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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