Mplus VERSION 8.8
MUTHEN & MUTHEN
04/19/2022 10:24 PM
INPUT INSTRUCTIONS
TITLE:
this is an example of a LCA with binary
latent class indicators and parameter
constraints
! this model is that of pp. 70-72 in
! McCutcheon (2002) in the Hagenaars & McCutcheon (2002)
! book Applied Latent Class Analysis (Cambridge Univ Press).
montecarlo:
names are u1-u4;
genclasses = c(2);
classes = c(2);
generate = u1-u4(1);
categorical = u1-u4;
nobs = 1000;
seed = 3454367;
nrep = 1;
save = ex7.13.dat;
ANALYSIS:
TYPE = MIXTURE;
MODEL POPULATION:
%OVERALL%
[c#1*-1];
%c#1%
[u1$1*-1];
[u2$1*-1];
[u3$1*-1];
[u4$1*-1];
%c#2%
[u1$1@-15];
[u2$1*1];
[u3$1*1];
[u4$1*1];
MODEL:
%OVERALL%
[c#1*-1];
%c#1%
[u1$1*-1];
[u2$1*-1] (1);
[u3$1*-1] (1);
[u4$1*-1] (p1);
%c#2%
[u1$1@-15];
! this gives the McCutcheon p. 72 eqn (13)
! deterministic restriction P(u1=1 |c=2) = 1
[u2$1*1] (2);
[u3$1*1] (2);
! this gives the McCutcheon p. 70 eqn (11)
! parallell indicators hypothesis
[u4$1*1] (p2);
MODEL CONSTRAINT:
p2 = - p1;
! this constraint gives the McCutcheon
! p. 71 eqn (12) equal error rate hypothesis
OUTPUT:
TECH8 tech9;
INPUT READING TERMINATED NORMALLY
this is an example of a LCA with binary
latent class indicators and parameter
constraints
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 1000
Number of replications
Requested 1
Completed 1
Value of seed 3454367
Number of dependent variables 4
Number of independent variables 0
Number of continuous latent variables 0
Number of categorical latent variables 1
Observed dependent variables
Binary and ordered categorical (ordinal)
U1 U2 U3 U4
Categorical latent variables
C
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-06
Relative loglikelihood change 0.100D-06
Derivative 0.100D-05
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-05
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-05
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
Link LOGIT
MODEL FIT INFORMATION
Number of Free Parameters 5
Loglikelihood
H0 Value
Mean -2208.165
Std Dev 0.000
Number of successful computations 1
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.000 -2208.165 -2208.165
0.980 0.000 -2208.165 -2208.165
0.950 0.000 -2208.165 -2208.165
0.900 0.000 -2208.165 -2208.165
0.800 0.000 -2208.165 -2208.165
0.700 0.000 -2208.165 -2208.165
0.500 0.000 -2208.165 -2208.165
0.300 0.000 -2208.165 -2208.165
0.200 0.000 -2208.165 -2208.165
0.100 0.000 -2208.165 -2208.165
0.050 0.000 -2208.165 -2208.165
0.020 0.000 -2208.165 -2208.165
0.010 0.000 -2208.165 -2208.165
Information Criteria
Akaike (AIC)
Mean 4426.329
Std Dev 0.000
Number of successful computations 1
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.000 4426.329 4426.329
0.980 0.000 4426.329 4426.329
0.950 0.000 4426.329 4426.329
0.900 0.000 4426.329 4426.329
0.800 0.000 4426.329 4426.329
0.700 0.000 4426.329 4426.329
0.500 0.000 4426.329 4426.329
0.300 0.000 4426.329 4426.329
0.200 0.000 4426.329 4426.329
0.100 0.000 4426.329 4426.329
0.050 0.000 4426.329 4426.329
0.020 0.000 4426.329 4426.329
0.010 0.000 4426.329 4426.329
Bayesian (BIC)
Mean 4450.868
Std Dev 0.000
Number of successful computations 1
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.000 4450.868 4450.868
0.980 0.000 4450.868 4450.868
0.950 0.000 4450.868 4450.868
0.900 0.000 4450.868 4450.868
0.800 0.000 4450.868 4450.868
0.700 0.000 4450.868 4450.868
0.500 0.000 4450.868 4450.868
0.300 0.000 4450.868 4450.868
0.200 0.000 4450.868 4450.868
0.100 0.000 4450.868 4450.868
0.050 0.000 4450.868 4450.868
0.020 0.000 4450.868 4450.868
0.010 0.000 4450.868 4450.868
Sample-Size Adjusted BIC (n* = (n + 2) / 24)
Mean 4434.988
Std Dev 0.000
Number of successful computations 1
Proportions Percentiles
Expected Observed Expected Observed
0.990 0.000 4434.988 4434.988
0.980 0.000 4434.988 4434.988
0.950 0.000 4434.988 4434.988
0.900 0.000 4434.988 4434.988
0.800 0.000 4434.988 4434.988
0.700 0.000 4434.988 4434.988
0.500 0.000 4434.988 4434.988
0.300 0.000 4434.988 4434.988
0.200 0.000 4434.988 4434.988
0.100 0.000 4434.988 4434.988
0.050 0.000 4434.988 4434.988
0.020 0.000 4434.988 4434.988
0.010 0.000 4434.988 4434.988
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Mean 18.173
Std Dev 0.000
Degrees of freedom 10
Number of successful computations 1
Proportions Percentiles
Expected Observed Expected Observed
0.990 1.000 2.558 18.173
0.980 1.000 3.059 18.173
0.950 1.000 3.940 18.173
0.900 1.000 4.865 18.173
0.800 1.000 6.179 18.173
0.700 1.000 7.267 18.173
0.500 1.000 9.342 18.173
0.300 1.000 11.781 18.173
0.200 1.000 13.442 18.173
0.100 1.000 15.987 18.173
0.050 0.000 18.307 18.173
0.020 0.000 21.161 18.173
0.010 0.000 23.209 18.173
Likelihood Ratio Chi-Square
Mean 18.994
Std Dev 0.000
Degrees of freedom 10
Number of successful computations 1
Proportions Percentiles
Expected Observed Expected Observed
0.990 1.000 2.558 18.994
0.980 1.000 3.059 18.994
0.950 1.000 3.940 18.994
0.900 1.000 4.865 18.994
0.800 1.000 6.179 18.994
0.700 1.000 7.267 18.994
0.500 1.000 9.342 18.994
0.300 1.000 11.781 18.994
0.200 1.000 13.442 18.994
0.100 1.000 15.987 18.994
0.050 1.000 18.307 18.994
0.020 0.000 21.161 18.994
0.010 0.000 23.209 18.994
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL
Latent
Classes
1 247.83178 0.24783
2 752.16822 0.75217
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON ESTIMATED POSTERIOR PROBABILITIES
Latent
Classes
1 247.83178 0.24783
2 752.16822 0.75217
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP
Class Counts and Proportions
Latent
Classes
1 176 0.17600
2 824 0.82400
CLASSIFICATION QUALITY
Entropy 0.636
Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)
1 2
1 0.912 0.088
2 0.106 0.894
Classification Probabilities for the Most Likely Latent Class Membership (Column)
by Latent Class (Row)
1 2
1 0.648 0.352
2 0.021 0.979
Logits for the Classification Probabilities for the Most Likely Latent Class Membership (Column)
by Latent Class (Row)
1 2
1 0.609 0.000
2 -3.862 0.000
MODEL RESULTS
ESTIMATES S. E. M. S. E. 95% % Sig
Population Average Std. Dev. Average Cover Coeff
Latent Class 1
Thresholds
U1$1 -1.000 -0.5097 0.0000 0.1750 0.2404 0.000 1.000
U2$1 -1.000 -1.1236 0.0000 0.1572 0.0153 1.000 1.000
U3$1 -1.000 -1.1236 0.0000 0.1572 0.0153 1.000 1.000
U4$1 -1.000 -1.0633 0.0000 0.0988 0.0040 1.000 1.000
Latent Class 2
Thresholds
U1$1 -15.000 -15.0000 0.0000 0.0000 0.0000 1.000 0.000
U2$1 1.000 0.9487 0.0000 0.0777 0.0026 1.000 1.000
U3$1 1.000 0.9487 0.0000 0.0777 0.0026 1.000 1.000
U4$1 1.000 1.0633 0.0000 0.0988 0.0040 1.000 1.000
Categorical Latent Variables
Means
C#1 -1.000 -1.1102 0.0000 0.1234 0.0121 1.000 1.000
QUALITY OF NUMERICAL RESULTS
Average Condition Number for the Information Matrix 0.377E-02
(ratio of smallest to largest eigenvalue)
TECHNICAL 1 OUTPUT
PARAMETER SPECIFICATION FOR LATENT CLASS 1
PARAMETER SPECIFICATION FOR LATENT CLASS 2
PARAMETER SPECIFICATION FOR LATENT CLASS INDICATOR MODEL PART
TAU(U) FOR LATENT CLASS 1
U1$1 U2$1 U3$1 U4$1
________ ________ ________ ________
1 2 2 3
TAU(U) FOR LATENT CLASS 2
U1$1 U2$1 U3$1 U4$1
________ ________ ________ ________
0 4 4 5
PARAMETER SPECIFICATION FOR LATENT CLASS REGRESSION MODEL PART
ALPHA(C)
C#1 C#2
________ ________
6 0
STARTING VALUES FOR LATENT CLASS 1
STARTING VALUES FOR LATENT CLASS 2
STARTING VALUES FOR LATENT CLASS INDICATOR MODEL PART
TAU(U) FOR LATENT CLASS 1
U1$1 U2$1 U3$1 U4$1
________ ________ ________ ________
-1.000 -1.000 -1.000 -1.000
TAU(U) FOR LATENT CLASS 2
U1$1 U2$1 U3$1 U4$1
________ ________ ________ ________
-15.000 1.000 1.000 1.000
STARTING VALUES FOR LATENT CLASS REGRESSION MODEL PART
ALPHA(C)
C#1 C#2
________ ________
-1.000 0.000
POPULATION VALUES FOR LATENT CLASS 1
POPULATION VALUES FOR LATENT CLASS 2
POPULATION VALUES FOR LATENT CLASS INDICATOR MODEL PART
TAU(U) FOR LATENT CLASS 1
U1$1 U2$1 U3$1 U4$1
________ ________ ________ ________
-1.000 -1.000 -1.000 -1.000
TAU(U) FOR LATENT CLASS 2
U1$1 U2$1 U3$1 U4$1
________ ________ ________ ________
-15.000 1.000 1.000 1.000
POPULATION VALUES FOR LATENT CLASS REGRESSION MODEL PART
ALPHA(C)
C#1 C#2
________ ________
-1.000 0.000
TECHNICAL 8 OUTPUT
TECHNICAL 8 OUTPUT FOR REPLICATION 1
E STEP ITER LOGLIKELIHOOD ABS CHANGE REL CHANGE ALGORITHM
1 -0.22130073D+04 0.0000000 0.0000000 EM
2 -0.22089547D+04 4.0525914 0.0018313 EM
3 -0.22085912D+04 0.3635104 0.0001646 EM
4 -0.22084197D+04 0.1714460 0.0000776 EM
5 -0.22083218D+04 0.0979384 0.0000443 EM
6 -0.22082625D+04 0.0592391 0.0000268 EM
7 -0.22082259D+04 0.0366112 0.0000166 EM
8 -0.22082031D+04 0.0228309 0.0000103 EM
9 -0.22081888D+04 0.0142960 0.0000065 EM
10 -0.22081798D+04 0.0089701 0.0000041 EM
11 -0.22081742D+04 0.0056346 0.0000026 EM
12 -0.22081707D+04 0.0035419 0.0000016 EM
13 -0.22081684D+04 0.0022274 0.0000010 EM
14 -0.22081670D+04 0.0014013 0.0000006 EM
15 -0.22081661D+04 0.0008818 0.0000004 EM
16 -0.22081656D+04 0.0005550 0.0000003 EM
17 -0.22081652D+04 0.0003494 0.0000002 EM
18 -0.22081650D+04 0.0002200 0.0000001 EM
19 -0.22081649D+04 0.0001385 0.0000001 EM
20 -0.22081648D+04 0.0000872 0.0000000 EM
21 -0.22081647D+04 0.0000549 0.0000000 EM
22 -0.22081647D+04 0.0000346 0.0000000 EM
23 -0.22081647D+04 0.0000218 0.0000000 EM
24 -0.22081647D+04 0.0000137 0.0000000 EM
25 -0.22081647D+04 0.0000086 0.0000000 EM
26 -0.22081647D+04 0.0000054 0.0000000 EM
27 -0.22081646D+04 0.0000034 0.0000000 EM
28 -0.22081646D+04 0.0000022 0.0000000 EM
29 -0.22081646D+04 0.0000014 0.0000000 EM
30 -0.22081646D+04 0.0000009 0.0000000 EM
31 -0.22081646D+04 0.0000005 0.0000000 EM
32 -0.22081646D+04 0.0000003 0.0000000 EM
33 -0.22081646D+04 0.0000002 0.0000000 EM
34 -0.22081646D+04 0.0000001 0.0000000 EM
35 -0.22081646D+04 0.0000001 0.0000000 EM
36 -0.22081646D+04 0.0000001 0.0000000 FS
37 -0.22081646D+04 0.0000000 0.0000000 FS
TECHNICAL 9 OUTPUT
Error messages for each replication (if any)
SAVEDATA INFORMATION
Order of variables
U1
U2
U3
U4
C
Save file
ex7.13.dat
Save file format Free
Save file record length 10000
Beginning Time: 22:24:28
Ending Time: 22:24:28
Elapsed Time: 00:00:00
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