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Table 2 on page 60 using data set page59_a.dat.
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL
Latent
Classes
1 155.68287 0.72075
2 60.31713 0.27925
RESULTS IN PROBABILITY SCALE
Latent Class 1
A
Category 1 0.286 0.042 6.841
Category 2 0.714 0.042 17.045
B
Category 1 0.646 0.049 13.175
Category 2 0.354 0.049 7.220
C
Category 1 0.670 0.051 13.140
Category 2 0.330 0.051 6.461
D
Category 1 0.868 0.039 22.325
Category 2 0.132 0.039 3.406
Latent Class 2
A
Category 1 0.007 0.025 0.269
Category 2 0.993 0.025 39.267
B
Category 1 0.073 0.068 1.088
Category 2 0.927 0.068 13.716
C
Category 1 0.060 0.067 0.896
Category 2 0.940 0.067 13.985
D
Category 1 0.231 0.098 2.351
Category 2 0.769 0.098 7.833
Table 3 on page 62. The output in the book is produced by LEM and in LEM the default coding scheme is effect coding. On the other hand, the only scheme possible in Mplus is dummy coding. The results obtained from two types of coding are equivalent to each other. Here we show how to convert the results using dummy coding to the results using effect coding.
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
MODEL RESULTS
Estimates S.E. Est./S.E.
Latent Class 1
Thresholds
A$1 -0.913 0.205 -4.457
B$1 0.601 0.214 2.805
C$1 0.710 0.231 3.075
D$1 1.880 0.338 5.556
Latent Class 2
Thresholds
A$1 -4.983 3.741 -1.332
B$1 -2.535 0.992 -2.554
C$1 -2.747 1.187 -2.314
D$1 -1.203 0.553 -2.176
Categorical Latent Variables
Means
X#1 0.948 0.300 3.162
The parameter for X in the book is .948/2 = .474. The rest can be converted as follows. The relationship between the parameters in the book for single variable (S) and two variable (T) with the parameters from Mplus, thresholds for latent class 1 (L1) and thresholds for latent class 2 (L2) is
S + T = L1/2
S - T = L2/2
We did the calculation in Stata:
. list, clean
s t
1. -1.472 1.016
2. -.483 .784
3. -.509 .864
4. .169 .771
. gen l1 = (s+t)*2
. gen l2 = (s-t)*2
. list, clean
s t l1 l2
1. -1.472 1.016 -.9119999 -4.976
2. -.483 .784 .6019999 -2.534
3. -.509 .864 .71 -2.746
4. .169 .771 1.88 -1.204
Table 4 on page 69 using Ego's Dilemma Data, page59_a.dat. Notice that the AIC and BIC from Mplus output are computed using different formulae than those computed in the book. Even though they are different, but the difference of AIC's between two models are the same regardless which way they are computed. For example, the difference of AIC's of the two models in Table 4 is 59.08-(-9.28) = 68.36 based on the output from the book. It is 1095.300 -1026.935 = 68.365 based on the Mplus output. That is they are the same in terms of difference of models and that is how AIC's are used.
More precisely, the formulae for AIC and BIC from the book are
AIC = G2 - 2*df
BIC = G2- df*[ln(N)],
where df is the number of degrees of freedom and N is the sample size.
The formulae for AIC and BIC from Mplus are
AIC = -2*logL + 2*r
BIC = -2*logL + r*[ln(N)],
where r is the number of free model parameters and N is the sample size.
Model I: Independence
Data:
File is c:\alca\page59_a.dat;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(1);
Analysis:
Type = mixture ;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -543.650
Information Criteria
Number of Free Parameters 4
Akaike (AIC) 1095.300
Bayesian (BIC) 1108.801
Sample-Size Adjusted BIC 1096.125
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 104.107
Degrees of Freedom 11
P-Value 0.0000
Likelihood Ratio Chi-Square
Value 81.084
Degrees of Freedom 11
P-Value 0.0000
Model II: Two-Class LCM
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -504.468
Information Criteria
Number of Free Parameters 9
Akaike (AIC) 1026.935
Bayesian (BIC) 1057.313
Sample-Size Adjusted BIC 1028.793
(n* = (n + 2) / 24)
Entropy 0.719
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 2.720
Degrees of Freedom 6
P-Value 0.8431
Likelihood Ratio Chi-Square
Value 2.720
Degrees of Freedom 6
P-Value 0.8431
Table 5 on page 71 using Ego's Dilemma Data, page59_a.dat.
Model H1: two-class LCM
This is the model above.
Model H2: H1 + B & C parallel indicators
Data:
File is c:\alca\page59_a.dat;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
model:
%overall%
[b$1 c$1] (1);
%x#1%
[b$1 c$1] (2);
TESTS OF MODEL FIT
Loglikelihood
H0 Value -504.551
Information Criteria
Number of Free Parameters 7
Akaike (AIC) 1023.101
Bayesian (BIC) 1046.728
Sample-Size Adjusted BIC 1024.546
(n* = (n + 2) / 24)
Entropy 0.720
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 2.838
Degrees of Freedom 8
P-Value 0.9441
Likelihood Ratio Chi-Square
Value 2.886
Degrees of Freedom 8
P-Value 0.9413
Model H3: H2 + D equal error rate
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
model:
%overall%
[b$1 c$1] (1);
[d$1] (p1);
%x#1%
[b$1 c$1] (2);
[d$1] (q1);
model constraint:
p1 = -q1;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -504.933
Information Criteria
Number of Free Parameters 6
Akaike (AIC) 1021.866
Bayesian (BIC) 1042.117
Sample-Size Adjusted BIC 1023.104
(n* = (n + 2) / 24)
Entropy 0.759
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 3.603
Degrees of Freedom 9
P-Value 0.9356
Likelihood Ratio Chi-Square
Value 3.650
Degrees of Freedom 9
P-Value 0.9329
Model H4: H3 + A as perfect indicator for class 2
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
model:
%overall%
[b$1 c$1] (1);
[d$1] (p1);
[a$1@-15];
%x#1%
[b$1 c$1] (2);
[d$1] (q1);
[a$1];
model constraint:
p1 = -q1;TESTS OF MODEL FIT
Loglikelihood
H0 Value -504.937
Information Criteria
Number of Free Parameters 5
Akaike (AIC) 1019.874
Bayesian (BIC) 1036.750
Sample-Size Adjusted BIC 1020.906
(n* = (n + 2) / 24)
Entropy 0.763
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 3.605
Degrees of Freedom 10
P-Value 0.9634
Likelihood Ratio Chi-Square
Value 3.659
Degrees of Freedom 10
P-Value 0.9614
Table 6 on page 72 based on Model H4.
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
model:
%overall%
[b$1 c$1] (1);
[d$1] (p1);
[a$1@-15];
%x#1%
[b$1 c$1] (2);
[d$1] (q1);
[a$1];
model constraint:
p1 = -q1;
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL
Latent
Classes
1 163.60581 0.75743
2 52.39419 0.24257
RESULTS IN PROBABILITY SCALE
Latent Class 1
A
Category 1 0.275 0.037 7.506
Category 2 0.725 0.037 19.783
B
Category 1 0.636 0.031 20.591
Category 2 0.364 0.031 11.777
C
Category 1 0.636 0.031 20.591
Category 2 0.364 0.031 11.777
D
Category 1 0.852 0.033 25.909
Category 2 0.148 0.033 4.489
Latent Class 2
A
Category 1 0.000 0.000 0.000
Category 2 1.000 0.000 0.000
B
Category 1 0.046 0.046 1.012
Category 2 0.954 0.046 20.894
C
Category 1 0.046 0.046 1.012
Category 2 0.954 0.046 20.894
D
Category 1 0.148 0.033 4.489
Category 2 0.852 0.033 25.909
Table 8 on page 75 using abortion approval data, page75.dat.
Model H1: two-class LCM
Data:
File is c:\alca\page75.dat ;
Variable:
Names are
a b c d freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(2);
Analysis:
Type = mixture ;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -2773.793
Information Criteria
Number of Free Parameters 9
Akaike (AIC) 5565.586
Bayesian (BIC) 5614.323
Sample-Size Adjusted BIC 5585.731
(n* = (n + 2) / 24)
Entropy 0.925
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 214.746
Degrees of Freedom 6
P-Value 0.0000
Likelihood Ratio Chi-Square
Value 179.853
Degrees of Freedom 6
P-Value 0.0000
Model H2: three-class model with linear restrictions.
Data:
File is c:\alca\page75.dat;
Variable:
Names are
a b c d freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(3);
Analysis:
Type = mixture ;
model:
%overall%
[a$1*-1] (a11);
[b$1*-1] (b11);
[c$1*-1] (c11);
[d$1*-1] (d11);
%x#2%
[a$1*0] (a12);
[b$1*0] (b12);
[c$1*0] (c12);
[d$1*0] (d12);
%x#3%
[a$1*1] (a13);
[b$1*1] (b13);
[c$1*1] (c13);
[d$1*1] (d13);
model constraint:
a13 = 2*a12 - a11;
b13 = 2*b12 - b11;
c13 = 2*c12 - c11;
d13 = 2*d12 - d11;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -2685.032
Information Criteria
Number of Free Parameters 10
Akaike (AIC) 5390.063
Bayesian (BIC) 5444.215
Sample-Size Adjusted BIC 5412.447
(n* = (n + 2) / 24)
Entropy 0.824
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 2.339
Degrees of Freedom 5
P-Value 0.8005
Likelihood Ratio Chi-Square
Value 2.331
Degrees of Freedom 5
P-Value 0.8017
Model H3: H2 + A, B, C restricted to equal association
Data:
File is c:\alca\page75.dat ;
Variable:
Names are
a b c d freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(3);
Analysis:
Type = mixture ;
starts = 50 5;
miteration = 10000;
mciterations = 10;
iterations =10000;
model:
%overall% !for x#1
[a$1] (a11);
[b$1] (b11);
[c$1] (c11);
[d$1] (d11);
%x#2%
[a$1] (a12);
[b$1] (b12);
[c$1] (c12);
[d$1] (d12);
%x#3%
[a$1] (a13);
[b$1] (b13);
[c$1] (c13);
[d$1] (d13);
model constraint:
a13 = 2*a12 - a11;
b12 = b11 + a12 - a11;
b13 = b11 + 2*(a12-a11);
c12 = c11 + a12 - a11;
c13 = c11 + 2*(a12 - a11);
d13 = 2*d12 - d11;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -2685.653
Information Criteria
Number of Free Parameters 8
Akaike (AIC) 5387.305
Bayesian (BIC) 5430.627
Sample-Size Adjusted BIC 5405.212
(n* = (n + 2) / 24)
Entropy 0.824
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 3.553
Degrees of Freedom 7
P-Value 0.8296
Likelihood Ratio Chi-Square
Value 3.573
Degrees of Freedom 7
P-Value 0.8275
Table 9 on page 77 using model H3 from the example above. As discussed for the output of Table 3, the output produced by Mplus 3 will be different from the book because of the difference in coding scheme.
Data:
File is c:\alca\page75.dat ;
Variable:
Names are
a b c d freq;
Missing are all (-9999) ;
usev are a b c d freq;
weight is freq (freq);
categorical are a b c d;
classes = x(3);
Analysis:
Type = mixture ;
starts = 50 5;
miteration = 10000;
mciterations = 10;
iterations =10000;
model:
%overall% !for x#1
[a$1] (a11);
[b$1] (b11);
[c$1] (c11);
[d$1] (d11);
%x#2%
[a$1] (a12);
[b$1] (b12);
[c$1] (c12);
[d$1] (d12);
%x#3%
[a$1] (a13);
[b$1] (b13);
[c$1] (c13);
[d$1] (d13);
model constraint:
a13 = 2*a12 - a11;
b12 = b11 + a12 - a11;
b13 = b11 + 2*(a12-a11);
c12 = c11 + a12 - a11;
c13 = c11 + 2*(a12 - a11);
d13 = 2*d12 - d11;
MODEL RESULTS
Estimates S.E. Est./S.E.
Latent Class 1
Thresholds
A$1 -4.301 0.288 -14.942
B$1 -3.847 0.277 -13.870
C$1 -5.335 0.309 -17.266
D$1 -10.045 0.383 -26.215
Latent Class 2
Thresholds
A$1 -0.189 0.154 -1.224
B$1 0.265 0.153 1.727
C$1 -1.223 0.169 -7.244
D$1 -0.144 0.192 -0.754
Latent Class 3
Thresholds
A$1 3.922 0.219 17.934
B$1 4.376 0.230 18.989
C$1 2.888 0.211 13.684
D$1 9.756 0.010 988.995
Categorical Latent Variables
Means
X#1 0.162 0.068 2.365
X#2 -0.581 0.092 -6.346
Table 10 on page 79.
Model H1:
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d group freq;
weight is freq (freq);
categorical are a b c d group;
classes = g(2) x(2);
Analysis:
Type = mixture ;
model:
%overall%
x#1 on g#1;
[x#1];
model g:
%g#1%
[group$1@-15];
%g#2%
[group$1@15];
THE MODEL ESTIMATION TERMINATED NORMALLY
TESTS OF MODEL FIT
Loglikelihood
H0 Value -1324.989
Information Criteria
Number of Free Parameters 19
Akaike (AIC) 2687.978
Bayesian (BIC) 2765.278
Sample-Size Adjusted BIC 2704.982
(n* = (n + 2) / 24)
Entropy 0.863
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 9.063
Degrees of Freedom 12
P-Value 0.6976
Likelihood Ratio Chi-Square
Value 8.253
Degrees of Freedom 12
P-Value 0.7650
Model H2:
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d group freq;
weight is freq (freq);
categorical are a b c d group;
classes = g(2) x(2);
Analysis:
Type = mixture ;
model:
%overall%
x#1 on g#1;
[x#1];
model g:
%g#1%
[group$1@-15];
%g#2%
[group$1@15];
model x:
%x#1%
[a$1 b$1 c$1 d$1];
%x#2%
[a$1 b$1 c$1 d$1];
TESTS OF MODEL FIT
Loglikelihood
H0 Value -1332.597
Information Criteria
Number of Free Parameters 11
Akaike (AIC) 2687.194
Bayesian (BIC) 2731.946
Sample-Size Adjusted BIC 2697.039
(n* = (n + 2) / 24)
Entropy 0.853
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 24.774
Degrees of Freedom 20
P-Value 0.2102
Likelihood Ratio Chi-Square
Value 23.469
Degrees of Freedom 20
P-Value 0.2663
Model H3:
Data:
File is c:\alca\59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d group freq;
weight is freq (freq);
categorical are a b c d group;
classes = g(2) x(2);
Analysis:
Type = mixture ;
model g:
%g#1%
[group$1@-15];
%g#2%
[group$1@15];
model x:
%x#1%
[a$1 b$1 c$1 d$1];
%x#2%
[a$1 b$1 c$1 d$1];
TESTS OF MODEL FIT
Loglikelihood
H0 Value -1332.603
Information Criteria
Number of Free Parameters 10
Akaike (AIC) 2685.205
Bayesian (BIC) 2725.889
Sample-Size Adjusted BIC 2694.155
(n* = (n + 2) / 24)
Entropy 0.853
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 24.815
Degrees of Freedom 21
P-Value 0.2553
Likelihood Ratio Chi-Square
Value 23.481
Degrees of Freedom 21
P-Value 0.3189
Table 11 on page 80 using Model H3 from previous example.
Data:
File is c:\alca\page59_a.dat ;
Variable:
Names are
a b c d group freq;
Missing are all (-9999) ;
usev are a b c d group freq;
weight is freq (freq);
categorical are a b c d group;
classes = g(2) x(2);
Analysis:
Type = mixture ;
model g:
%g#1%
[group$1@-15];
%g#2%
[group$1@15];
model x:
%x#1%
[a$1 b$1 c$1 d$1];
%x#2%
[a$1 b$1 c$1 d$1];
LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL
G Classes (Rows) by X Classes (Columns)
1 2
1 0.292 0.708
2 0.292 0.708
RESULTS IN PROBABILITY SCALE
Latent Class Pattern 1 1
A
Category 1 0.010 0.023 0.454
Category 2 0.990 0.023 42.848
B
Category 1 0.108 0.057 1.895
Category 2 0.892 0.057 15.711
C
Category 1 0.021 0.058 0.355
Category 2 0.979 0.058 16.940
D
Category 1 0.319 0.072 4.434
Category 2 0.681 0.072 9.474
GROUP
Category 1 0.000 0.000 0.000
Category 2 1.000 0.000 0.000
Latent Class Pattern 1 2
A
Category 1 0.345 0.034 10.072
Category 2 0.655 0.034 19.096
B
Category 1 0.567 0.034 16.808
Category 2 0.433 0.034 12.850
C
Category 1 0.717 0.043 16.491
Category 2 0.283 0.043 6.511
D
Category 1 0.849 0.029 28.966
Category 2 0.151 0.029 5.150
GROUP
Category 1 0.000 0.000 0.000
Category 2 1.000 0.000 0.000
Latent Class Pattern 2 1
A
Category 1 0.010 0.023 0.454
Category 2 0.990 0.023 42.848
B
Category 1 0.108 0.057 1.895
Category 2 0.892 0.057 15.711
C
Category 1 0.021 0.058 0.355
Category 2 0.979 0.058 16.940
D
Category 1 0.319 0.072 4.434
Category 2 0.681 0.072 9.474
GROUP
Category 1 1.000 0.000 0.000
Category 2 0.000 0.000 0.000
Latent Class Pattern 2 2
A
Category 1 0.345 0.034 10.072
Category 2 0.655 0.034 19.096
B
Category 1 0.567 0.034 16.808
Category 2 0.433 0.034 12.850
C
Category 1 0.717 0.043 16.491
Category 2 0.283 0.043 6.511
D
Category 1 0.849 0.029 28.966
Category 2 0.151 0.029 5.150
GROUP
Category 1 1.000 0.000 0.000
Category 2 0.000 0.000 0.000
Table 12 on page 81 using model H3 from previous example. Notice that because of the difference in terms of coding schemes, the results from Mplus 3 are different from the results in the book. But they are equivalent and can be converted from each other.
MODEL RESULTS
Estimates S.E. Est./S.E.
Latent Class Pattern 1 1
Thresholds
A$1 -4.548 2.227 -2.042
B$1 -2.115 0.591 -3.576
C$1 -3.865 2.875 -1.344
D$1 -0.759 0.331 -2.293
GROUP$1 -15.000 0.000 0.000
Latent Class Pattern 1 2
Thresholds
A$1 -0.640 0.152 -4.218
B$1 0.269 0.137 1.955
C$1 0.929 0.214 4.338
D$1 1.727 0.229 7.552
GROUP$1 -15.000 0.000 0.000
Latent Class Pattern 2 1
Thresholds
A$1 -4.548 2.227 -2.042
B$1 -2.115 0.591 -3.576
C$1 -3.865 2.875 -1.344
D$1 -0.759 0.331 -2.293
GROUP$1 15.000 0.000 0.000
Latent Class Pattern 2 2
Thresholds
A$1 -0.640 0.152 -4.218
B$1 0.269 0.137 1.955
C$1 0.929 0.214 4.338
D$1 1.727 0.229 7.552
GROUP$1 15.000 0.000 0.000
Categorical Latent Variables
Means
G#1 0.000 0.096 0.000
X#1 -0.888 0.245 -3.629
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