P1253814517DamFl

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Meeting 8 Multiple Group Latent Class Analysis
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Multiple Group Latent Class Model
Model for Group h
Class 1, Group h
Class T, Group h
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Model of Complete Homogeneity
Latent class proportions are equal across groups
Conditional probabilities are equal across groups
Also, see Equation 6.2 in LCSA
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Model of Complete Heterogeneity
Latent class proportions can be different across
groups
Conditional probabilities can be different across
groups
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Model of Partial Homogeneity
Latent class proportions can be different equal
across groups
Conditional probabilities are equal across groups
Note Equality of conditional probabilities means
that the interpretation of the latent classes is
the same across groups although the sizes of
the latent classes are different across groups.
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INPUT Cheat 4 data by M/F 2
class model Heterogeneous model lat 1 man 5
dim 2 2 2 2 2 2 lab X S A B C D mod S XS
AXS BXS CXS DXS dat
99 18 1 2 1 1 0 1 5 1 1 1 3 2 1 0 107 28 6
3 11 3 1 1 5 2 0 1 8 2 0 2 nR2
STATISTICS Number of iterations 329
Converge criterion 0.0000009935 Seed random
values 1668 X-squared 12.8281
(0.3817) L-squared 15.0583
(0.2383) Cressie-Read 12.9989
(0.3691) Dissimilarity index 0.0328
Degrees of freedom 12 Log-likelihood
-646.18676 Number of parameters 19 (1)
Sample size 317.0 BIC(L-squared)
-54.0485 AIC(L-squared) -8.9417
BIC(log-likelihood) 1401.7927
AIC(log-likelihood) 1330.3735 Eigenvalues
information matrix 380.1473 311.1798
214.5029 153.2716 124.7565 113.9862
110.8429 73.9976 69.5910 64.3118
59.2551 51.1920 44.0038 32.7376
8.1185 7.1598 2.1630 1.4117
0.0000
Males Females Heterogeneous Model
Note that the latent classes, X, are conditional
on S (sex) and that the conditional
probabilities for A, B, C, D are
conditional on both X and S. Thus, this
is a heterogeneous model.
Note the 0 eigenvalue model is not
identified.
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(CONDITIONAL) PROBABILITIES P(S) 1
0.4322 (0.0278) 2
0.5678 (0.0278) P(XS) 1 1
0.1454 (0.0593) 1 2 0.8534
(0.1022) 2 1 0.8546 (0.0593) 2
2 0.1466 (0.1022) P(AXS) 1 1
1 0.4296 (0.1701) 2 1 1 0.5704
(0.1701) 1 1 2 0.9802 (0.0421) 2
1 2 0.0198 (0.0421) 1 2 1
0.9775 (0.0281) 2 2 1 0.0225
(0.0281) 1 2 2 0.3570 (0.2974) 2
2 2 0.6430 (0.2974) P(BXS) 1 1
1 0.5481 (0.1942) 2 1 1 0.4519
(0.1942) 1 1 2 0.9365 (0.0514) 2
1 2 0.0635 (0.0514) 1 2 1
1.0000 (0.0000) 2 2 1 0.0000
(0.0000) 1 2 2 0.3085 (0.2614) 2
2 2 0.6915 (0.2614) P(CXS) 1 1
1 0.6816 (0.1341) 2 1 1 0.3184
(0.1341) 1 1 2 0.9411 (0.0214) 2
1 2 0.0589 (0.0214) 1 2 1
0.9944 (0.0184) 2 2 1 0.0056
(0.0184) 1 2 2 0.8126 (0.1053) 2
2 2 0.1874 (0.1053) P(DXS) 1 1
1 0.5456 (0.1425) 2 1 1 0.4544
(0.1425) 1 1 2 0.7909 (0.0355) 2
1 2 0.2091 (0.0355) 1 2 1
0.8552 (0.0364) 2 2 1 0.1448
(0.0364) 1 2 2 0.6255 (0.1312) 2
2 2 0.3745 (0.1312)
Group proportions 139/317 180/317
M F
M F M F
LC Proportions per group
M M F F M M F F
NOTE response 1 No and response 2 Yes
in LCSA, 0 No and 1 Yes LEM requires
1 ,2, etc. responses for variables
Conditional Probs per group the 0 CP results
from identification problems.
Compare to separate group runs in next set of
slides. Note that classes 1 and 2 are
reversed for M and F
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Males
Cheat 4 data by Males only 2 class model
heterogeneous model lat 1 man 4 dim 2 2 2 2
2 lab X A B C D mod X AX BX
CX DX dat 99 18 1 2 1 1 0 1 5 1
1 1 3 2 1 0 nR2 STATISTICS
Number of iterations 139 Converge criterion
0.0000009927 Seed random values 5679
X-squared 5.5236 (0.4786)
L-squared 6.3978 (0.3801)
Cressie-Read 5.5771 (0.4722)
Dissimilarity index 0.0284 Degrees of
freedom 6 Log-likelihood
-156.17714 Number of parameters 9 (1)
Sample size 137.0 BIC(L-squared)
-23.1220 AIC(L-squared) -5.6022
BIC(log-likelihood) 356.6341
AIC(log-likelihood) 330.3543 Eigenvalues
information matrix 107.1777 57.0205
37.0367 32.2004 29.6518 16.3625
4.0309 0.6916 -0.0000 WARNING 1 (nearly)
boundary or non-identified (log-linear) parameters
LATENT CLASS OUTPUT X 1 X
2 0.8545 0.1455 A 1 0.9775
0.4299 A 2 0.0225 0.5701 B 1 1.0000
0.5486 B 2 0.0000 0.4514 C 1 0.9944
0.6817 C 2 0.0056 0.3183 D 1 0.8553
0.5457 D 2 0.1447 0.4543
NOTE These results differ somewhat from those
presented in Table 6.2 of LCSA. The LEM
L-squared value is smaller and the
estimates vary quite a bit. This is
partly due to the better
computational accuracy of the more modern
software but is also affected that the lack
of identification of the model for
this group (see eigenvalue of 0).
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INPUT Cheat 4 data by Female only
2 class model heterogeneous model lat 1 man
4 dim 2 2 2 2 2 lab X A B C D mod X
AX BX CX DX dat 107
28 6 3 11 3 1 1 5 2 0 1 8 2 0 2 nR2
STATISTICS Number of iterations 365
Converge criterion 0.0000009810 Seed random
values 2425 X-squared 7.3026
(0.2938) L-squared 8.6605
(0.1936) Cressie-Read 7.4208
(0.2837) Dissimilarity index 0.0362
Degrees of freedom 6 Log-likelihood
-273.20739 Number of parameters 9 (1)
Sample size 180.0 BIC(L-squared)
-22.4972 AIC(L-squared) -3.3395
BIC(log-likelihood) 593.1514
AIC(log-likelihood) 564.4148 Eigenvalues
information matrix 190.0721 76.6448
62.3757 55.4232 34.7954 25.5990
22.0059 3.5875 1.0827
Females
LATENT CLASS OUTPUT X 1 X
2 0.1466 0.8534 A 1 0.3570
0.9802 A 2 0.6430 0.0198 B 1 0.3085
0.9365 B 2 0.6915 0.0635 C 1 0.8126
0.9411 C 2 0.1874 0.0589 D 1 0.6255
0.7909 D 2 0.3745 0.2091
NOTE These results differ VERY LITTLE from
those presented in Table 6.2 of LCSA.
Unlike the MALE group, all
eigenvalues are positive.
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MALES FEMALES P(AXS)
1 1 1 0.4296 (0.1701) 2 1 1
0.5704 (0.1701) 1 1 2 0.9802
(0.0421) 2 1 2 0.0198 (0.0421) 1
2 1 0.9775 (0.0281) 2 2 1
0.0225 (0.0281) 1 2 2 0.3570
(0.2974) 2 2 2 0.6430 (0.2974)
MALES LATENT CLASS OUTPUT
X 1 X 2 0.8545 0.1455 A 1
0.9775 0.4299 A 2 0.0225 0.5701
CP SE
M M F F
M M F F
FEMALES LATENT CLASS OUTPUT
X 1 X 2 0.1466 0.8534 A
1 0.3570 0.9802 A 2 0.6430 0.0198
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MALES FEMALES
MALES LATENT CLASS OUTPUT
X 1 X 2 0.8545 0.1455
CP SE
P(XS) 1 1 0.1454 (0.0593) 1
2 0.8534 (0.1022) 2 1
0.8546 (0.0593) 2 2 0.1466 (0.1022)
M F M F
FEMALES LATENT CLASS OUTPUT
X 1 X 2 0.1466
0.8534
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INPUT Cheat 4 data by M/F 2
class model Homogeneous model lat 1 man 5
dim 2 2 2 2 2 2 lab X S A B C D mod S X
AX BX CX DX dat 99 18 1
2 1 1 0 1 5 1 1 1 3 2 1 0 107 28 6 3 11 3 1
1 5 2 0 1 8 2 0 2 nR2 STATISTICS
Number of iterations 343 Converge criterion
0.0000009591 Seed random values 2354
X-squared 24.8712 (0.2528)
L-squared 28.8872 (0.1167)
Cressie-Read 25.4289 (0.2291)
Dissimilarity index 0.0856 Degrees of
freedom 21 Log-likelihood
-653.10120 Number of parameters 10 (1)
Sample size 317.0 BIC(L-squared)
-92.0497 AIC(L-squared) -13.1128
BIC(log-likelihood) 1363.7914
AIC(log-likelihood) 1326.2024 Eigenvalues
information matrix 311.1798 291.7924
140.4972 85.7408 74.3713 69.9178
46.5648 28.1696 5.1108 1.5924
Males Females Homogeneous Model
Note that neither the LC proportions nor
conditional probabilities for A, B, C, D
are conditional on S. Thus, this is a
homogeneous model.
This model is identified all eigenvalues are
positive.
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LATENT CLASS OUTPUT X 1 X
2 0.1647 0.8353 S 1 0.4322
0.4322 S 2 0.5678 0.5678 A 1 0.4314
0.9837 A 2 0.5686 0.0163 B 1 0.4128
0.9761 B 2 0.5872 0.0239 C 1 0.7858
0.9629 C 2 0.2142 0.0371 D 1 0.6236
0.8174 D 2 0.3764 0.1826
Compare to Table 3.3 in LCSA. The LC proportions
are within rounding error the conditional
probs vary somewhat this is due to the use
of the more modern LEM program.
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Comparison of Chi-Square Fit Statistics and DF
STATISTICS Number of iterations 343
Converge criterion 0.0000009591 Seed
random values 2354 X-squared
24.8712 (0.2528) L-squared 28.8872
(0.1167) Cressie-Read 25.4289
(0.2291) Dissimilarity index 0.0856
Degrees of freedom 21 Log-likelihood
-653.10120 Number of parameters 10 (1)
Sample size 317.0
In LCSA, L-squared (G2) is reported to be 7.764
with DF 6. That model was fitted to the 24
16 cells and ignores the M/F groups. There
are 9 independent parameters estimated so
the DF are 16 9 1 6 (see page 31 and
Table 3.3). This analysis is based on 2X24
32 cells that includes the M/F grouping.
There are still 9 independent parameters
estimated but the sizes of the M/F groups
are fixed so there is one additional
restraint on the estimated cell frequencies.
Thus, the DF are 32 9 2 21.
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Comparison of Fit Statistics
Homogeneous Model for M/F
Model Without M/F Groups
FREQUENCIES S A B C D observed
estimated std. res. 1 1 1 1 1 99.000
88.455 1.121 1 1 1 1 2 18.000
20.507 -0.554 1 1 1 2 1 1.000
3.865 -1.457 1 1 1 2 2 2.000
1.068 0.902 1 1 2 1 1 1.000
4.922 -1.768 1 1 2 1 2 1.000
2.164 -0.791 1 1 2 2 1 0.000
0.845 -0.919 1 1 2 2 2 1.000
0.479 0.752 1 2 1 1 1 5.000
4.026 0.485 1 2 1 1 2 1.000
1.886 -0.645 1 2 1 2 1 1.000
0.763 0.272 1 2 1 2 2 1.000
0.439 0.846 1 2 2 1 1 3.000
3.727 -0.377 1 2 2 1 2 2.000
2.236 -0.158 1 2 2 2 1 1.000
1.008 -0.008 1 2 2 2 2 0.000
0.608 -0.780 2 1 1 1 1 107.000
116.218 -0.855 2 1 1 1 2 28.000
26.944 0.203 2 1 1 2 1 6.000
5.079 0.409 2 1 1 2 2 3.000
1.403 1.349 2 1 2 1 1 11.000
6.467 1.783 2 1 2 1 2 3.000
2.844 0.093 2 1 2 2 1 1.000
1.111 -0.105 2 1 2 2 2 1.000
0.630 0.467 2 2 1 1 1 5.000
5.290 -0.126 2 2 1 1 2 2.000
2.478 -0.304 2 2 1 2 1 0.000
1.002 -1.001 2 2 1 2 2 1.000
0.577 0.556 2 2 2 1 1 8.000
4.897 1.402 2 2 2 1 2 2.000
2.938 -0.547 2 2 2 2 1 0.000
1.324 -1.151 2 2 2 2 2 2.000
0.799 1.344
FREQUENCIES A B C D observed
estimated std. res. 1 1 1 1 207.000
205.718 0.089 1 1 1 2 46.000 47.412
-0.205 1 1 2 1 7.000 8.955
-0.653 1 1 2 2 5.000 2.450
1.629 1 2 1 1 13.000 12.300 0.200
1 2 1 2 4.000 5.118 -0.494 1 2
2 1 1.000 1.956 -0.684 1 2 2 2
2.000 1.091 0.871 2 1 1 1
10.000 9.334 0.218 2 1 1 2
3.000 4.343 -0.644 2 1 2 1 1.000
1.770 -0.579 2 1 2 2 2.000
1.017 0.974 2 2 1 1 11.000 8.619
0.811 2 2 1 2 4.000 5.156
-0.509 2 2 2 1 1.000 2.347
-0.879 2 2 2 2 2.000 1.413 0.494
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INPUT Cheat 4 data by M/F 2
class model Partially Heterogeneous model lat
1 man 5 dim 2 2 2 2 2 2 lab X S A B C D
mod S XS AX BX CX
DX dat 99 18 1 2 1 1 0 1 5 1 1 1 3 2 1 0
107 28 6 3 11 3 1 1 5 2 0 1 8 2 0 2 nR2
STATISTICS Number of iterations 143
Converge criterion 0.0000009059 Seed random
values 1575 X-squared 21.9414
(0.3437) L-squared 24.4410
(0.2237) Cressie-Read 22.0180
(0.3395) Dissimilarity index 0.0681
Degrees of freedom 20 Log-likelihood
-650.87808 Number of parameters 11 (1)
Sample size 317.0 BIC(L-squared)
-90.7371 AIC(L-squared) -15.5590
BIC(log-likelihood) 1365.1041
AIC(log-likelihood) 1323.7562 Eigenvalues
information matrix 311.1799 276.9004
207.2852 112.3006 90.5815 82.0939
75.2587 45.9881 23.2329 4.6714
-0.0000 WARNING 1 (nearly) boundary or
non-identified (log-linear) parameters
Males Females Partial Heterogeneous Model
Note that only the LC proportions are conditional
on S. Thus, this is a partial
heterogeneous model.
Note the 0 eigenvalue model is not
identified.
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(CONDITIONAL) PROBABILITIES P(S)
1 0.4322 (0.0278) 2
0.5678 (0.0278) P(XS) 1 1
0.1270 (0.0528) 1 2 0.2483
(0.0649) 2 1 0.8730 (0.0528) 2
2 0.7517 (0.0649) P(AX) 1 1
0.5375 (0.0877) 2 1 0.4625
(0.0877) 1 2 0.9793 (0.0222) 2
2 0.0207 (0.0222) P(BX) 1 1
0.4041 (0.1608) 2 1 0.5959
(0.1608) 1 2 1.0000 (0.0000) 2
2 0.0000 (0.0000) P(CX) 1
1 0.8017 (0.0612) 2 1
0.1983 (0.0612) 1 2 0.9659
(0.0170) 2 2 0.0341 (0.0170)
P(DX) 1 1 0.6447 (0.0725) 2
1 0.3553 (0.0725) 1 2
0.8198 (0.0274) 2 2 0.1802 (0.0274)
M F M F
Note that the LC proportions are different for
Males and Females.
However, the conditional probabilities do not
vary for Males and Female.
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Note that profiles of conditional probabilities
(CP) are very similar but not identical.
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Chi-Square Difference Tests
Note that Pearson (X) and Likelihood ratio
(L) chi-square statistics lead to different
decisions using a .05 significance level.
Pearson Chi-Square decision Homogeneous model
fits no worse that more complex models
therefore, interpret homogeneous
model. Likelihood-Ratio Chi-Square decision
Homogeneous models provides worse fit than
partial heterogeneous model therefore,
interpret partial heterogeneous model.
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Information Measures
AIC decision Partial heterogeneous model is best
approximating model. BIC decision Homogeneous
model is best approximating model.