#### Mplus Textbook ExamplesApplied Latent Class Analysis Chapter 11 Latent Markov Chains by Rolf Langeheine and Frank van de Pol

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:\alca\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:\alca\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
(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:\alca\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
(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
(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
(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

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