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LEM Textbook Examples
Applied Latent Class Analysis
Chapter 2 Basic Concepts and Procedures in Single- and Multiple-Group Latent Class Analysis by Allan L. McCutcheon


Table 2 and Table 3 using roll conflict data.

  lat 1
  man 4
  dim 2 2 2 2 2 
  mod X
      A|X 
      B|X 
      C|X
      D|X  
    
  dat [20 2
        6 1
        9 2
        4 1
       38 7
       25 6
       24 6
       23 42]


*** STATISTICS ***

  Number of iterations = 77
  Converge criterion   = 0.0000009342
  Seed random values   = 2131

  X-squared            = 2.7200 (0.8431)
  L-squared            = 2.7199 (0.8431)
  Cressie-Read         = 2.7174 (0.8434)
  Dissimilarity index  = 0.0386
  Degrees of freedom   = 6
  Log-likelihood       = -504.46767
  Number of parameters = 9 (+1)
  Sample size          = 216.0
  BIC(L-squared)       = -29.5317
  AIC(L-squared)       = -9.2801
  BIC(log-likelihood)  = 1057.3129
  AIC(log-likelihood)  = 1026.9353

*** LOG-LINEAR PARAMETERS ***

* TABLE X [or P(X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  X 
   1            -0.4737   0.1443   -3.283      0.6227 
   2             0.4737                        1.6060    10.78   1 0.001

* TABLE XA [or P(A|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  A 
   1            -1.4724   0.9301   -1.583      0.2294 
   2             1.4724                        4.3599     2.51   1 0.113
  XA 
   1 1          -1.0161   0.9337   -1.088      0.3620 
   1 2           1.0161                        2.7624 
   2 1           1.0161                        2.7624 
   2 2          -1.0161                        0.3620     1.18   1 0.276

* TABLE XB [or P(B|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  B 
   1            -0.4828   0.2502   -1.930      0.6171 
   2             0.4828                        1.6206     3.72   1 0.054
  XB 
   1 1          -0.7837   0.2428   -3.228      0.4567 
   1 2           0.7837                        2.1895 
   2 1           0.7837                        2.1895 
   2 2          -0.7837                        0.4567    10.42   1 0.001

* TABLE XC [or P(C|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  C 
   1            -0.5087   0.3002   -1.694      0.6013 
   2             0.5087                        1.6632     2.87   1 0.090
  XC 
   1 1          -0.8638   0.2920   -2.958      0.4215 
   1 2           0.8638                        2.3723 
   2 1           0.8638                        2.3723 
   2 2          -0.8638                        0.4215     8.75   1 0.003

* TABLE XD [or P(D|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  D 
   1             0.1695   0.1641    1.033      1.1848 
   2            -0.1695                        0.8440     1.07   1 0.302
  XD 
   1 1          -0.7708   0.1512   -5.098      0.4626 
   1 2           0.7708                        2.1615 
   2 1           0.7708                        2.1615 
   2 2          -0.7708                        0.4626    25.99   1 0.000


*** (CONDITIONAL) PROBABILITIES ***

* P(X) *

  1              0.2794  (0.0581)
  2              0.7206  (0.0581)

* P(A|X) *

  1 | 1          0.0068  (0.0253)
  2 | 1          0.9932  (0.0253)
  1 | 2          0.2865  (0.0404)
  2 | 2          0.7135  (0.0404)

* P(B|X) *

  1 | 1          0.0736  (0.0656)
  2 | 1          0.9264  (0.0656)
  1 | 2          0.6461  (0.0486)
  2 | 2          0.3539  (0.0486)

* P(C|X) *

  1 | 1          0.0604  (0.0660)
  2 | 1          0.9396  (0.0660)
  1 | 2          0.6705  (0.0497)
  2 | 2          0.3295  (0.0497)

* P(D|X) *

  1 | 1          0.2310  (0.0952)
  2 | 1          0.7690  (0.0952)
  1 | 2          0.8677  (0.0383)
  2 | 2          0.1323  (0.0383)

Table 4 on page 69

Independence Model:

  lat 1
  man 4
  dim 1 2 2 2 2 
  lab X A B C D
  mod 
      X 
      A|X
      B|X
      C|X
      D|X

  dat [20 2
        6 1
        9 2
        4 1
       38 7
       25 6
       24 6
       23 42]
*** STATISTICS ***
  Number of iterations = 2
  Converge criterion   = 0.0000000000
  Seed random values   = 1595
  X-squared            = 104.1071 (0.0000)
  L-squared            = 81.0842 (0.0000)
  Cressie-Read         = 94.3049 (0.0000)
  Dissimilarity index  = 0.2369
  Degrees of freedom   = 11
  Log-likelihood       = -543.64982
  Number of parameters = 4 (+1)
  Sample size          = 216.0
  BIC(L-squared)       = 21.9562
  AIC(L-squared)       = 59.0842
  BIC(log-likelihood)  = 1108.8008
  AIC(log-likelihood)  = 1095.2996

Two-class LCM: See the output in the previous example


Table 5 on page 71.

Model H1: two-class LCM. See previous example.

Model H2: H1 + B & C parallel indicators

  lat 1
  man 4
  dim 2 2 2 2 2 
  lab X A B C D
  mod 
      X 
      A|X
      B|X 
      C|X eq1 B|X
      D|X
  dat [20 2
        6 1
        9 2
        4 1
       38 7
       25 6
       24 6
       23 42]
*** STATISTICS ***
  Number of iterations = 74
  Converge criterion   = 0.0000008459
  Seed random values   = 2955
  X-squared            = 2.8379 (0.9441)
  L-squared            = 2.8857 (0.9413)
  Cressie-Read         = 2.8500 (0.9434)
  Dissimilarity index  = 0.0401
  Degrees of freedom   = 8
  Log-likelihood       = -504.55057
  Number of parameters = 7 (+1)
  Sample size          = 216.0
  BIC(L-squared)       = -40.1165
  AIC(L-squared)       = -13.1143
  BIC(log-likelihood)  = 1046.7281
  AIC(log-likelihood)  = 1023.1011

Model H3: H2 + D equal error rate

  lat 1
  man 4
  dim 2 2 2 2 2 
  lab X A B C D
  mod 
      X 
      A|X
      B|X 
      C|X eq1 B|X
      D|X eq2
  
  des [1 0 0 1]
  
  dat [20 2
        6 1
        9 2
        4 1
       38 7
       25 6
       24 6
       23 42]

*** STATISTICS ***

  Number of iterations = 60
  Converge criterion   = 0.0000009450
  Seed random values   = 5864

  X-squared            = 3.6029 (0.9356)
  L-squared            = 3.6504 (0.9329)
  Cressie-Read         = 3.6133 (0.9350)
  Dissimilarity index  = 0.0431
  Degrees of freedom   = 9
  Log-likelihood       = -504.93290
  Number of parameters = 6 (+1)
  Sample size          = 216.0
  BIC(L-squared)       = -44.7271
  AIC(L-squared)       = -14.3496
  BIC(log-likelihood)  = 1042.1175
  AIC(log-likelihood)  = 1021.8658

Model H4: H3 + A as perfect indicator for class 2

lat 1
  man 4
  dim 2 2 2 2 2 
  lab X A B C D
  mod 
      X 
      A|X eq2
      B|X 
      C|X eq1 B|X
      D|X eq2
  
  des [1 0 0 -1
       2 0 0 2]
  sta A|X [.2 .8 0 1]
  
  dat [20 2
        6 1
        9 2
        4 1
       38 7
       25 6
       24 6
       23 42]
  
*** STATISTICS ***

  Number of iterations = 38
  Converge criterion   = 0.0000009584
  Seed random values   = 5607

  X-squared            = 3.6055 (0.9634)
  L-squared            = 3.6586 (0.9614)
  Cressie-Read         = 3.6177 (0.9629)
  Dissimilarity index  = 0.0430
  Degrees of freedom   = 10
  Log-likelihood       = -504.93703
  Number of parameters = 5 (+1)
  Sample size          = 216.0
  BIC(L-squared)       = -50.0941
  AIC(L-squared)       = -16.3414
  BIC(log-likelihood)  = 1036.7505
  AIC(log-likelihood)  = 1019.8741

Table 6 on page 72 based on H4:

*** LATENT CLASS OUTPUT ***
          X  1    X  2
         0.7574  0.2426
  A  1   0.2751  0.0000
  A  2   0.7249  1.0000
  B  1   0.6362  0.0463
  B  2   0.3638  0.9537
  C  1   0.6362  0.0463
  C  2   0.3638  0.9537
  D  1   0.8524  0.1476
  D  2   0.1476  0.8524

Table 8 on page 75 based on data presented in Table 7.

Model H1: two-class LCM

 lat 1
  man 4 
  dim 2 2 2 2 2
  lab X A B C D
  mod X   
      A|X 
      B|X 
      C|X 
      D|X 
  dat [567 11 62 32 17 10 22 38 18 13 42 62 9 11 28 719]
*** STATISTICS ***
  Number of iterations = 67
  Converge criterion   = 0.0000008203
  Seed random values   = 2814
  X-squared            = 214.7577 (0.0000)
  L-squared            = 179.8535 (0.0000)
  Cressie-Read         = 199.6028 (0.0000)
  Dissimilarity index  = 0.0963
  Degrees of freedom   = 6
  Log-likelihood       = -2773.79306
  Number of parameters = 9 (+1)
  Sample size          = 1661.0
  BIC(L-squared)       = 135.3624
  AIC(L-squared)       = 167.8535
  BIC(log-likelihood)  = 5614.3227
  AIC(log-likelihood)  = 5565.5861

Model H2: three-class model with linear restrictions

lat 1
man 4 
dim 3 2 2 2 2
lab X A B C D
mod X   {X}
    A|X {A,spe(XA,1b)}
    B|X {B,spe(XB,1b)}
    C|X {C,spe(XC,1b)}
    D|X {D,spe(XD,1b)}
dat [567 11 62 32 17 10 22 38 18 13 42 62 9 11 28 719]
see 12357
ite 10000

*** STATISTICS ***

  Number of iterations = 8587
  Converge criterion   = 0.0000009999
  Seed random values   = 12357

  X-squared            = 2.3618 (0.7972)
  L-squared            = 2.3451 (0.7996)
  Cressie-Read         = 2.3554 (0.7981)
  Dissimilarity index  = 0.0069
  Degrees of freedom   = 5
  Log-likelihood       = -2685.03886
  Number of parameters = 10 (+1)
  Sample size          = 1661.0
  BIC(L-squared)       = -34.7308
  AIC(L-squared)       = -7.6549
  BIC(log-likelihood)  = 5444.2295
  AIC(log-likelihood)  = 5390.0777

Model H3: H2 + A, B, C restricted to equal association

 lat 1
  man 4 
  dim 3 2 2 2 2
  lab X A B C D
  mod X   {X}
      ABC|X {A,B,C,spe(XA,XB,XC,1b)}
      D|X {D,spe(XD,1b)}
  dat [567 11 62 32 17 10 22 38 18 13 42 62 9 11 28 719]
  ite 10000

*** STATISTICS ***

  Number of iterations = 9128
  Converge criterion   = 0.0000009999
  Seed random values   = 1532

  X-squared            = 3.5802 (0.8267)
  L-squared            = 3.5897 (0.8256)
  Cressie-Read         = 3.5816 (0.8265)
  Dissimilarity index  = 0.0089
  Degrees of freedom   = 7
  Log-likelihood       = -2685.66117
  Number of parameters = 8 (+1)
  Sample size          = 1661.0
  BIC(L-squared)       = -48.3165
  AIC(L-squared)       = -10.4103
  BIC(log-likelihood)  = 5430.6437
  AIC(log-likelihood)  = 5387.3223

Table 9 on page 77 based on model H3 in previous table

  lat 1
  man 4 
  dim 3 2 2 2 2
  lab X A B C D
  mod X   {X}
      ABC|X {A,B,C,spe(XA,XB,XC,1b)}
      D|X {D,spe(XD,1b)}
  dat [567 11 62 32 17 10 22 38 18 13 42 62 9 11 28 719]
  ite 10000
  

*** LOG-LINEAR PARAMETERS ***

* TABLE X [or P(X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  X 
   1             0.3024   0.0489    6.186      1.3531 
   2            -0.4448   0.0613   -7.255      0.6409 
   3             0.1424                        1.1531    55.16   2 0.000

* TABLE XABC [or P(ABC|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  A 
   1            -0.0969   0.0827   -1.172      0.9077 
   2             0.0969                        1.1017     1.37   1 0.241
  B 
   1             0.1306   0.0839    1.558      1.1395 
   2            -0.1306                        0.8776     2.43   1 0.119
  C 
   1            -0.6152   0.0899   -6.841      0.5405 
   2             0.6152                        1.8501    46.80   1 0.000
  spe(XA,.,XC,1b)
   1            -4.1065   0.2247  -18.279      0.0165   334.13   1 0.000

* TABLE XD [or P(D|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  D 
   1            -0.0749   0.1008   -0.743      0.9279 
   2             0.0749                        1.0777     0.55   1 0.458
  spe(XD,1b)
   1            -7.2381   ******    *****  7.19E-0004     0.00   1 1.000

Table 10 on page 79

Model H1: {AXG, BXG, CXG, DXG}

  lat 1
  man 5 
  dim 2 2 2 2 2 2
  lab X G A B C D
  mod G
      X|G   
      A|XG 
      B|XG 
      C|XG
      D|XG 
  dat [20 2 6 1  9 2 4 1 38 7 25 6 24 6 23 42
       20 3 4 3 23 3 4 2 25 6 15 6 29 5 31 37]
*** STATISTICS ***
  Number of iterations = 341
  Converge criterion   = 0.0000009878
  Seed random values   = 3036
  X-squared            = 9.0628 (0.6976)
  L-squared            = 8.2533 (0.7650)
  Cressie-Read         = 8.7323 (0.7256)
  Dissimilarity index  = 0.0443
  Degrees of freedom   = 12
  Log-likelihood       = -1324.98883
  Number of parameters = 19 (+1)
  Sample size          = 432.0
  BIC(L-squared)       = -64.5678
  AIC(L-squared)       = -15.7467
  BIC(log-likelihood)  = 2765.2777
  AIC(log-likelihood)  = 2687.9777

Model H2: {XG, AX, BX, CX, DX}

  lat 1
  man 5 
  dim 2 2 2 2 2 2
  lab X G A B C D
  mod G
      X|G   
      A|X
      B|X 
      C|X
      D|X 
  dat [20 2 6 1  9 2 4 1 38 7 25 6 24 6 23 42
       20 3 4 3 23 3 4 2 25 6 15 6 29 5 31 37]
  

*** STATISTICS ***

  Number of iterations = 135
  Converge criterion   = 0.0000009252
  Seed random values   = 3122

  X-squared            = 24.7761 (0.2101)
  L-squared            = 23.4693 (0.2663)
  Cressie-Read         = 24.2359 (0.2322)
  Dissimilarity index  = 0.0871
  Degrees of freedom   = 20
  Log-likelihood       = -1332.59683
  Number of parameters = 11 (+1)
  Sample size          = 432.0
  BIC(L-squared)       = -97.8992
  AIC(L-squared)       = -16.5307
  BIC(log-likelihood)  = 2731.9463
  AIC(log-likelihood)  = 2687.1937

Model H3: {G, AX, BX, CX, DX}

 lat 1
  man 5 
  dim 2 2 2 2 2 2
  lab X G A B C D
  mod G
      X   
      A|X
      B|X 
      C|X
      D|X 
  dat [20 2 6 1  9 2 4 1 38 7 25 6 24 6 23 42
       20 3 4 3 23 3 4 2 25 6 15 6 29 5 31 37]
  

*** STATISTICS ***

  Number of iterations = 133
  Converge criterion   = 0.0000009886
  Seed random values   = 4424

  X-squared            = 24.8170 (0.2552)
  L-squared            = 23.4808 (0.3189)
  Cressie-Read         = 24.2661 (0.2803)
  Dissimilarity index  = 0.0875
  Degrees of freedom   = 21
  Log-likelihood       = -1332.60258
  Number of parameters = 10 (+1)
  Sample size          = 432.0
  BIC(L-squared)       = -103.9561
  AIC(L-squared)       = -18.5192
  BIC(log-likelihood)  = 2725.8894
  AIC(log-likelihood)  = 2685.2052

Table 11 on page 80 based on Model H3 from previous table

lat 1
  man 5 
  dim 2 2 2 2 2 2
  lab X G A B C D
  mod G
      X   
      A|X
      B|X 
      C|X
      D|X 
  dat [20 2 6 1  9 2 4 1 38 7 25 6 24 6 23 42
       20 3 4 3 23 3 4 2 25 6 15 6 29 5 31 37]
*** LATENT CLASS OUTPUT ***

          X  1    X  2
         0.2917  0.7083
  G  1   0.5000  0.5000
  G  2   0.5000  0.5000
  A  1   0.0105  0.3454
  A  2   0.9895  0.6546
  B  1   0.1077  0.5668
  B  2   0.8923  0.4332
  C  1   0.0208  0.7170
  C  2   0.9792  0.2830
  D  1   0.3189  0.8491
  D  2   0.6811  0.1509

Table 12 on page 81 based on Model H3 from the previous example

lat 1
  man 5 
  dim 2 2 2 2 2 2
  lab X G A B C D
  mod G
      X   
      A|X
      B|X 
      C|X
      D|X 
  dat [20 2 6 1  9 2 4 1 38 7 25 6 24 6 23 42
       20 3 4 3 23 3 4 2 25 6 15 6 29 5 31 37]
*** LOG-LINEAR PARAMETERS ***

* TABLE G [or P(G)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  G 
   1             0.0000   0.0481    0.000      1.0000 
   2            -0.0000                        1.0000     0.00   1 1.000

* TABLE X [or P(X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  X 
   1            -0.4435   0.1145   -3.874      0.6418 
   2             0.4435                        1.5582    15.01   1 0.000

* TABLE XA [or P(A|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  A 
   1            -1.2959   0.5554   -2.333      0.2737 
   2             1.2959                        3.6542     5.44   1 0.020
  XA 
   1 1          -0.9761   0.5542   -1.761      0.3768 
   1 2           0.9761                        2.6542 
   2 1           0.9761                        2.6542 
   2 2          -0.9761                        0.3768     3.10   1 0.078

* TABLE XB [or P(B|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  B 
   1            -0.4614   0.1398   -3.301      0.6304 
   2             0.4614                        1.5863    10.90   1 0.001
  XB 
   1 1          -0.5958   0.1376   -4.330      0.5511 
   1 2           0.5958                        1.8146 
   2 1           0.5958                        1.8146 
   2 2          -0.5958                        0.5511    18.75   1 0.000

* TABLE XC [or P(C|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  C 
   1            -0.7306   0.7089   -1.031      0.4816 
   2             0.7306                        2.0764     1.06   1 0.303
  XC 
   1 1          -1.1954   0.6980   -1.712      0.3026 
   1 2           1.1954                        3.3049 
   2 1           1.1954                        3.3049 
   2 2          -1.1954                        0.3026     2.93   1 0.087

* TABLE XD [or P(D|X)] *

  effect           beta  std err  z-value   exp(beta)     Wald  df  prob
  D 
   1             0.2422   0.0983    2.464      1.2741 
   2            -0.2422                        0.7849     6.07   1 0.014
  XD 
   1 1          -0.6216   0.0916   -6.784      0.5371 
   1 2           0.6216                        1.8620 
   2 1           0.6216                        1.8620 
   2 2          -0.6216                        0.5371    46.02   1 0.000

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