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

INPUT INSTRUCTIONS

  TITLE:	this is an example of non-linear
  	constraint on the logit parameters of an
  	unordered categorical (nominal) variable.
  	the example draws on the classic Rao animal
  	genetics example on page 2 of Dempster,
  	Laird & Rubin (1977) in JRSS B
  MONTECARLO:			
  	NAMES = u;
  	NOBSERVATIONS = 500;
  	generate = u(n 3);
  	nominal = u;
  	NREPS = 1;
  	SEED = 53487;
  	SAVE = ex3.10.dat;
  MODEL POPULATION:
  	[u#1*1.433 u#2*-.519 u#3*-.519];
  MODEL:	
  	[u#1*1.433] (p1);
  	[u#2*-.519] (p2);
  	[u#3*-.519] (p2);
  MODEL CONSTRAINT:
  	p2 = log ((exp (p1) - 1)/2 - 1);
  OUTPUT:
  	TECH8 TECH9;



INPUT READING TERMINATED NORMALLY



this is an example of non-linear
constraint on the logit parameters of an
unordered categorical (nominal) variable.
the example draws on the classic Rao animal
genetics example on page 2 of Dempster,
Laird & Rubin (1977) in JRSS B

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                         500

Number of replications
    Requested                                                    1
    Completed                                                    1
Value of seed                                                53487

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

Observed dependent variables

  Unordered categorical (nominal)
   U


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                            15
  Minimum value for logit thresholds                           -15
  Minimum expected cell size for chi-square              0.100D-01
Optimization algorithm                                         EMA
Integration Specifications
  Type                                                    STANDARD
  Number of integration points                                  15
  Dimensions of numerical integration                            0
  Adaptive quadrature                                           ON
Cholesky                                                       OFF





MODEL FIT INFORMATION

Number of Free Parameters                        1

Loglikelihood

    H0 Value

        Mean                              -536.568
        Std Dev                              0.000
        Number of successful computations        1

             Proportions                   Percentiles
        Expected    Observed         Expected       Observed
           0.990       0.000         -536.568       -536.568
           0.980       0.000         -536.568       -536.568
           0.950       0.000         -536.568       -536.568
           0.900       0.000         -536.568       -536.568
           0.800       0.000         -536.568       -536.568
           0.700       0.000         -536.568       -536.568
           0.500       0.000         -536.568       -536.568
           0.300       0.000         -536.568       -536.568
           0.200       0.000         -536.568       -536.568
           0.100       0.000         -536.568       -536.568
           0.050       0.000         -536.568       -536.568
           0.020       0.000         -536.568       -536.568
           0.010       0.000         -536.568       -536.568

Information Criteria

    Akaike (AIC)

        Mean                              1075.136
        Std Dev                              0.000
        Number of successful computations        1

             Proportions                   Percentiles
        Expected    Observed         Expected       Observed
           0.990       0.000         1075.136       1075.136
           0.980       0.000         1075.136       1075.136
           0.950       0.000         1075.136       1075.136
           0.900       0.000         1075.136       1075.136
           0.800       0.000         1075.136       1075.136
           0.700       0.000         1075.136       1075.136
           0.500       0.000         1075.136       1075.136
           0.300       0.000         1075.136       1075.136
           0.200       0.000         1075.136       1075.136
           0.100       0.000         1075.136       1075.136
           0.050       0.000         1075.136       1075.136
           0.020       0.000         1075.136       1075.136
           0.010       0.000         1075.136       1075.136

    Bayesian (BIC)

        Mean                              1079.351
        Std Dev                              0.000
        Number of successful computations        1

             Proportions                   Percentiles
        Expected    Observed         Expected       Observed
           0.990       0.000         1079.351       1079.351
           0.980       0.000         1079.351       1079.351
           0.950       0.000         1079.351       1079.351
           0.900       0.000         1079.351       1079.351
           0.800       0.000         1079.351       1079.351
           0.700       0.000         1079.351       1079.351
           0.500       0.000         1079.351       1079.351
           0.300       0.000         1079.351       1079.351
           0.200       0.000         1079.351       1079.351
           0.100       0.000         1079.351       1079.351
           0.050       0.000         1079.351       1079.351
           0.020       0.000         1079.351       1079.351
           0.010       0.000         1079.351       1079.351

    Sample-Size Adjusted BIC (n* = (n + 2) / 24)

        Mean                              1076.176
        Std Dev                              0.000
        Number of successful computations        1

             Proportions                   Percentiles
        Expected    Observed         Expected       Observed
           0.990       0.000         1076.176       1076.176
           0.980       0.000         1076.176       1076.176
           0.950       0.000         1076.176       1076.176
           0.900       0.000         1076.176       1076.176
           0.800       0.000         1076.176       1076.176
           0.700       0.000         1076.176       1076.176
           0.500       0.000         1076.176       1076.176
           0.300       0.000         1076.176       1076.176
           0.200       0.000         1076.176       1076.176
           0.100       0.000         1076.176       1076.176
           0.050       0.000         1076.176       1076.176
           0.020       0.000         1076.176       1076.176
           0.010       0.000         1076.176       1076.176



MODEL RESULTS

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

 Means
  U#1                 1.433     1.4685     0.0000     0.0398     0.0013 1.000 1.000
  U#2                -0.519    -0.3983     0.0000     0.1286     0.0146 1.000 1.000
  U#3                -0.519    -0.3983     0.0000     0.1286     0.0146 1.000 1.000


QUALITY OF NUMERICAL RESULTS

     Average Condition Number for the Information Matrix      0.167E-02
       (ratio of smallest to largest eigenvalue)


TECHNICAL 1 OUTPUT


     PARAMETER SPECIFICATION


           NU
              U#1           U#2           U#3
              ________      ________      ________
                    1             2             2


           THETA
              U#1           U#2           U#3
              ________      ________      ________
 U#1                0
 U#2                0             0
 U#3                0             0             0


     STARTING VALUES


           NU
              U#1           U#2           U#3
              ________      ________      ________
                1.433        -0.519        -0.519


           THETA
              U#1           U#2           U#3
              ________      ________      ________
 U#1            0.000
 U#2            0.000         0.000
 U#3            0.000         0.000         0.000


     POPULATION VALUES


           NU
              U#1           U#2           U#3
              ________      ________      ________
                1.433        -0.519        -0.519


           THETA
              U#1           U#2           U#3
              ________      ________      ________
 U#1            0.000
 U#2            0.000         0.000
 U#3            0.000         0.000         0.000


TECHNICAL 8 OUTPUT


  TECHNICAL 8 OUTPUT FOR REPLICATION 1


   E STEP  ITER  LOGLIKELIHOOD    ABS CHANGE   REL CHANGE  ALGORITHM
              1 -0.53695534D+03    0.0000000    0.0000000  EM
              2 -0.53657285D+03    0.3824912    0.0007123  EM
              3 -0.53656797D+03    0.0048792    0.0000091  EM
              4 -0.53656797D+03    0.0000009    0.0000000  EM


TECHNICAL 9 OUTPUT

  Error messages for each replication (if any)



SAVEDATA INFORMATION

  Order of variables

    U

  Save file
    ex3.10.dat

  Save file format           Free
  Save file record length    10000


     Beginning Time:  22:22:32
        Ending Time:  22:22:32
       Elapsed Time:  00:00:00



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