
  cesditems.dat
Input data format  FREE


SUMMARY OF DATA

     Number of missing data patterns            22
     Number of y missing data patterns           0
     Number of u missing data patterns          22


COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     PROPORTION OF DATA PRESENT FOR U

           Covariance Coverage
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            0.993
 BLUES          0.987         0.993
 DEPRESS        0.991         0.989         0.996
 HAPPY          0.987         0.982         0.986         0.989
 ENJOY          0.986         0.981         0.986         0.981         0.988
 HOPEFUL        0.927         0.925         0.929         0.924         0.923

           Covariance Coverage
              HOPEFUL
              ________
 HOPEFUL        0.931


UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES

    SAD
      Category 1    0.653         1768.000
      Category 2    0.274          743.000
      Category 3    0.035           96.000
      Category 4    0.038          102.000
    BLUES
      Category 1    0.774         2097.000
      Category 2    0.150          406.000
      Category 3    0.032           87.000
      Category 4    0.044          118.000
    DEPRESS
      Category 1    0.635         1726.000
      Category 2    0.267          725.000
      Category 3    0.037          101.000
      Category 4    0.060          164.000
    HAPPY
      Category 1    0.662         1786.000
      Category 2    0.100          270.000
      Category 3    0.163          440.000
      Category 4    0.075          201.000
    ENJOY
      Category 1    0.759         2045.000
      Category 2    0.075          202.000
      Category 3    0.107          289.000
      Category 4    0.059          158.000
    HOPEFUL
      Category 1    0.569         1444.000
      Category 2    0.069          175.000
      Category 3    0.158          402.000
      Category 4    0.204          518.000



THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       24

Loglikelihood

          H0 Value                      -12954.570
          H0 Scaling Correction Factor      1.0539
            for MLR

Information Criteria

          Akaike (AIC)                   25957.139
          Bayesian (BIC)                 26099.011
          Sample-Size Adjusted BIC       26022.756
            (n* = (n + 2) / 24)

Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes**

          Pearson Chi-Square

          Value                           5306.008
          Degrees of Freedom                  4050
          P-Value                           0.0000

          Likelihood Ratio Chi-Square

          Value                           2398.965
          Degrees of Freedom                  4050
          P-Value                           1.0000

** Of the 12328 cells in the frequency table, 21
   were deleted in the calculation of chi-square due to extreme values.

Chi-Square Test for MCAR under the Unrestricted Latent Class Indicator Model

          Pearson Chi-Square

          Value                           1154.010
          Degrees of Freedom                  8211
          P-Value                           1.0000

          Likelihood Ratio Chi-Square

          Value                            429.717
          Degrees of Freedom                  8211
          P-Value                           1.0000



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                1.243      0.000    999.000    999.000
    BLUES              1.348      0.100     13.413      0.000
    DEPRESS            1.553      0.109     14.295      0.000
    HAPPY              0.986      0.081     12.202      0.000
    ENJOY              0.991      0.084     11.758      0.000
    HOPEFUL            0.520      0.049     10.632      0.000

 Thresholds
    SAD$1              0.620      0.044     14.192      0.000
    SAD$2              2.304      0.082     28.056      0.000
    SAD$3              2.839      0.104     27.375      0.000
    BLUES$1            1.257      0.065     19.226      0.000
    BLUES$2            2.391      0.103     23.104      0.000
    BLUES$3            2.860      0.123     23.270      0.000
    DEPRESS$1          0.643      0.054     12.010      0.000
    DEPRESS$2          2.377      0.111     21.362      0.000
    DEPRESS$3          2.843      0.130     21.842      0.000
    HAPPY$1            0.577      0.038     15.185      0.000
    HAPPY$2            0.992      0.043     22.887      0.000
    HAPPY$3            2.019      0.069     29.205      0.000
    ENJOY$1            0.978      0.045     21.893      0.000
    ENJOY$2            1.352      0.051     26.363      0.000
    ENJOY$3            2.195      0.076     28.946      0.000
    HOPEFUL$1          0.198      0.028      6.996      0.000
    HOPEFUL$2          0.401      0.029     13.766      0.000
    HOPEFUL$3          0.933      0.034     27.296      0.000

 Variances
    F                  1.000      0.106      9.398      0.000


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.312E-02
       (ratio of smallest to largest eigenvalue)


RESULTS IN PROBABILITY SCALE

                    Estimate

 SAD
    Category 1         0.651
    Category 2         0.274
    Category 3         0.037
    Category 4         0.038
 BLUES
    Category 1         0.773
    Category 2         0.150
    Category 3         0.033
    Category 4         0.044
 DEPRESS
    Category 1         0.636
    Category 2         0.265
    Category 3         0.037
    Category 4         0.062
 HAPPY
    Category 1         0.659
    Category 2         0.101
    Category 3         0.165
    Category 4         0.075
 ENJOY
    Category 1         0.756
    Category 2         0.075
    Category 3         0.109
    Category 4         0.060
 HOPEFUL
    Category 1         0.570
    Category 2         0.069
    Category 3         0.157
    Category 4         0.204


TECHNICAL 1 OUTPUT

     PARAMETER SPECIFICATION

           TAU
              SAD$1         SAD$2         SAD$3         BLUES$1       BLUES$2
              ________      ________      ________      ________      ________
                    7             8             9            10            11

           TAU
              BLUES$3       DEPRESS$      DEPRESS$      DEPRESS$      HAPPY$1
              ________      ________      ________      ________      ________
                   12            13            14            15            16

           TAU
              HAPPY$2       HAPPY$3       ENJOY$1       ENJOY$2       ENJOY$3
              ________      ________      ________      ________      ________
                   17            18            19            20            21

           TAU
              HOPEFUL$      HOPEFUL$      HOPEFUL$
              ________      ________      ________
                   22            23            24

           NU
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
                    0             0             0             0             0

           NU
              HOPEFUL
              ________
                    0

           LAMBDA
              F
              ________
 SAD                0
 BLUES              1
 DEPRESS            2
 HAPPY              3
 ENJOY              4
 HOPEFUL            5

           THETA
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD                0
 BLUES              0             0
 DEPRESS            0             0             0
 HAPPY              0             0             0             0
 ENJOY              0             0             0             0             0
 HOPEFUL            0             0             0             0             0

           THETA
              HOPEFUL
              ________
 HOPEFUL            0

           ALPHA
              F
              ________
                    0

           BETA
              F
              ________
 F                  0

           PSI
              F
              ________
 F                  6

     STARTING VALUES

           TAU
              SAD$1         SAD$2         SAD$3         BLUES$1       BLUES$2
              ________      ________      ________      ________      ________
                0.350         1.411         1.801         0.685         1.390

           TAU
              BLUES$3       DEPRESS$      DEPRESS$      DEPRESS$      HAPPY$1
              ________      ________      ________      ________      ________
                1.716         0.309         1.236         1.525         0.374

           TAU
              HAPPY$2       HAPPY$3       ENJOY$1       ENJOY$2       ENJOY$3
              ________      ________      ________      ________      ________
                0.647         1.400         0.638         0.897         1.542

           TAU
              HOPEFUL$      HOPEFUL$      HOPEFUL$
              ________      ________      ________
                0.154         0.314         0.756

           NU
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
                0.000         0.000         0.000         0.000         0.000

           NU
              HOPEFUL
              ________
                0.000

           LAMBDA
              F
              ________
 SAD            1.243
 BLUES          1.000
 DEPRESS        1.000
 HAPPY          1.000
 ENJOY          1.000
 HOPEFUL        1.000

           THETA
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            1.000
 BLUES          0.000         1.000
 DEPRESS        0.000         0.000         1.000
 HAPPY          0.000         0.000         0.000         1.000
 ENJOY          0.000         0.000         0.000         0.000         1.000
 HOPEFUL        0.000         0.000         0.000         0.000         0.000

           THETA
              HOPEFUL
              ________
 HOPEFUL        1.000

           ALPHA
              F
              ________
                0.000

           BETA
              F
              ________
 F              0.000

           PSI
              F
              ________
 F              1.000

     Beginning Time:  11:54:13
        Ending Time:  11:54:14
       Elapsed Time:  00:00:01



MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA  90066

Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com

Copyright (c) 1998-2024 Muthen & Muthen

Save output from Mplus Engine

file.copy("formplus.out", "model22.out", overwrite = TRUE)
file.copy("formplus.inp", "model22.inp", overwrite = TRUE)

Categorical item CFA of CESD at baseline WITH COVARIATES

Step 1. Note the identification constraint and the OUTPUT options.

TITLE:    Categorical item CFA of CESD at baseline
DATA:     FILE = cesditems.dat ;
VARIABLE: NAMES = sad blues depress happy enjoy hopeful
                  agec70 female; 
          CATEGORICAL are sad-hopeful ;
          MISSING = . ;
ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
MODEL:    f by sad-hopeful*; 
          f@1;
          f on agec70 female ;
OUTPUT:   STDYX; TECH1 ; TECH4; !SVALUES ;
Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025  11:54 AM

INPUT INSTRUCTIONS

  TITLE:    Categorical item CFA of CESD at baseline
  DATA:     FILE = cesditems.dat ;
  VARIABLE: NAMES = sad blues depress happy enjoy hopeful
                    agec70 female;
            CATEGORICAL are sad-hopeful ;
            MISSING = . ;
  ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
  MODEL:    f by sad-hopeful*;
            f@1;
            f on agec70 female ;
  OUTPUT:   STDYX; TECH1 ; TECH4; !SVALUES ;



INPUT READING TERMINATED NORMALLY



Categorical item CFA of CESD at baseline

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        2728

Number of dependent variables                                    6
Number of independent variables                                  2
Number of continuous latent variables                            1

Observed dependent variables

  Binary and ordered categorical (ordinal)
   SAD         BLUES       DEPRESS     HAPPY       ENJOY       HOPEFUL

Observed independent variables
   AGEC70      FEMALE

Continuous latent variables
   F


Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                 100
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-02
    Relative loglikelihood change                        0.100D-05
    Derivative                                           0.100D-02
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            10
  Minimum value for logit thresholds                           -10
  Minimum expected cell size for chi-square              0.100D-01
Maximum number of iterations for H1                           2000
Convergence criterion for H1                             0.100D-03
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            1
  Adaptive quadrature                                           ON
Link                                                        PROBIT
Cholesky                                                        ON

Input data file(s)
  cesditems.dat
Input data format  FREE


SUMMARY OF DATA

     Number of missing data patterns            22
     Number of y missing data patterns           0
     Number of u missing data patterns          22


COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     PROPORTION OF DATA PRESENT FOR U

           Covariance Coverage
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            0.993
 BLUES          0.987         0.993
 DEPRESS        0.991         0.989         0.996
 HAPPY          0.987         0.982         0.986         0.989
 ENJOY          0.986         0.981         0.986         0.981         0.988
 HOPEFUL        0.927         0.925         0.929         0.924         0.923

           Covariance Coverage
              HOPEFUL
              ________
 HOPEFUL        0.931


UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES

    SAD
      Category 1    0.653         1768.000
      Category 2    0.274          743.000
      Category 3    0.035           96.000
      Category 4    0.038          102.000
    BLUES
      Category 1    0.774         2097.000
      Category 2    0.150          406.000
      Category 3    0.032           87.000
      Category 4    0.044          118.000
    DEPRESS
      Category 1    0.635         1726.000
      Category 2    0.267          725.000
      Category 3    0.037          101.000
      Category 4    0.060          164.000
    HAPPY
      Category 1    0.662         1786.000
      Category 2    0.100          270.000
      Category 3    0.163          440.000
      Category 4    0.075          201.000
    ENJOY
      Category 1    0.759         2045.000
      Category 2    0.075          202.000
      Category 3    0.107          289.000
      Category 4    0.059          158.000
    HOPEFUL
      Category 1    0.569         1444.000
      Category 2    0.069          175.000
      Category 3    0.158          402.000
      Category 4    0.204          518.000



UNIVARIATE SAMPLE STATISTICS

     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     AGEC70                0.363       0.675      -0.300   34.60%      -0.300      0.200      0.200
            2728.000       0.418      -0.674       1.700    8.72%       0.200      1.200
     FEMALE                0.585      -0.345       0.000   41.50%       0.000      0.000      1.000
            2728.000       0.243      -1.881       1.000   58.50%       1.000      1.000


THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       26

Loglikelihood

          H0 Value                      -12922.703
          H0 Scaling Correction Factor      1.0499
            for MLR

Information Criteria

          Akaike (AIC)                   25897.405
          Bayesian (BIC)                 26051.100
          Sample-Size Adjusted BIC       25968.489
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                1.222      0.065     18.771      0.000
    BLUES              1.328      0.085     15.646      0.000
    DEPRESS            1.518      0.091     16.686      0.000
    HAPPY              0.973      0.056     17.312      0.000
    ENJOY              0.981      0.061     16.195      0.000
    HOPEFUL            0.511      0.037     13.980      0.000

 F          ON
    AGEC70             0.143      0.035      4.090      0.000
    FEMALE             0.311      0.046      6.718      0.000

 Thresholds
    SAD$1              0.906      0.062     14.515      0.000
    SAD$2              2.589      0.099     26.176      0.000
    SAD$3              3.124      0.120     26.039      0.000
    BLUES$1            1.568      0.090     17.475      0.000
    BLUES$2            2.702      0.126     21.389      0.000
    BLUES$3            3.172      0.145     21.838      0.000
    DEPRESS$1          0.996      0.079     12.604      0.000
    DEPRESS$2          2.721      0.134     20.246      0.000
    DEPRESS$3          3.185      0.152     20.917      0.000
    HAPPY$1            0.805      0.053     15.274      0.000
    HAPPY$2            1.221      0.059     20.795      0.000
    HAPPY$3            2.249      0.083     26.975      0.000
    ENJOY$1            1.210      0.061     19.762      0.000
    ENJOY$2            1.585      0.068     23.237      0.000
    ENJOY$3            2.429      0.092     26.482      0.000
    HOPEFUL$1          0.317      0.034      9.409      0.000
    HOPEFUL$2          0.520      0.035     14.876      0.000
    HOPEFUL$3          1.051      0.040     26.108      0.000

 Residual Variances
    F                  1.000      0.000    999.000    999.000


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.317E-02
       (ratio of smallest to largest eigenvalue)


RESULTS IN PROBABILITY SCALE

                    Estimate

 SAD
    Category 1         0.651
    Category 2         0.274
    Category 3         0.037
    Category 4         0.038
 BLUES
    Category 1         0.773
    Category 2         0.150
    Category 3         0.033
    Category 4         0.044
 DEPRESS
    Category 1         0.636
    Category 2         0.265
    Category 3         0.037
    Category 4         0.062
 HAPPY
    Category 1         0.660
    Category 2         0.101
    Category 3         0.165
    Category 4         0.075
 ENJOY
    Category 1         0.756
    Category 2         0.075
    Category 3         0.109
    Category 4         0.060
 HOPEFUL
    Category 1         0.570
    Category 2         0.069
    Category 3         0.157
    Category 4         0.204


STANDARDIZED MODEL RESULTS


STDYX Standardization

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                0.779      0.016     47.857      0.000
    BLUES              0.803      0.018     44.122      0.000
    DEPRESS            0.839      0.015     56.578      0.000
    HAPPY              0.703      0.021     34.256      0.000
    ENJOY              0.706      0.022     32.258      0.000
    HOPEFUL            0.461      0.026     17.809      0.000

 F          ON
    AGEC70             0.091      0.022      4.116      0.000
    FEMALE             0.151      0.022      6.844      0.000

 Thresholds
    SAD$1              0.568      0.033     17.394      0.000
    SAD$2              1.624      0.040     40.243      0.000
    SAD$3              1.959      0.049     39.826      0.000
    BLUES$1            0.934      0.034     27.339      0.000
    BLUES$2            1.609      0.040     39.869      0.000
    BLUES$3            1.889      0.047     40.272      0.000
    DEPRESS$1          0.541      0.034     16.098      0.000
    DEPRESS$2          1.480      0.040     37.337      0.000
    DEPRESS$3          1.732      0.044     39.106      0.000
    HAPPY$1            0.573      0.032     18.141      0.000
    HAPPY$2            0.869      0.032     26.843      0.000
    HAPPY$3            1.599      0.040     40.084      0.000
    ENJOY$1            0.857      0.032     26.521      0.000
    ENJOY$2            1.122      0.034     33.152      0.000
    ENJOY$3            1.720      0.042     40.871      0.000
    HOPEFUL$1          0.281      0.029      9.808      0.000
    HOPEFUL$2          0.462      0.029     15.995      0.000
    HOPEFUL$3          0.933      0.031     30.296      0.000

 Residual Variances
    F                  0.968      0.008    124.714      0.000


R-SQUARE

    Observed                                        Two-Tailed
    Variable        Estimate       S.E.  Est./S.E.    P-Value

    SAD                0.607      0.025     23.929      0.000
    BLUES              0.645      0.029     22.061      0.000
    DEPRESS            0.704      0.025     28.289      0.000
    HAPPY              0.494      0.029     17.128      0.000
    ENJOY              0.499      0.031     16.129      0.000
    HOPEFUL            0.212      0.024      8.904      0.000

     Latent                                         Two-Tailed
    Variable        Estimate       S.E.  Est./S.E.    P-Value

    F                  0.032      0.008      4.131      0.000


TECHNICAL 1 OUTPUT

     PARAMETER SPECIFICATION

           TAU
              SAD$1         SAD$2         SAD$3         BLUES$1       BLUES$2
              ________      ________      ________      ________      ________
                    9            10            11            12            13

           TAU
              BLUES$3       DEPRESS$      DEPRESS$      DEPRESS$      HAPPY$1
              ________      ________      ________      ________      ________
                   14            15            16            17            18

           TAU
              HAPPY$2       HAPPY$3       ENJOY$1       ENJOY$2       ENJOY$3
              ________      ________      ________      ________      ________
                   19            20            21            22            23

           TAU
              HOPEFUL$      HOPEFUL$      HOPEFUL$
              ________      ________      ________
                   24            25            26

           NU
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
                    0             0             0             0             0

           NU
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
                    0             0             0

           LAMBDA
              F             AGEC70        FEMALE
              ________      ________      ________
 SAD                1             0             0
 BLUES              2             0             0
 DEPRESS            3             0             0
 HAPPY              4             0             0
 ENJOY              5             0             0
 HOPEFUL            6             0             0
 AGEC70             0             0             0
 FEMALE             0             0             0

           THETA
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD                0
 BLUES              0             0
 DEPRESS            0             0             0
 HAPPY              0             0             0             0
 ENJOY              0             0             0             0             0
 HOPEFUL            0             0             0             0             0
 AGEC70             0             0             0             0             0
 FEMALE             0             0             0             0             0

           THETA
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
 HOPEFUL            0
 AGEC70             0             0
 FEMALE             0             0             0

           ALPHA
              F             AGEC70        FEMALE
              ________      ________      ________
                    0             0             0

           BETA
              F             AGEC70        FEMALE
              ________      ________      ________
 F                  0             7             8
 AGEC70             0             0             0
 FEMALE             0             0             0

           PSI
              F             AGEC70        FEMALE
              ________      ________      ________
 F                  0
 AGEC70             0             0
 FEMALE             0             0             0

     STARTING VALUES

           TAU
              SAD$1         SAD$2         SAD$3         BLUES$1       BLUES$2
              ________      ________      ________      ________      ________
                0.350         1.411         1.801         0.685         1.390

           TAU
              BLUES$3       DEPRESS$      DEPRESS$      DEPRESS$      HAPPY$1
              ________      ________      ________      ________      ________
                1.716         0.309         1.236         1.525         0.374

           TAU
              HAPPY$2       HAPPY$3       ENJOY$1       ENJOY$2       ENJOY$3
              ________      ________      ________      ________      ________
                0.647         1.400         0.638         0.897         1.542

           TAU
              HOPEFUL$      HOPEFUL$      HOPEFUL$
              ________      ________      ________
                0.154         0.314         0.756

           NU
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
                0.000         0.000         0.000         0.000         0.000

           NU
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
                0.000         0.000         0.000

           LAMBDA
              F             AGEC70        FEMALE
              ________      ________      ________
 SAD            1.000         0.000         0.000
 BLUES          1.000         0.000         0.000
 DEPRESS        1.000         0.000         0.000
 HAPPY          1.000         0.000         0.000
 ENJOY          1.000         0.000         0.000
 HOPEFUL        1.000         0.000         0.000
 AGEC70         0.000         1.000         0.000
 FEMALE         0.000         0.000         1.000

           THETA
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            1.000
 BLUES          0.000         1.000
 DEPRESS        0.000         0.000         1.000
 HAPPY          0.000         0.000         0.000         1.000
 ENJOY          0.000         0.000         0.000         0.000         1.000
 HOPEFUL        0.000         0.000         0.000         0.000         0.000
 AGEC70         0.000         0.000         0.000         0.000         0.000
 FEMALE         0.000         0.000         0.000         0.000         0.000

           THETA
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
 HOPEFUL        1.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.000         0.000

           ALPHA
              F             AGEC70        FEMALE
              ________      ________      ________
                0.000         0.000         0.000

           BETA
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000         0.000         0.000
 AGEC70         0.000         0.000         0.000
 FEMALE         0.000         0.000         0.000

           PSI
              F             AGEC70        FEMALE
              ________      ________      ________
 F              1.000
 AGEC70         0.000         0.209
 FEMALE         0.000         0.000         0.121


TECHNICAL 4 OUTPUT

     ESTIMATES DERIVED FROM THE MODEL

           ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.234         0.363         0.585

           S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.030         0.012         0.009

           EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                7.914        29.327        62.018

           TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.000         0.000         0.000

           ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              1.033
 AGEC70         0.064         0.418
 FEMALE         0.077         0.013         0.243

           S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.008
 AGEC70         0.015         0.011
 FEMALE         0.011         0.006         0.007

           EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F            124.714
 AGEC70         4.302        36.932
 FEMALE         6.753         2.099        36.932

           TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.036         0.000

           ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              1.000
 AGEC70         0.097         1.000
 FEMALE         0.154         0.040         1.000

           S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.022         0.000
 FEMALE         0.022         0.019         0.000

           EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F            999.000
 AGEC70         4.364       999.000
 FEMALE         7.004         2.104       999.000

           TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.035         0.000

     Beginning Time:  11:54:14
        Ending Time:  11:54:14
       Elapsed Time:  00:00:00



MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA  90066

Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com

Copyright (c) 1998-2024 Muthen & Muthen

Notice:
Note the r-squared for latent factor f.
Note the variance for the latent factor f in TECH4.

Save output from Mplus Engine

file.copy("formplus.out", "model23step1.out", overwrite = TRUE)
file.copy("formplus.inp", "model23step1.inp", overwrite = TRUE)

Step 2. Note the identification constraint and OUTPUT options.

TITLE:    Categorical item CFA of CESD at baseline
DATA:     FILE = cesditems.dat ;
VARIABLE: NAMES = sad blues depress happy enjoy hopeful
                  agec70 female; 
          CATEGORICAL are sad-hopeful ;
          MISSING = . ;
ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
MODEL:    f by sad-hopeful*; 
          f@0.968; ! 0.968 = 1 - .032
          f on agec70 female ;
OUTPUT:   STDYX; TECH1 ; TECH4; SVALUES ;
Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025  11:54 AM

INPUT INSTRUCTIONS

  TITLE:    Categorical item CFA of CESD at baseline
  DATA:     FILE = cesditems.dat ;
  VARIABLE: NAMES = sad blues depress happy enjoy hopeful
                    agec70 female;
            CATEGORICAL are sad-hopeful ;
            MISSING = . ;
  ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
  MODEL:    f by sad-hopeful*;
            f@0.968; ! 0.968 = 1 - .032
            f on agec70 female ;
  OUTPUT:   STDYX; TECH1 ; TECH4; SVALUES ;



INPUT READING TERMINATED NORMALLY



Categorical item CFA of CESD at baseline

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        2728

Number of dependent variables                                    6
Number of independent variables                                  2
Number of continuous latent variables                            1

Observed dependent variables

  Binary and ordered categorical (ordinal)
   SAD         BLUES       DEPRESS     HAPPY       ENJOY       HOPEFUL

Observed independent variables
   AGEC70      FEMALE

Continuous latent variables
   F


Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                 100
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-02
    Relative loglikelihood change                        0.100D-05
    Derivative                                           0.100D-02
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            10
  Minimum value for logit thresholds                           -10
  Minimum expected cell size for chi-square              0.100D-01
Maximum number of iterations for H1                           2000
Convergence criterion for H1                             0.100D-03
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            1
  Adaptive quadrature                                           ON
Link                                                        PROBIT
Cholesky                                                        ON

Input data file(s)
  cesditems.dat
Input data format  FREE


SUMMARY OF DATA

     Number of missing data patterns            22
     Number of y missing data patterns           0
     Number of u missing data patterns          22


COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     PROPORTION OF DATA PRESENT FOR U

           Covariance Coverage
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            0.993
 BLUES          0.987         0.993
 DEPRESS        0.991         0.989         0.996
 HAPPY          0.987         0.982         0.986         0.989
 ENJOY          0.986         0.981         0.986         0.981         0.988
 HOPEFUL        0.927         0.925         0.929         0.924         0.923

           Covariance Coverage
              HOPEFUL
              ________
 HOPEFUL        0.931


UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES

    SAD
      Category 1    0.653         1768.000
      Category 2    0.274          743.000
      Category 3    0.035           96.000
      Category 4    0.038          102.000
    BLUES
      Category 1    0.774         2097.000
      Category 2    0.150          406.000
      Category 3    0.032           87.000
      Category 4    0.044          118.000
    DEPRESS
      Category 1    0.635         1726.000
      Category 2    0.267          725.000
      Category 3    0.037          101.000
      Category 4    0.060          164.000
    HAPPY
      Category 1    0.662         1786.000
      Category 2    0.100          270.000
      Category 3    0.163          440.000
      Category 4    0.075          201.000
    ENJOY
      Category 1    0.759         2045.000
      Category 2    0.075          202.000
      Category 3    0.107          289.000
      Category 4    0.059          158.000
    HOPEFUL
      Category 1    0.569         1444.000
      Category 2    0.069          175.000
      Category 3    0.158          402.000
      Category 4    0.204          518.000



UNIVARIATE SAMPLE STATISTICS

     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     AGEC70                0.363       0.675      -0.300   34.60%      -0.300      0.200      0.200
            2728.000       0.418      -0.674       1.700    8.72%       0.200      1.200
     FEMALE                0.585      -0.345       0.000   41.50%       0.000      0.000      1.000
            2728.000       0.243      -1.881       1.000   58.50%       1.000      1.000


THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       26

Loglikelihood

          H0 Value                      -12922.703
          H0 Scaling Correction Factor      1.0499
            for MLR

Information Criteria

          Akaike (AIC)                   25897.405
          Bayesian (BIC)                 26051.100
          Sample-Size Adjusted BIC       25968.489
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                1.242      0.066     18.771      0.000
    BLUES              1.349      0.086     15.646      0.000
    DEPRESS            1.543      0.092     16.686      0.000
    HAPPY              0.989      0.057     17.312      0.000
    ENJOY              0.997      0.062     16.195      0.000
    HOPEFUL            0.519      0.037     13.980      0.000

 F          ON
    AGEC70             0.141      0.034      4.090      0.000
    FEMALE             0.306      0.046      6.718      0.000

 Thresholds
    SAD$1              0.906      0.062     14.515      0.000
    SAD$2              2.589      0.099     26.176      0.000
    SAD$3              3.124      0.120     26.039      0.000
    BLUES$1            1.568      0.090     17.475      0.000
    BLUES$2            2.702      0.126     21.389      0.000
    BLUES$3            3.172      0.145     21.838      0.000
    DEPRESS$1          0.996      0.079     12.604      0.000
    DEPRESS$2          2.721      0.134     20.246      0.000
    DEPRESS$3          3.185      0.152     20.917      0.000
    HAPPY$1            0.805      0.053     15.273      0.000
    HAPPY$2            1.221      0.059     20.795      0.000
    HAPPY$3            2.249      0.083     26.975      0.000
    ENJOY$1            1.210      0.061     19.762      0.000
    ENJOY$2            1.585      0.068     23.237      0.000
    ENJOY$3            2.429      0.092     26.482      0.000
    HOPEFUL$1          0.317      0.034      9.409      0.000
    HOPEFUL$2          0.520      0.035     14.876      0.000
    HOPEFUL$3          1.051      0.040     26.108      0.000

 Residual Variances
    F                  0.968      0.000    999.000    999.000


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.316E-02
       (ratio of smallest to largest eigenvalue)


RESULTS IN PROBABILITY SCALE

                    Estimate

 SAD
    Category 1         0.651
    Category 2         0.274
    Category 3         0.037
    Category 4         0.038
 BLUES
    Category 1         0.773
    Category 2         0.150
    Category 3         0.033
    Category 4         0.044
 DEPRESS
    Category 1         0.636
    Category 2         0.265
    Category 3         0.037
    Category 4         0.062
 HAPPY
    Category 1         0.660
    Category 2         0.101
    Category 3         0.165
    Category 4         0.075
 ENJOY
    Category 1         0.756
    Category 2         0.075
    Category 3         0.109
    Category 4         0.060
 HOPEFUL
    Category 1         0.570
    Category 2         0.069
    Category 3         0.157
    Category 4         0.204


STANDARDIZED MODEL RESULTS


STDYX Standardization

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                0.779      0.016     47.857      0.000
    BLUES              0.803      0.018     44.122      0.000
    DEPRESS            0.839      0.015     56.577      0.000
    HAPPY              0.703      0.021     34.257      0.000
    ENJOY              0.706      0.022     32.257      0.000
    HOPEFUL            0.461      0.026     17.809      0.000

 F          ON
    AGEC70             0.091      0.022      4.116      0.000
    FEMALE             0.151      0.022      6.844      0.000

 Thresholds
    SAD$1              0.568      0.033     17.394      0.000
    SAD$2              1.624      0.040     40.243      0.000
    SAD$3              1.959      0.049     39.826      0.000
    BLUES$1            0.934      0.034     27.339      0.000
    BLUES$2            1.609      0.040     39.869      0.000
    BLUES$3            1.889      0.047     40.272      0.000
    DEPRESS$1          0.541      0.034     16.098      0.000
    DEPRESS$2          1.480      0.040     37.337      0.000
    DEPRESS$3          1.732      0.044     39.106      0.000
    HAPPY$1            0.573      0.032     18.141      0.000
    HAPPY$2            0.869      0.032     26.843      0.000
    HAPPY$3            1.599      0.040     40.084      0.000
    ENJOY$1            0.857      0.032     26.521      0.000
    ENJOY$2            1.122      0.034     33.152      0.000
    ENJOY$3            1.720      0.042     40.871      0.000
    HOPEFUL$1          0.281      0.029      9.808      0.000
    HOPEFUL$2          0.462      0.029     15.995      0.000
    HOPEFUL$3          0.933      0.031     30.296      0.000

 Residual Variances
    F                  0.968      0.008    124.714      0.000


R-SQUARE

    Observed                                        Two-Tailed
    Variable        Estimate       S.E.  Est./S.E.    P-Value

    SAD                0.607      0.025     23.929      0.000
    BLUES              0.645      0.029     22.061      0.000
    DEPRESS            0.704      0.025     28.289      0.000
    HAPPY              0.494      0.029     17.128      0.000
    ENJOY              0.499      0.031     16.129      0.000
    HOPEFUL            0.212      0.024      8.904      0.000

     Latent                                         Two-Tailed
    Variable        Estimate       S.E.  Est./S.E.    P-Value

    F                  0.032      0.008      4.131      0.000


MODEL COMMAND WITH FINAL ESTIMATES USED AS STARTING VALUES

     f BY sad*1.24246;
     f BY blues*1.34935;
     f BY depress*1.54319;
     f BY happy*0.98851;
     f BY enjoy*0.99714;
     f BY hopeful*0.51918;

     f ON agec70*0.14057;
     f ON female*0.30581;

     [ sad$1*0.90573 ];
     [ sad$2*2.58945 ];
     [ sad$3*3.12399 ];
     [ blues$1*1.56819 ];
     [ blues$2*2.70245 ];
     [ blues$3*3.17185 ];
     [ depress$1*0.99558 ];
     [ depress$2*2.72132 ];
     [ depress$3*3.18523 ];
     [ happy$1*0.80544 ];
     [ happy$2*1.22139 ];
     [ happy$3*2.24855 ];
     [ enjoy$1*1.21014 ];
     [ enjoy$2*1.58509 ];
     [ enjoy$3*2.42945 ];
     [ hopeful$1*0.31681 ];
     [ hopeful$2*0.52039 ];
     [ hopeful$3*1.05122 ];

     f@0.96800;



TECHNICAL 1 OUTPUT

     PARAMETER SPECIFICATION

           TAU
              SAD$1         SAD$2         SAD$3         BLUES$1       BLUES$2
              ________      ________      ________      ________      ________
                    9            10            11            12            13

           TAU
              BLUES$3       DEPRESS$      DEPRESS$      DEPRESS$      HAPPY$1
              ________      ________      ________      ________      ________
                   14            15            16            17            18

           TAU
              HAPPY$2       HAPPY$3       ENJOY$1       ENJOY$2       ENJOY$3
              ________      ________      ________      ________      ________
                   19            20            21            22            23

           TAU
              HOPEFUL$      HOPEFUL$      HOPEFUL$
              ________      ________      ________
                   24            25            26

           NU
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
                    0             0             0             0             0

           NU
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
                    0             0             0

           LAMBDA
              F             AGEC70        FEMALE
              ________      ________      ________
 SAD                1             0             0
 BLUES              2             0             0
 DEPRESS            3             0             0
 HAPPY              4             0             0
 ENJOY              5             0             0
 HOPEFUL            6             0             0
 AGEC70             0             0             0
 FEMALE             0             0             0

           THETA
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD                0
 BLUES              0             0
 DEPRESS            0             0             0
 HAPPY              0             0             0             0
 ENJOY              0             0             0             0             0
 HOPEFUL            0             0             0             0             0
 AGEC70             0             0             0             0             0
 FEMALE             0             0             0             0             0

           THETA
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
 HOPEFUL            0
 AGEC70             0             0
 FEMALE             0             0             0

           ALPHA
              F             AGEC70        FEMALE
              ________      ________      ________
                    0             0             0

           BETA
              F             AGEC70        FEMALE
              ________      ________      ________
 F                  0             7             8
 AGEC70             0             0             0
 FEMALE             0             0             0

           PSI
              F             AGEC70        FEMALE
              ________      ________      ________
 F                  0
 AGEC70             0             0
 FEMALE             0             0             0

     STARTING VALUES

           TAU
              SAD$1         SAD$2         SAD$3         BLUES$1       BLUES$2
              ________      ________      ________      ________      ________
                0.350         1.411         1.801         0.685         1.390

           TAU
              BLUES$3       DEPRESS$      DEPRESS$      DEPRESS$      HAPPY$1
              ________      ________      ________      ________      ________
                1.716         0.309         1.236         1.525         0.374

           TAU
              HAPPY$2       HAPPY$3       ENJOY$1       ENJOY$2       ENJOY$3
              ________      ________      ________      ________      ________
                0.647         1.400         0.638         0.897         1.542

           TAU
              HOPEFUL$      HOPEFUL$      HOPEFUL$
              ________      ________      ________
                0.154         0.314         0.756

           NU
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
                0.000         0.000         0.000         0.000         0.000

           NU
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
                0.000         0.000         0.000

           LAMBDA
              F             AGEC70        FEMALE
              ________      ________      ________
 SAD            1.000         0.000         0.000
 BLUES          1.000         0.000         0.000
 DEPRESS        1.000         0.000         0.000
 HAPPY          1.000         0.000         0.000
 ENJOY          1.000         0.000         0.000
 HOPEFUL        1.000         0.000         0.000
 AGEC70         0.000         1.000         0.000
 FEMALE         0.000         0.000         1.000

           THETA
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            1.000
 BLUES          0.000         1.000
 DEPRESS        0.000         0.000         1.000
 HAPPY          0.000         0.000         0.000         1.000
 ENJOY          0.000         0.000         0.000         0.000         1.000
 HOPEFUL        0.000         0.000         0.000         0.000         0.000
 AGEC70         0.000         0.000         0.000         0.000         0.000
 FEMALE         0.000         0.000         0.000         0.000         0.000

           THETA
              HOPEFUL       AGEC70        FEMALE
              ________      ________      ________
 HOPEFUL        1.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.000         0.000

           ALPHA
              F             AGEC70        FEMALE
              ________      ________      ________
                0.000         0.000         0.000

           BETA
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000         0.000         0.000
 AGEC70         0.000         0.000         0.000
 FEMALE         0.000         0.000         0.000

           PSI
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.968
 AGEC70         0.000         0.209
 FEMALE         0.000         0.000         0.121


TECHNICAL 4 OUTPUT

     ESTIMATES DERIVED FROM THE MODEL

           ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.230         0.363         0.585

           S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.029         0.012         0.009

           EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                7.914        29.327        62.018

           TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.000         0.000         0.000

           ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              1.000
 AGEC70         0.063         0.418
 FEMALE         0.076         0.013         0.243

           S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.008
 AGEC70         0.015         0.011
 FEMALE         0.011         0.006         0.007

           EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F            124.714
 AGEC70         4.302        36.932
 FEMALE         6.753         2.099        36.932

           TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.036         0.000

           ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              1.000
 AGEC70         0.097         1.000
 FEMALE         0.154         0.040         1.000

           S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.022         0.000
 FEMALE         0.022         0.019         0.000

           EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F            999.000
 AGEC70         4.364       999.000
 FEMALE         7.004         2.104       999.000

           TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.035         0.000

     Beginning Time:  11:54:14
        Ending Time:  11:54:14
       Elapsed Time:  00:00:00



MUTHEN & MUTHEN
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Los Angeles, CA  90066

Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com

Copyright (c) 1998-2024 Muthen & Muthen

Notice:
Note the variance for the latent factor f in TECH4.
Note the unstandardized factor loading for sad, in MODEL RESULTS and in STARTING VALUES.

Save output from Mplus Engine

file.copy("formplus.out", "model23step2.out", overwrite = TRUE)
file.copy("formplus.inp", "model23step2.inp", overwrite = TRUE)

Step 3. Final baseline parameterization (and check variance of f).

TITLE:    Categorical item CFA of CESD at baseline
DATA:     FILE = cesditems.dat ;
VARIABLE: NAMES = sad blues depress happy enjoy hopeful
                  agec70 female; 
          CATEGORICAL are sad-hopeful ;
          MISSING = . ;
ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
MODEL:    f BY sad@1.24246; ! now fixed
          f by blues-hopeful*; 
          f* ; ! now estimated
          f on agec70 female ;
OUTPUT:   TECH4;
Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025  11:54 AM

INPUT INSTRUCTIONS

  TITLE:    Categorical item CFA of CESD at baseline
  DATA:     FILE = cesditems.dat ;
  VARIABLE: NAMES = sad blues depress happy enjoy hopeful
                    agec70 female;
            CATEGORICAL are sad-hopeful ;
            MISSING = . ;
  ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
  MODEL:    f BY sad@1.24246; ! now fixed
            f by blues-hopeful*;
            f* ; ! now estimated
            f on agec70 female ;
  OUTPUT:   TECH4;



INPUT READING TERMINATED NORMALLY



Categorical item CFA of CESD at baseline

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        2728

Number of dependent variables                                    6
Number of independent variables                                  2
Number of continuous latent variables                            1

Observed dependent variables

  Binary and ordered categorical (ordinal)
   SAD         BLUES       DEPRESS     HAPPY       ENJOY       HOPEFUL

Observed independent variables
   AGEC70      FEMALE

Continuous latent variables
   F


Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                 100
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-02
    Relative loglikelihood change                        0.100D-05
    Derivative                                           0.100D-02
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            10
  Minimum value for logit thresholds                           -10
  Minimum expected cell size for chi-square              0.100D-01
Maximum number of iterations for H1                           2000
Convergence criterion for H1                             0.100D-03
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            1
  Adaptive quadrature                                           ON
Link                                                        PROBIT
Cholesky                                                        ON

Input data file(s)
  cesditems.dat
Input data format  FREE


SUMMARY OF DATA

     Number of missing data patterns            22
     Number of y missing data patterns           0
     Number of u missing data patterns          22


COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     PROPORTION OF DATA PRESENT FOR U

           Covariance Coverage
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            0.993
 BLUES          0.987         0.993
 DEPRESS        0.991         0.989         0.996
 HAPPY          0.987         0.982         0.986         0.989
 ENJOY          0.986         0.981         0.986         0.981         0.988
 HOPEFUL        0.927         0.925         0.929         0.924         0.923

           Covariance Coverage
              HOPEFUL
              ________
 HOPEFUL        0.931


UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES

    SAD
      Category 1    0.653         1768.000
      Category 2    0.274          743.000
      Category 3    0.035           96.000
      Category 4    0.038          102.000
    BLUES
      Category 1    0.774         2097.000
      Category 2    0.150          406.000
      Category 3    0.032           87.000
      Category 4    0.044          118.000
    DEPRESS
      Category 1    0.635         1726.000
      Category 2    0.267          725.000
      Category 3    0.037          101.000
      Category 4    0.060          164.000
    HAPPY
      Category 1    0.662         1786.000
      Category 2    0.100          270.000
      Category 3    0.163          440.000
      Category 4    0.075          201.000
    ENJOY
      Category 1    0.759         2045.000
      Category 2    0.075          202.000
      Category 3    0.107          289.000
      Category 4    0.059          158.000
    HOPEFUL
      Category 1    0.569         1444.000
      Category 2    0.069          175.000
      Category 3    0.158          402.000
      Category 4    0.204          518.000



UNIVARIATE SAMPLE STATISTICS

     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     AGEC70                0.363       0.675      -0.300   34.60%      -0.300      0.200      0.200
            2728.000       0.418      -0.674       1.700    8.72%       0.200      1.200
     FEMALE                0.585      -0.345       0.000   41.50%       0.000      0.000      1.000
            2728.000       0.243      -1.881       1.000   58.50%       1.000      1.000


THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       26

Loglikelihood

          H0 Value                      -12922.702
          H0 Scaling Correction Factor      1.0500
            for MLR

Information Criteria

          Akaike (AIC)                   25897.405
          Bayesian (BIC)                 26051.099
          Sample-Size Adjusted BIC       25968.489
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F        BY
    SAD                1.242      0.000    999.000    999.000
    BLUES              1.349      0.100     13.445      0.000
    DEPRESS            1.543      0.107     14.475      0.000
    HAPPY              0.989      0.081     12.234      0.000
    ENJOY              0.997      0.085     11.768      0.000
    HOPEFUL            0.519      0.049     10.587      0.000

 F          ON
    AGEC70             0.141      0.035      4.057      0.000
    FEMALE             0.306      0.048      6.393      0.000

 Thresholds
    SAD$1              0.906      0.062     14.510      0.000
    SAD$2              2.589      0.099     26.163      0.000
    SAD$3              3.124      0.120     26.027      0.000
    BLUES$1            1.568      0.090     17.473      0.000
    BLUES$2            2.703      0.126     21.387      0.000
    BLUES$3            3.172      0.145     21.837      0.000
    DEPRESS$1          0.996      0.079     12.603      0.000
    DEPRESS$2          2.721      0.134     20.246      0.000
    DEPRESS$3          3.185      0.152     20.916      0.000
    HAPPY$1            0.805      0.053     15.270      0.000
    HAPPY$2            1.221      0.059     20.790      0.000
    HAPPY$3            2.249      0.083     26.970      0.000
    ENJOY$1            1.210      0.061     19.759      0.000
    ENJOY$2            1.585      0.068     23.234      0.000
    ENJOY$3            2.429      0.092     26.479      0.000
    HOPEFUL$1          0.317      0.034      9.408      0.000
    HOPEFUL$2          0.520      0.035     14.874      0.000
    HOPEFUL$3          1.051      0.040     26.104      0.000

 Residual Variances
    F                  0.968      0.103      9.378      0.000


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.202E-02
       (ratio of smallest to largest eigenvalue)


RESULTS IN PROBABILITY SCALE

                    Estimate

 SAD
    Category 1         0.651
    Category 2         0.274
    Category 3         0.037
    Category 4         0.038
 BLUES
    Category 1         0.773
    Category 2         0.150
    Category 3         0.033
    Category 4         0.044
 DEPRESS
    Category 1         0.636
    Category 2         0.265
    Category 3         0.037
    Category 4         0.062
 HAPPY
    Category 1         0.659
    Category 2         0.101
    Category 3         0.165
    Category 4         0.075
 ENJOY
    Category 1         0.756
    Category 2         0.075
    Category 3         0.109
    Category 4         0.060
 HOPEFUL
    Category 1         0.570
    Category 2         0.069
    Category 3         0.157
    Category 4         0.204


TECHNICAL 4 OUTPUT

     ESTIMATES DERIVED FROM THE MODEL

           ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.230         0.363         0.585

           S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.031         0.012         0.009

           EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                7.458        29.327        62.018

           TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
                0.000         0.000         0.000

           ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              1.000
 AGEC70         0.063         0.418
 FEMALE         0.076         0.013         0.243

           S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.106
 AGEC70         0.015         0.011
 FEMALE         0.012         0.006         0.007

           EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              9.399
 AGEC70         4.260        36.932
 FEMALE         6.427         2.099        36.932

           TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.036         0.000

           ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              1.000
 AGEC70         0.097         1.000
 FEMALE         0.154         0.040         1.000

           S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.022         0.000
 FEMALE         0.022         0.019         0.000

           EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F            999.000
 AGEC70         4.364       999.000
 FEMALE         7.003         2.104       999.000

           TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F             AGEC70        FEMALE
              ________      ________      ________
 F              0.000
 AGEC70         0.000         0.000
 FEMALE         0.000         0.035         0.000

     Beginning Time:  11:54:14
        Ending Time:  11:54:15
       Elapsed Time:  00:00:01



MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA  90066

Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com

Copyright (c) 1998-2024 Muthen & Muthen

Notice:
Note the variance for the latent factor f in TECH4.

Save output from Mplus Engine

file.copy("formplus.out", "model23step3.out", overwrite = TRUE)
file.copy("formplus.inp", "model23step3.inp", overwrite = TRUE)

Mplus Multiple Indicator LGC model, categorical items

MILGCM

TITLE:    Categorical item multiple indicator LGCM of CESD
DATA:     FILE = "cesdwide.dat";
VARIABLE: NAMES = id female agec70 
          u01 u02 u03 u04 u05 u06 
          u31 u32 u33 u34 u35 u36 
          u61 u62 u63 u64 u65 u66; 
          IDVARIABLE = id ;
          MISSING = . ;
          CATEGORICAL are u01-u66 ;
ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
MODEL:    f0 by u01@1.281 u02-u06* (l1-l6); ! = 1.24246*sqrt(1.063)
          f3 by u31@1.281 u32-u36* (l1-l6); ! where 1.063 is estimate
          f6 by u61@1.281 u62-u66* (l1-l6); ! of VAR(F0) when l1 = 1.24246
          f0 f3 f6 (psi) ; ! for comparability with MLGOF
          ! Also no item-level resid covar for comparability with MLGOF
          [u01$1 u31$1 u61$1] (t11) ;
          [u01$2 u31$2 u61$2] (t12) ;
          [u01$3 u31$3 u61$3] (t13) ;                  
          [u02$1 u32$1 u62$1] (t21) ;
          [u02$2 u32$2 u62$2] (t22) ;
          [u02$3 u32$3 u62$3] (t23) ;                  
          [u03$1 u33$1 u63$1] (t31) ;
          [u03$2 u33$2 u63$2] (t32) ;
          [u03$3 u33$3 u63$3] (t33) ;                  
          [u04$1 u34$1 u64$1] (t41) ;
          [u04$2 u34$2 u64$2] (t42) ;
          [u04$3 u34$3 u64$3] (t43) ;                  
          [u05$1 u35$1 u65$1] (t51) ;
          [u05$2 u35$2 u65$2] (t52) ;
          [u05$3 u35$3 u65$3] (t53) ;                  
          [u06$1 u36$1 u66$1] (t61) ;
          [u06$2 u36$2 u66$2] (t62) ;
          [u06$3 u36$3 u66$3] (t63) ;
          i by f0@1 f3@1 f6@1 ;
          i* ;
          [i@0] ;
          s by f0@0 f3@0.3 f6@0.6 ;
          s* ;
          [s*] ;
          i with s ;
          i s on female agec70 ;
OUTPUT: TECH4;

Note: Data set up in “Multilevel Curve of Factors” presentation.

Note in this model we have:

MODEL:    f0 by u01@1.281 u02-u06* (l1-l6); ! = 1.24246*sqrt(1.063)
          f3 by u31@1.281 u32-u36* (l1-l6); ! where 1.063 is estimate in TECH4
          f6 by u61@1.281 u62-u66* (l1-l6); ! of VAR(F0) when l1 = 1.24246

On the first run of this mode, the variance of f0 was 1.063 in the TECH4 output. It should be exected that the constraint on u01 or sad derived from the one time point model would not perfectly produce a 1.0 variance in the longitudinal model. The longitudinal model we have forces equal residual variance on the time-specific latent traits (f0, f3, and f6) and these will come out as a weighted average. It is not reasonable to assume this parameter would be the same as the model that only included f1. Also, the f3 and f6 factors are predicted by latent variable S, but f0 is not.

The implication of this is we could probably have skipped the three step procedure using the baseline model to get a latent variable variance of 1 in the MILGCM and just run two MILGM models (as was done here).

You will see in this model, the TECH4 variance estimate for f0 is 1.0.

Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025  11:54 AM

INPUT INSTRUCTIONS

  TITLE:    Categorical item multiple indicator LGCM of CESD
  DATA:     FILE = "cesdwide.dat";
  VARIABLE: NAMES = id female agec70
            u01 u02 u03 u04 u05 u06
            u31 u32 u33 u34 u35 u36
            u61 u62 u63 u64 u65 u66;
            IDVARIABLE = id ;
            MISSING = . ;
            CATEGORICAL are u01-u66 ;
  ANALYSIS: ESTIMATOR = MLR ; LINK = PROBIT ;
  MODEL:    f0 by u01@1.281 u02-u06* (l1-l6); ! = 1.24246*sqrt(1.063)
            f3 by u31@1.281 u32-u36* (l1-l6); ! where 1.063 is estimate
            f6 by u61@1.281 u62-u66* (l1-l6); ! of VAR(F0) when l1 = 1.24246
            f0 f3 f6 (psi) ; ! for comparability with MLGOF
            ! Also no item-level resid covar for comparability with MLGOF
            [u01$1 u31$1 u61$1] (t11) ;
            [u01$2 u31$2 u61$2] (t12) ;
            [u01$3 u31$3 u61$3] (t13) ;
            [u02$1 u32$1 u62$1] (t21) ;
            [u02$2 u32$2 u62$2] (t22) ;
            [u02$3 u32$3 u62$3] (t23) ;
            [u03$1 u33$1 u63$1] (t31) ;
            [u03$2 u33$2 u63$2] (t32) ;
            [u03$3 u33$3 u63$3] (t33) ;
            [u04$1 u34$1 u64$1] (t41) ;
            [u04$2 u34$2 u64$2] (t42) ;
            [u04$3 u34$3 u64$3] (t43) ;
            [u05$1 u35$1 u65$1] (t51) ;
            [u05$2 u35$2 u65$2] (t52) ;
            [u05$3 u35$3 u65$3] (t53) ;
            [u06$1 u36$1 u66$1] (t61) ;
            [u06$2 u36$2 u66$2] (t62) ;
            [u06$3 u36$3 u66$3] (t63) ;
            i by f0@1 f3@1 f6@1 ;
            i* ;
            [i@0] ;
            s by f0@0 f3@0.3 f6@0.6 ;
            s* ;
            [s*] ;
            i with s ;
            i s on female agec70 ;
  OUTPUT: TECH4;



INPUT READING TERMINATED NORMALLY



Categorical item multiple indicator LGCM of CESD

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        2762

Number of dependent variables                                   18
Number of independent variables                                  2
Number of continuous latent variables                            5

Observed dependent variables

  Binary and ordered categorical (ordinal)
   U01         U02         U03         U04         U05         U06
   U31         U32         U33         U34         U35         U36
   U61         U62         U63         U64         U65         U66

Observed independent variables
   FEMALE      AGEC70

Continuous latent variables
   F0          F3          F6          I           S

Variables with special functions

  ID variable           ID

Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                 100
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-02
    Relative loglikelihood change                        0.100D-05
    Derivative                                           0.100D-02
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            10
  Minimum value for logit thresholds                           -10
  Minimum expected cell size for chi-square              0.100D-01
Maximum number of iterations for H1                           2000
Convergence criterion for H1                             0.100D-03
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            3
  Adaptive quadrature                                           ON
Link                                                        PROBIT
Cholesky                                                        ON

Input data file(s)
  cesdwide.dat
Input data format  FREE


SUMMARY OF DATA

     Number of missing data patterns           140
     Number of y missing data patterns           0
     Number of u missing data patterns         140


COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     PROPORTION OF DATA PRESENT FOR U

           Covariance Coverage
              U01           U02           U03           U04           U05
              ________      ________      ________      ________      ________
 U01            0.981
 U02            0.975         0.980
 U03            0.979         0.977         0.983
 U04            0.975         0.970         0.974         0.976
 U05            0.974         0.969         0.974         0.969         0.975
 U06            0.915         0.914         0.917         0.913         0.911
 U31            0.711         0.711         0.711         0.708         0.707
 U32            0.707         0.706         0.706         0.703         0.702
 U33            0.711         0.711         0.711         0.708         0.707
 U34            0.701         0.700         0.701         0.698         0.696
 U35            0.699         0.699         0.699         0.696         0.694
 U36            0.642         0.642         0.642         0.639         0.638
 U61            0.501         0.500         0.501         0.499         0.498
 U62            0.502         0.501         0.502         0.500         0.499
 U63            0.501         0.501         0.501         0.499         0.499
 U64            0.498         0.498         0.499         0.496         0.496
 U65            0.498         0.498         0.499         0.496         0.496
 U66            0.479         0.478         0.479         0.477         0.476

           Covariance Coverage
              U06           U31           U32           U33           U34
              ________      ________      ________      ________      ________
 U06            0.919
 U31            0.665         0.724
 U32            0.662         0.715         0.720
 U33            0.665         0.722         0.716         0.725
 U34            0.657         0.712         0.706         0.711         0.714
 U35            0.655         0.710         0.703         0.710         0.702
 U36            0.603         0.652         0.648         0.652         0.646
 U61            0.467         0.485         0.484         0.485         0.481
 U62            0.468         0.487         0.485         0.486         0.482
 U63            0.467         0.486         0.485         0.486         0.482
 U64            0.465         0.483         0.482         0.483         0.479
 U65            0.465         0.484         0.482         0.483         0.480
 U66            0.447         0.464         0.462         0.463         0.459

           Covariance Coverage
              U35           U36           U61           U62           U63
              ________      ________      ________      ________      ________
 U35            0.712
 U36            0.644         0.654
 U61            0.479         0.446         0.510
 U62            0.480         0.447         0.507         0.512
 U63            0.479         0.446         0.508         0.508         0.511
 U64            0.476         0.444         0.504         0.505         0.505
 U65            0.478         0.445         0.506         0.505         0.505
 U66            0.457         0.429         0.485         0.486         0.487

           Covariance Coverage
              U64           U65           U66
              ________      ________      ________
 U64            0.508
 U65            0.504         0.508
 U66            0.483         0.483         0.488


UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES

    U01
      Category 1    0.653         1768.000
      Category 2    0.274          743.000
      Category 3    0.035           96.000
      Category 4    0.038          102.000
    U02
      Category 1    0.774         2097.000
      Category 2    0.150          406.000
      Category 3    0.032           87.000
      Category 4    0.044          118.000
    U03
      Category 1    0.635         1726.000
      Category 2    0.267          725.000
      Category 3    0.037          101.000
      Category 4    0.060          164.000
    U04
      Category 1    0.662         1786.000
      Category 2    0.100          270.000
      Category 3    0.163          440.000
      Category 4    0.075          201.000
    U05
      Category 1    0.759         2045.000
      Category 2    0.075          202.000
      Category 3    0.107          289.000
      Category 4    0.059          158.000
    U06
      Category 1    0.569         1444.000
      Category 2    0.069          175.000
      Category 3    0.158          402.000
      Category 4    0.204          518.000
    U31
      Category 1    0.651         1302.000
      Category 2    0.273          547.000
      Category 3    0.032           65.000
      Category 4    0.043           87.000
    U32
      Category 1    0.749         1489.000
      Category 2    0.178          353.000
      Category 3    0.030           60.000
      Category 4    0.043           86.000
    U33
      Category 1    0.651         1303.000
      Category 2    0.254          508.000
      Category 3    0.039           79.000
      Category 4    0.056          113.000
    U34
      Category 1    0.598         1179.000
      Category 2    0.141          279.000
      Category 3    0.200          395.000
      Category 4    0.060          119.000
    U35
      Category 1    0.675         1328.000
      Category 2    0.115          227.000
      Category 3    0.154          303.000
      Category 4    0.055          108.000
    U36
      Category 1    0.568         1026.000
      Category 2    0.131          237.000
      Category 3    0.163          294.000
      Category 4    0.138          249.000
    U61
      Category 1    0.616          868.000
      Category 2    0.308          434.000
      Category 3    0.031           43.000
      Category 4    0.045           64.000
    U62
      Category 1    0.769         1087.000
      Category 2    0.156          220.000
      Category 3    0.031           44.000
      Category 4    0.044           62.000
    U63
      Category 1    0.608          858.000
      Category 2    0.290          409.000
      Category 3    0.048           68.000
      Category 4    0.054           76.000
    U64
      Category 1    0.616          863.000
      Category 2    0.101          142.000
      Category 3    0.202          283.000
      Category 4    0.081          114.000
    U65
      Category 1    0.674          946.000
      Category 2    0.103          145.000
      Category 3    0.140          197.000
      Category 4    0.082          115.000
    U66
      Category 1    0.551          743.000
      Category 2    0.111          150.000
      Category 3    0.183          247.000
      Category 4    0.154          208.000



UNIVARIATE SAMPLE STATISTICS

     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     FEMALE                0.584      -0.342       0.000   41.56%       0.000      0.000      1.000
            2762.000       0.243      -1.883       1.000   58.44%       1.000      1.000
     AGEC70                0.364       0.673      -0.300   34.65%      -0.300      0.200      0.200
            2762.000       0.419      -0.681       1.700    8.76%       0.200      1.200


THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       32

Loglikelihood

          H0 Value                      -29743.061
          H0 Scaling Correction Factor      1.1084
            for MLR

Information Criteria

          Akaike (AIC)                   59550.123
          Bayesian (BIC)                 59739.682
          Sample-Size Adjusted BIC       59638.007
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 F0       BY
    U01                1.281      0.000    999.000    999.000
    U02                1.419      0.068     20.838      0.000
    U03                1.583      0.072     21.863      0.000
    U04                1.042      0.058     17.890      0.000
    U05                1.045      0.060     17.494      0.000
    U06                0.499      0.033     14.933      0.000

 F3       BY
    U31                1.281      0.000    999.000    999.000
    U32                1.419      0.068     20.838      0.000
    U33                1.583      0.072     21.863      0.000
    U34                1.042      0.058     17.890      0.000
    U35                1.045      0.060     17.494      0.000
    U36                0.499      0.033     14.933      0.000

 F6       BY
    U61                1.281      0.000    999.000    999.000
    U62                1.419      0.068     20.838      0.000
    U63                1.583      0.072     21.863      0.000
    U64                1.042      0.058     17.890      0.000
    U65                1.045      0.060     17.494      0.000
    U66                0.499      0.033     14.933      0.000

 I        BY
    F0                 1.000      0.000    999.000    999.000
    F3                 1.000      0.000    999.000    999.000
    F6                 1.000      0.000    999.000    999.000

 S        BY
    F0                 0.000      0.000    999.000    999.000
    F3                 0.300      0.000    999.000    999.000
    F6                 0.600      0.000    999.000    999.000

 I          ON
    FEMALE             0.293      0.044      6.584      0.000
    AGEC70             0.128      0.033      3.857      0.000

 S          ON
    FEMALE             0.005      0.105      0.043      0.965
    AGEC70             0.134      0.089      1.498      0.134

 I        WITH
    S                 -0.165      0.086     -1.914      0.056

 Intercepts
    I                  0.000      0.000    999.000    999.000
    S                  0.255      0.087      2.913      0.004

 Thresholds
    U01$1              0.926      0.054     17.074      0.000
    U01$2              2.648      0.076     34.710      0.000
    U01$3              3.131      0.087     35.909      0.000
    U02$1              1.607      0.071     22.530      0.000
    U02$2              2.836      0.093     30.459      0.000
    U02$3              3.305      0.104     31.646      0.000
    U03$1              1.054      0.069     15.377      0.000
    U03$2              2.797      0.101     27.798      0.000
    U03$3              3.322      0.114     29.240      0.000
    U04$1              0.747      0.045     16.725      0.000
    U04$2              1.210      0.049     24.789      0.000
    U04$3              2.367      0.064     36.892      0.000
    U05$1              1.069      0.048     22.448      0.000
    U05$2              1.509      0.053     28.659      0.000
    U05$3              2.461      0.068     36.014      0.000
    U06$1              0.316      0.027     11.781      0.000
    U06$2              0.606      0.028     21.667      0.000
    U06$3              1.186      0.032     37.273      0.000
    U31$1              0.926      0.054     17.074      0.000
    U31$2              2.648      0.076     34.710      0.000
    U31$3              3.131      0.087     35.909      0.000
    U32$1              1.607      0.071     22.530      0.000
    U32$2              2.836      0.093     30.459      0.000
    U32$3              3.305      0.104     31.646      0.000
    U33$1              1.054      0.069     15.377      0.000
    U33$2              2.797      0.101     27.798      0.000
    U33$3              3.322      0.114     29.240      0.000
    U34$1              0.747      0.045     16.725      0.000
    U34$2              1.210      0.049     24.789      0.000
    U34$3              2.367      0.064     36.892      0.000
    U35$1              1.069      0.048     22.448      0.000
    U35$2              1.509      0.053     28.659      0.000
    U35$3              2.461      0.068     36.014      0.000
    U36$1              0.316      0.027     11.781      0.000
    U36$2              0.606      0.028     21.667      0.000
    U36$3              1.186      0.032     37.273      0.000
    U61$1              0.926      0.054     17.074      0.000
    U61$2              2.648      0.076     34.710      0.000
    U61$3              3.131      0.087     35.909      0.000
    U62$1              1.607      0.071     22.530      0.000
    U62$2              2.836      0.093     30.459      0.000
    U62$3              3.305      0.104     31.646      0.000
    U63$1              1.054      0.069     15.377      0.000
    U63$2              2.797      0.101     27.798      0.000
    U63$3              3.322      0.114     29.240      0.000
    U64$1              0.747      0.045     16.725      0.000
    U64$2              1.210      0.049     24.789      0.000
    U64$3              2.367      0.064     36.892      0.000
    U65$1              1.069      0.048     22.448      0.000
    U65$2              1.509      0.053     28.659      0.000
    U65$3              2.461      0.068     36.014      0.000
    U66$1              0.316      0.027     11.781      0.000
    U66$2              0.606      0.028     21.667      0.000
    U66$3              1.186      0.032     37.273      0.000

 Residual Variances
    F0                 0.433      0.042     10.425      0.000
    F3                 0.433      0.042     10.425      0.000
    F6                 0.433      0.042     10.425      0.000
    I                  0.537      0.057      9.466      0.000
    S                  0.338      0.237      1.425      0.154


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.120E-04
       (ratio of smallest to largest eigenvalue)


RESULTS IN PROBABILITY SCALE

                    Estimate

 U01
    Category 1         0.655
    Category 2         0.273
    Category 3         0.033
    Category 4         0.040
 U02
    Category 1         0.773
    Category 2         0.155
    Category 3         0.031
    Category 4         0.042
 U03
    Category 1         0.648
    Category 2         0.257
    Category 3         0.039
    Category 4         0.056
 U04
    Category 1         0.641
    Category 2         0.111
    Category 3         0.179
    Category 4         0.069
 U05
    Category 1         0.720
    Category 2         0.093
    Category 3         0.127
    Category 4         0.061
 U06
    Category 1         0.573
    Category 2         0.098
    Category 3         0.161
    Category 4         0.167
 U31
    Category 1         0.630
    Category 2         0.291
    Category 3         0.036
    Category 4         0.043
 U32
    Category 1         0.754
    Category 2         0.167
    Category 3         0.033
    Category 4         0.046
 U33
    Category 1         0.621
    Category 2         0.275
    Category 3         0.043
    Category 4         0.061
 U34
    Category 1         0.617
    Category 2         0.117
    Category 3         0.191
    Category 4         0.075
 U35
    Category 1         0.700
    Category 2         0.098
    Category 3         0.136
    Category 4         0.066
 U36
    Category 1         0.558
    Category 2         0.100
    Category 3         0.166
    Category 4         0.176
 U61
    Category 1         0.602
    Category 2         0.308
    Category 3         0.040
    Category 4         0.050
 U62
    Category 1         0.729
    Category 2         0.180
    Category 3         0.037
    Category 4         0.054
 U63
    Category 1         0.591
    Category 2         0.291
    Category 3         0.048
    Category 4         0.071
 U64
    Category 1         0.592
    Category 2         0.120
    Category 3         0.204
    Category 4         0.085
 U65
    Category 1         0.676
    Category 2         0.102
    Category 3         0.146
    Category 4         0.076
 U66
    Category 1         0.541
    Category 2         0.101
    Category 3         0.170
    Category 4         0.187


TECHNICAL 4 OUTPUT

     ESTIMATES DERIVED FROM THE MODEL

           ESTIMATED MEANS FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
                0.218         0.309         0.401         0.218         0.306

           ESTIMATED MEANS FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
                0.584         0.364

           S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
                0.028         0.034         0.045         0.028         0.056

           S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
                0.009         0.012

           EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
                7.712         9.135         8.831         7.712         5.456

           EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
               62.315        29.528

           TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
                0.000         0.000         0.000         0.000         0.000

           TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
                0.000         0.000

           ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0             1.000
 F3             0.519         0.937
 F6             0.472         0.487         0.936
 I              0.566         0.519         0.472         0.566
 S             -0.157        -0.053         0.051        -0.157         0.345
 FEMALE         0.073         0.074         0.075         0.073         0.003
 AGEC70         0.058         0.074         0.091         0.058         0.056

           ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE         0.243
 AGEC70         0.014         0.419

           S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0             0.078
 F3             0.045         0.071
 F6             0.044         0.046         0.078
 I              0.058         0.045         0.044         0.058
 S              0.087         0.055         0.093         0.087         0.238
 FEMALE         0.011         0.010         0.013         0.011         0.025
 AGEC70         0.014         0.013         0.020         0.014         0.037

           S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE         0.007
 AGEC70         0.006         0.011

           EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0            12.825
 F3            11.579        13.264
 F6            10.661        10.505        11.932
 I              9.693        11.579        10.661         9.693
 S             -1.806        -0.973         0.545        -1.806         1.453
 FEMALE         6.637         7.659         5.554         6.637         0.119
 AGEC70         4.105         5.717         4.610         4.105         1.502

           EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE        37.162
 AGEC70         2.346        37.162

           TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0             0.000
 F3             0.000         0.000
 F6             0.000         0.000         0.000
 I              0.000         0.000         0.000         0.000
 S              0.071         0.331         0.586         0.071         0.146
 FEMALE         0.000         0.000         0.000         0.000         0.906
 AGEC70         0.000         0.000         0.000         0.000         0.133

           TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE         0.000
 AGEC70         0.019         0.000

           ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0             1.000
 F3             0.537         1.000
 F6             0.488         0.521         1.000
 I              0.753         0.713         0.649         1.000
 S             -0.267        -0.093         0.089        -0.354         1.000
 FEMALE         0.148         0.155         0.157         0.197         0.010
 AGEC70         0.089         0.119         0.146         0.118         0.147

           ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE         1.000
 AGEC70         0.045         1.000

           S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0             0.000
 F3             0.022         0.000
 F6             0.029         0.024         0.000
 I              0.021         0.015         0.044         0.000
 S              0.091         0.103         0.138         0.116         0.000
 FEMALE         0.021         0.019         0.027         0.028         0.088
 AGEC70         0.021         0.020         0.030         0.029         0.107

           S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE         0.000
 AGEC70         0.019         0.000

           EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0           999.000
 F3            24.501       999.000
 F6            16.832        21.423       999.000
 I             36.424        47.019        14.857       999.000
 S             -2.917        -0.905         0.644        -3.044       999.000
 FEMALE         7.032         8.225         5.774         6.941         0.118
 AGEC70         4.155         5.936         4.826         4.129         1.376

           EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE       999.000
 AGEC70         2.353       999.000

           TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              F0            F3            F6            I             S
              ________      ________      ________      ________      ________
 F0             0.000
 F3             0.000         0.000
 F6             0.000         0.000         0.000
 I              0.000         0.000         0.000         0.000
 S              0.004         0.366         0.519         0.002         0.000
 FEMALE         0.000         0.000         0.000         0.000         0.906
 AGEC70         0.000         0.000         0.000         0.000         0.169

           TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
              FEMALE        AGEC70
              ________      ________
 FEMALE         0.000
 AGEC70         0.019         0.000

     Beginning Time:  11:54:15
        Ending Time:  11:57:57
       Elapsed Time:  00:03:42



MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA  90066

Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com

Copyright (c) 1998-2024 Muthen & Muthen

Save output from Mplus Engine

file.copy("formplus.out", "model24.out", overwrite = TRUE)
file.copy("formplus.inp", "model24.inp", overwrite = TRUE)

Mplus Multilevel Growth of Factors, categorical items

MLCOF

TITLE:    Categorical item multilevel growth of factors ;
DATA:     FILE = cesdlongk.dat ;
          VARIANCES = NOCHECK ;
VARIABLE: NAMES = id td female agec70 sad blues depress happy enjoy hopeful k ; 
          CATEGORICAL are sad blues depress happy enjoy hopeful ;
          MISSING = . ;
          WITHIN = td k ;
          BETWEEN = agec70 female ; 
          CLUSTER = id ;
ANALYSIS: TYPE = TWOLEVEL RANDOM ;
          ESTIMATOR = MLR ; 
          LINK = PROBIT ;
MODEL:    %WITHIN%
          fw by sad@1.281 blues-hopeful*; 
          s | fw on td ;
          i | fw on k ;
          %BETWEEN%
          !sad-hopeful@0 ; ! This would be included in continuous variable case
          [i@0] ;
          i with s ;
          i s on female agec70 ;

Note: The scale for fw :

MODEL:    %WITHIN%
          fw by sad@1.281 blues-hopeful*;

Was borrowed from the MILGM model. It was not possible to run TECH4 in model25.inp, the MLCOF model. Mplus says:

*** WARNING in OUTPUT command
  TECH4 option is only available for TYPE=TWOLEVEL RANDOM with ESTIMATOR=BAYES.
  Request for TECH4 is ignored.

Switching to ESTIMATOR=BAYES is therefore an option, but even `tho I am using one of three common scale parameterizations (MLR/PROBIT) and the parameter estimates will be on the same scale in MLR/PROBIT and BAYES, they might not be equal enough to guarantee a variance of 1 at the first time point. But it is an option.

Also, if I have a follow-up model to run that includes sampling weights, that cannot be accommodated by ESTIMATOR=BAYES.

Therefore I also can’t confirm the variance of fw is 1.0 at baseline without calculating the variance “by hand” or moving to a Bayes estimator.

Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025  11:57 AM

INPUT INSTRUCTIONS

  TITLE:    Categorical item multilevel growth of factors ;
  DATA:     FILE = cesdlongk.dat ;
            VARIANCES = NOCHECK ;
  VARIABLE: NAMES = id td female agec70 sad blues depress happy enjoy hopeful k ;
            CATEGORICAL are sad blues depress happy enjoy hopeful ;
            MISSING = . ;
            WITHIN = td k ;
            BETWEEN = agec70 female ;
            CLUSTER = id ;
  ANALYSIS: TYPE = TWOLEVEL RANDOM ;
            ESTIMATOR = MLR ;
            LINK = PROBIT ;
  MODEL:    %WITHIN%
            fw by sad@1.281 blues-hopeful*;
            s | fw on td ;
            i | fw on k ;
            %BETWEEN%
            !sad-hopeful@0 ; ! This would be included in continuous variable case
            [i@0] ;
            i with s ;
            i s on female agec70 ;



*** WARNING in MODEL command
  A y-variable has been declared on the within level but not referred to on
  the between level.  Please check that this is what is intended.  If this is not intended,
  specify the variable as a within variable.  Problem with:  SAD
*** WARNING in MODEL command
  A y-variable has been declared on the within level but not referred to on
  the between level.  Please check that this is what is intended.  If this is not intended,
  specify the variable as a within variable.  Problem with:  BLUES
*** WARNING in MODEL command
  A y-variable has been declared on the within level but not referred to on
  the between level.  Please check that this is what is intended.  If this is not intended,
  specify the variable as a within variable.  Problem with:  DEPRESS
*** WARNING in MODEL command
  A y-variable has been declared on the within level but not referred to on
  the between level.  Please check that this is what is intended.  If this is not intended,
  specify the variable as a within variable.  Problem with:  HAPPY
*** WARNING in MODEL command
  A y-variable has been declared on the within level but not referred to on
  the between level.  Please check that this is what is intended.  If this is not intended,
  specify the variable as a within variable.  Problem with:  ENJOY
*** WARNING in MODEL command
  A y-variable has been declared on the within level but not referred to on
  the between level.  Please check that this is what is intended.  If this is not intended,
  specify the variable as a within variable.  Problem with:  HOPEFUL
*** WARNING
  One or more individual-level variables have no variation within a
  cluster for the following clusters.

     Variable   Cluster IDs with no within-cluster variation

      SAD         154 319 484 815 1046 1528 1791 2164 2451 557 98
      BLUES       687 793 1071 1618 1669 2096 2104 2369 557 2205
      DEPRESS     154 2369
      HAPPY       154 319 484 605 815 1012 1046 1528 1786 1849 2549 557 98
      ENJOY       154 223 319 484 1046 1528 1741 2164 2451 2528 2557 557 886 98
      HOPEFUL     29 67 143 154 239 341 360 373 448 510 553 605 687 707 933 938 979 1046 1115
                  1191 1282 1381 1406 1417 1479 1667 1687 1742 1756 1762 1764 1786 1809 1878 1895
                  1956 1987 2012 2096 2266 2292 2369 2375 2404 2428 2496 2523 2528 2756 557 802
                  2205 50 444 1783 905 952 1994 2452 180 2576 1325 1342

   7 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS



Categorical item multilevel growth of factors ;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        6165

Number of dependent variables                                    6
Number of independent variables                                  4
Number of continuous latent variables                            3

Observed dependent variables

  Binary and ordered categorical (ordinal)
   SAD         BLUES       DEPRESS     HAPPY       ENJOY       HOPEFUL

Observed independent variables
   TD          FEMALE      AGEC70      K

Continuous latent variables
   FW          S           I

Variables with special functions

  Cluster variable      ID

  Within variables
   TD          K

  Between variables
   FEMALE      AGEC70


Estimator                                                      MLR
Information matrix                                        OBSERVED
Optimization Specifications for the Quasi-Newton Algorithm for
Continuous Outcomes
  Maximum number of iterations                                 100
  Convergence criterion                                  0.100D-05
Optimization Specifications for the EM Algorithm
  Maximum number of iterations                                 500
  Convergence criteria
    Loglikelihood change                                 0.100D-02
    Relative loglikelihood change                        0.100D-05
    Derivative                                           0.100D-02
Optimization Specifications for the M step of the EM Algorithm for
Categorical Latent variables
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
Optimization Specifications for the M step of the EM Algorithm for
Censored, Binary or Ordered Categorical (Ordinal), Unordered
Categorical (Nominal) and Count Outcomes
  Number of M step iterations                                    1
  M step convergence criterion                           0.100D-02
  Basis for M step termination                           ITERATION
  Maximum value for logit thresholds                            10
  Minimum value for logit thresholds                           -10
  Minimum expected cell size for chi-square              0.100D-01
Maximum number of iterations for H1                           2000
Convergence criterion for H1                             0.100D-03
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            3
  Adaptive quadrature                                           ON
Link                                                        PROBIT
Cholesky                                                        ON

Input data file(s)
  cesdlongk.dat
Input data format  FREE


SUMMARY OF DATA

     Number of missing data patterns            32
     Number of y missing data patterns           0
     Number of u missing data patterns          32
     Number of clusters                       2762



COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value   0.100

     PROPORTION OF DATA PRESENT FOR U

           Covariance Coverage
              SAD           BLUES         DEPRESS       HAPPY         ENJOY
              ________      ________      ________      ________      ________
 SAD            0.993
 BLUES          0.984         0.991
 DEPRESS        0.989         0.986         0.994
 HAPPY          0.982         0.977         0.981         0.985
 ENJOY          0.981         0.976         0.981         0.975         0.983
 HOPEFUL        0.920         0.918         0.921         0.915         0.913

           Covariance Coverage
              HOPEFUL
              ________
 HOPEFUL        0.923


UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES

    SAD
      Category 1    0.644         3938.000
      Category 2    0.282         1724.000
      Category 3    0.033          204.000
      Category 4    0.041          253.000
    BLUES
      Category 1    0.765         4673.000
      Category 2    0.160          979.000
      Category 3    0.031          191.000
      Category 4    0.044          266.000
    DEPRESS
      Category 1    0.634         3887.000
      Category 2    0.268         1642.000
      Category 3    0.040          248.000
      Category 4    0.058          353.000
    HAPPY
      Category 1    0.631         3828.000
      Category 2    0.114          691.000
      Category 3    0.184         1118.000
      Category 4    0.071          434.000
    ENJOY
      Category 1    0.712         4319.000
      Category 2    0.095          574.000
      Category 3    0.130          789.000
      Category 4    0.063          381.000
    HOPEFUL
      Category 1    0.564         3213.000
      Category 2    0.099          562.000
      Category 3    0.166          943.000
      Category 4    0.171          975.000



UNIVARIATE SAMPLE STATISTICS

     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     TD                    0.236       0.395       0.000   44.25%       0.000      0.000      0.300
            6165.000       0.057      -1.306       0.600   23.07%       0.300      0.600
     K                     1.000       0.000       1.000  100.00%       1.000      1.000      1.000
            6165.000       0.000       0.000       1.000  100.00%       1.000      1.000
     FEMALE                0.584      -0.342       0.000   41.56%       0.000      0.000      1.000
            2762.000       0.243      -1.883       1.000   58.44%       1.000      1.000
     AGEC70                0.364       0.673      -0.300   34.65%      -0.300      0.200      0.200
            2762.000       0.419      -0.681       1.700    8.76%       0.200      1.200

     WARNING:  THE SAMPLE COVARIANCE OF THE INDEPENDENT VARIABLES IS SINGULAR.


THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       32

Loglikelihood

          H0 Value                      -29743.055
          H0 Scaling Correction Factor      1.1080
            for MLR

Information Criteria

          Akaike (AIC)                   59550.110
          Bayesian (BIC)                 59765.363
          Sample-Size Adjusted BIC       59663.675
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

Within Level

 FW       BY
    SAD                1.281      0.000    999.000    999.000
    BLUES              1.419      0.068     20.800      0.000
    DEPRESS            1.583      0.072     21.850      0.000
    HAPPY              1.043      0.059     17.803      0.000
    ENJOY              1.046      0.060     17.384      0.000
    HOPEFUL            0.500      0.034     14.877      0.000

 Residual Variances
    FW                 0.434      0.042     10.408      0.000

Between Level

 I          ON
    FEMALE             0.293      0.045      6.578      0.000
    AGEC70             0.128      0.033      3.854      0.000

 S          ON
    FEMALE             0.004      0.105      0.043      0.966
    AGEC70             0.134      0.089      1.498      0.134

 I        WITH
    S                 -0.165      0.085     -1.936      0.053

 Intercepts
    S                  0.255      0.087      2.914      0.004
    I                  0.000      0.000    999.000    999.000

 Thresholds
    SAD$1              0.926      0.054     17.047      0.000
    SAD$2              2.647      0.077     34.578      0.000
    SAD$3              3.131      0.088     35.769      0.000
    BLUES$1            1.606      0.071     22.523      0.000
    BLUES$2            2.835      0.093     30.452      0.000
    BLUES$3            3.304      0.104     31.640      0.000
    DEPRESS$1          1.054      0.069     15.364      0.000
    DEPRESS$2          2.796      0.101     27.778      0.000
    DEPRESS$3          3.321      0.114     29.224      0.000
    HAPPY$1            0.747      0.045     16.726      0.000
    HAPPY$2            1.210      0.049     24.792      0.000
    HAPPY$3            2.367      0.064     36.889      0.000
    ENJOY$1            1.069      0.048     22.451      0.000
    ENJOY$2            1.510      0.053     28.660      0.000
    ENJOY$3            2.462      0.068     36.005      0.000
    HOPEFUL$1          0.316      0.027     11.779      0.000
    HOPEFUL$2          0.606      0.028     21.666      0.000
    HOPEFUL$3          1.187      0.032     37.270      0.000

 Residual Variances
    S                  0.337      0.232      1.453      0.146
    I                  0.537      0.057      9.477      0.000


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.120E-04
       (ratio of smallest to largest eigenvalue)

     Beginning Time:  11:57:57
        Ending Time:  12:04:58
       Elapsed Time:  00:07:01



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Save output from Mplus Engine

file.copy("formplus.out", "model25.out", overwrite = TRUE)
file.copy("formplus.inp", "model25.inp", overwrite = TRUE)