Mplus Textbook Examples
Applied Latent Class Analysis
Chapter 11 Latent Markov Chains by Rolf Langeheine and Frank van de Pol
The data set can be downloaded following the link here.
Expected frequencies of Table 1 on page 310 based on the simple Markov model with time homogeneous transition probabilities.
data:
file is chap11_dat.txt ;
variable:
names are
a b c d e count group;
missing are all (-9999) ;
usevariables are a b c d e count;
weight is count (freq);
categorical are a b c d e;
classes = ca(2) cb(2) cc(2) cd(2) ce(2);
analysis:
type = mixture;
model:
%overall%
cb#1 on ca#1 (1);
cc#1 on cb#1 (1);
cd#1 on cc#1 (1);
ce#1 on cd#1 (1);
[ca#1];
[cb#1 cc#1 cd#1 ce#1] (2);
model ca:
%ca#1%
[a$1@15];
%ca#2%
[a$1@-15]; !for variable a;
model cb:
%cb#1%
[b$1@15];
%cb#2%
[b$1@-15]; !for variable b;
model cc:
%cc#1%
[c$1@15];
%cc#2%
[c$1@-15]; !for variable c;
model cd:
%cd#1%
[d$1@15];
%cd#2%
[d$1@-15]; !for variable d;
model ce:
%ce#1%
[e$1@15];
%ce#2%
[e$1@-15]; !for variable e;
output: tech10;
RESPONSE PATTERN FREQUENCIES AND CHI-SQUARE CONTRIBUTIONS
Response Frequency Standard Chi-square Contribution
Pattern Observed Estimated Residual Pearson Loglikelihood Deleted
1 891.00 553.84 15.17 205.25 847.29
2 237.00 288.85 3.14 9.31 -93.79
3 120.00 92.85 2.84 7.94 61.56
4 136.00 293.64 9.47 84.63 -209.36
5 111.00 92.85 1.90 3.55 39.64
6 80.00 48.43 4.56 20.59 80.32
7 54.00 94.39 4.20 17.28 -60.31
8 99.00 298.50 11.90 133.34 -218.53
9 119.00 92.85 2.74 7.37 59.06
10 68.00 48.43 2.83 7.91 46.17
11 51.00 15.57 8.99 80.66 121.05
12 64.00 49.23 2.12 4.43 33.59
13 52.00 94.39 4.40 19.04 -62.00
14 51.00 49.23 0.25 0.06 3.61
15 49.00 95.95 4.84 22.97 -65.86
16 172.00 303.45 7.78 56.94 -195.30
17 176.00 231.40 3.73 13.26 -96.33
18 107.00 120.69 1.26 1.55 -25.76
19 64.00 38.79 4.06 16.38 64.08
20 95.00 122.69 2.53 6.25 -48.59
21 60.00 38.79 3.42 11.59 52.33
22 75.00 20.23 12.20 148.25 196.53
23 50.00 39.44 1.69 2.83 23.73
24 165.00 124.72 3.65 13.01 92.36
25 106.00 235.23 8.63 71.00 -168.99
26 107.00 122.69 1.43 2.01 -29.28
27 67.00 39.44 4.41 19.27 71.02
28 187.00 124.72 5.65 31.10 151.49
29 92.00 239.13 9.74 90.53 -175.76
30 200.00 124.72 6.82 45.44 188.90
31 176.00 243.09 4.41 18.52 -113.68
32 1066.00 768.79 11.62 114.90 696.84
Table 2 based on the simple Markov chain model with time heterogeneous transition probabilities.
data:
file is d:alcahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/chap11.dat ;
variable:
names are
a b c d e count group;
missing are all (-9999) ;
usevariables are a b c d e count;
weight is count (freq);
categorical are a b c d e;
classes = ca(2) cb(2) cc(2) cd(2) ce(2);
analysis:
type = mixture;
model:
%overall%
cb#1 on ca#1 ;
cc#1 on cb#1 ;
cd#1 on cc#1 ;
ce#1 on cd#1 ;
[ca#1 cb#1 cc#1 cd#1 ce#1] ;
model ca:
%ca#1%
[a$1@15];
%ca#2%
[a$1@-15]; !for variable a;
model cb:
%cb#1%
[b$1@15];
%cb#2%
[b$1@-15]; !for variable b;
model cc:
%cc#1%
[c$1@15];
%cc#2%
[c$1@-15]; !for variable c;
model cd:
%cd#1%
[d$1@15];
%cd#2%
[d$1@-15]; !for variable d;
model ce:
%ce#1%
[e$1@15];
%ce#2%
[e$1@-15]; !for variable e;
FINAL CLASS COUNTS AND PROPORTIONS FOR EACH LATENT CLASS VARIABLE
BASED ON THE ESTIMATED MODEL
Latent Class
Variable Class
CA 1 2237.99976 0.43482
2 2909.00024 0.56518
CB 1 2531.99976 0.49194
2 2615.00000 0.50806
CC 1 2594.99976 0.50418
2 2552.00000 0.49582
CD 1 2520.00000 0.48961
2 2627.00000 0.51039
CE 1 2353.99976 0.45735
2 2793.00000 0.54265
LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL
CA Classes (Rows) by CB Classes (Columns)
1 2
1 0.718 0.282
2 0.318 0.682
CB Classes (Rows) by CC Classes (Columns)
1 2
1 0.715 0.285
2 0.300 0.700
CC Classes (Rows) by CD Classes (Columns)
1 2
1 0.704 0.296
2 0.272 0.728
CD Classes (Rows) by CE Classes (Columns)
1 2
1 0.686 0.314
2 0.238 0.762
Table 3 on page 315.
Model 1a (M): Simple Markov model with time homogeneous transition probabilities
data:
file is d:alcahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/chap11.dat ;
variable:
names are
a b c d e count group;
missing are all (-9999) ;
usevariables are a b c d e count;
weight is count (freq);
categorical are a b c d e;
classes = ca(2) cb(2) cc(2) cd(2) ce(2);
analysis:
type = mixture;
model:
%overall%
cb#1 on ca#1 (1);
cc#1 on cb#1 (1);
cd#1 on cc#1 (1);
ce#1 on cd#1 (1);
[ca#1];
[cb#1 cc#1 cd#1 ce#1] (2);
model ca:
%ca#1%
[a$1@15];
%ca#2%
[a$1@-15]; !for variable a;
model cb:
%cb#1%
[b$1@15];
%cb#2%
[b$1@-15]; !for variable b;
model cc:
%cc#1%
[c$1@15];
%cc#2%
[c$1@-15]; !for variable c;
model cd:
%cd#1%
[d$1@15];
%cd#2%
[d$1@-15]; !for variable d;
model ce:
%ce#1%
[e$1@15];
%ce#2%
[e$1@-15]; !for variable e;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -15893.523
Information Criteria
Number of Free Parameters 3
Akaike (AIC) 31793.046
Bayesian (BIC) 31812.685
Sample-Size Adjusted BIC 31803.152
(n* = (n + 2) / 24)
Entropy 1.000
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 1287.146
Degrees of Freedom 28
P-Value 0.0000
Likelihood Ratio Chi-Square
Value 1266.023
Degrees of Freedom 28
P-Value 0.0000
Model 1b (M): Simple Markov model with time heterogeneous transition probabilities
data:
file is d:alcahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/chap11.dat ;
variable:
names are
a b c d e count group;
missing are all (-9999) ;
usevariables are a b c d e count;
weight is count (freq);
categorical are a b c d e;
classes = ca(2) cb(2) cc(2) cd(2) ce(2);
analysis:
type = mixture;
model:
%overall%
cb#1 on ca#1 ;
cc#1 on cb#1 ;
cd#1 on cc#1 ;
ce#1 on cd#1 ;
! [ca#1 cb#1 cc#1 cd#1 ce#1] ;
model ca:
%ca#1%
[a$1@15];
%ca#2%
[a$1@-15]; !for variable a;
model cb:
%cb#1%
[b$1@15];
%cb#2%
[b$1@-15]; !for variable b;
model cc:
%cc#1%
[c$1@15];
%cc#2%
[c$1@-15]; !for variable c;
model cd:
%cd#1%
[d$1@15];
%cd#2%
[d$1@-15]; !for variable d;
model ce:
%ce#1%
[e$1@15];
%ce#2%
[e$1@-15]; !for variable e;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -15865.184
Information Criteria
Number of Free Parameters 9
Akaike (AIC) 31748.367
Bayesian (BIC) 31807.283
Sample-Size Adjusted BIC 31778.684
(n* = (n + 2) / 24)
Entropy 1.000
Chi-Square Test of Model Fit for the Binary and Ordered Categorical
(Ordinal) Outcomes
Pearson Chi-Square
Value 1239.474
Degrees of Freedom 22
P-Value 0.0000
Likelihood Ratio Chi-Square
Value 1209.345
Degrees of Freedom 22
P-Value 0.0000
Model 3a (MS): MS Mover-Stayer with time homogenous transition probabilities.
Model 7a (LM): Latent Markov with time homogeneous transition probabilities.
data:
file is chap11_dat.txt ;
variable:
names are
a b c d e count group;
missing are all (-9999) ;
usevariables are a b c d e count;
weight is count (freq);
categorical are a b c d e;
classes = ca(2) cb(2) cc(2) cd(2) ce(2);
analysis:
type = mixture;
model:
%overall%
cb#1 on ca#1 (1);
cc#1 on cb#1 (1);
cd#1 on cc#1 (1);
ce#1 on cd#1 (1);
[cb#1 cc#1 cd#1 ce#1] (2);
model ca:
%ca#1%
[a$1] (3);
%ca#2%
[a$1] (4); !for variable a;
model cb:
%cb#1%
[b$1] (3);
%cb#2%
[b$1] (4); !for variable b;
model cc:
%cc#1%
[c$1] (3);
%cc#2%
[c$1] (4); !for variable c;
model cd:
%cd#1%
[d$1] (3);
%cd#2%
[d$1] (4); !for variable d;
model ce:
%ce#1%
[e$1] (3);
%ce#2%
[e$1] (4); !for variable e;
output: tech10;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -15378.476
Information Criteria
Number of Free Parameters 5
Akaike (AIC) 30766.952
Bayesian (BIC) 30799.683
Sample-Size Adjusted BIC 30783.795
(n* = (n + 2) / 24)
Entropy 0.777
Chi-Square Test of Model Fit for the Binary and Ordered Categorical (Ordinal) Outcomes
Pearson Chi-Square
Value 243.841
Degrees of Freedom 26
P-Value 0.0000
Likelihood Ratio Chi-Square
Value 235.928
Degrees of Freedom 26
P-Value 0.0000
Model 7b (LM): Latent Markov with time heterogeneous transition probabilities.
data:
file is chap11_dat.txt ;
variable:
names are
a b c d e count group;
missing are all (-9999) ;
usevariables are a b c d e count;
weight is count (freq);
categorical are a b c d e;
classes = ca(2) cb(2) cc(2) cd(2) ce(2);
analysis:
type = mixture;
model:
%overall%
cb#1 on ca#1 ;
cc#1 on cb#1 ;
cd#1 on cc#1 ;
ce#1 on cd#1 ;
[cb#1 cc#1 cd#1 ce#1] ;
model ca:
%ca#1%
[a$1] (3);
%ca#2%
[a$1] (4); !for variable a;
model cb:
%cb#1%
[b$1] (3);
%cb#2%
[b$1] (4); !for variable b;
model cc:
%cc#1%
[c$1] (3);
%cc#2%
[c$1] (4); !for variable c;
model cd:
%cd#1%
[d$1] (3);
%cd#2%
[d$1] (4); !for variable d;
model ce:
%ce#1%
[e$1] (3);
%ce#2%
[e$1] (4); !for variable e;
output: tech10;
TESTS OF MODEL FIT
Loglikelihood
H0 Value -15325.635
Information Criteria
Number of Free Parameters 11
Akaike (AIC) 30673.271
Bayesian (BIC) 30745.279
Sample-Size Adjusted BIC 30710.324
(n* = (n + 2) / 24)
Entropy 0.802
Chi-Square Test of Model Fit for the Binary and Ordered Categorical (Ordinal) Outcomes
Pearson Chi-Square
Value 131.025
Degrees of Freedom 20
P-Value 0.0000
Likelihood Ratio Chi-Square
Value 130.246
Degrees of Freedom 20
P-Value 0.0000