Parallel Models

1
Parallel Models
2
Parallel Models

• Model two separate processes which run in tandem
• Bedwetting and daytime wetting
• 5 time points 4½, 5½, 6½ ,7½ 9½ yrs
• Binary measures
• Fit and n-class parallel model as an n²-class
  model with constraints

3
4 class model syntax pt 1

• title 4 class (un)constrained parallel model
• data file is 'day_and_night.txt'
• listwise on
• variable
• names sex bwt marr m_age parity educ tenure
• ne_kk ne_km ne_kp ne_kr ne_ku
• dw_kk dw_km dw_kp dw_kr dw_ku
• categorical dw_kk dw_km dw_kp dw_kr dw_ku
  ne_kk ne_km ne_kp ne_kr ne_ku
• usevariables dw_kk dw_km dw_kp dw_kr dw_ku
  ne_kk ne_km ne_kp ne_kr ne_ku
• missing are dw_kk dw_km dw_kp dw_kr dw_ku ne_kk
  ne_km ne_kp ne_kr ne_ku (-9)
• classes c (4)
• analysis
• type mixture
• starts 200 100

4
4 class UNconstrained model

• model
• OVERALL
• c1
• dw_kk1
• dw_km1
• dw_kp1
• dw_kr1
• dw_ku1
• ne_kk1
• ne_km1
• ne_kp1
• ne_kr1
• ne_ku1

• c3
• dw_kk1
• dw_km1
• dw_kp1
• dw_kr1
• dw_ku1
• ne_kk1
• ne_km1
• ne_kp1
• ne_kr1
• ne_ku1

• c4
• dw_kk1
• dw_km1
• dw_kp1
• dw_kr1
• dw_ku1
• ne_kk1
• ne_km1
• ne_kp1
• ne_kr1
• ne_ku1

c2 dw_kk1 dw_km1
dw_kp1 dw_kr1 dw_ku1
ne_kk1 ne_km1 ne_kp1
ne_kr1 ne_ku1
5
4 class UNconstrained model

• model
• OVERALL
• c1
• dw_kk1
• dw_km1
• dw_kp1
• dw_kr1
• dw_ku1
• ne_kk1
• ne_km1
• ne_kp1
• ne_kr1
• ne_ku1

• c3
• dw_kk1
• dw_km1
• dw_kp1
• dw_kr1
• dw_ku1
• ne_kk1
• ne_km1
• ne_kp1
• ne_kr1
• ne_ku1

• c4
• dw_kk1
• dw_km1
• dw_kp1
• dw_kr1
• dw_ku1
• ne_kk1
• ne_km1
• ne_kp1
• ne_kr1
• ne_ku1

c2 dw_kk1 dw_km1
dw_kp1 dw_kr1 dw_ku1
ne_kk1 ne_km1 ne_kp1
ne_kr1 ne_ku1
Red text Not necessary, but useful for
comparison
6
Output for 4-class Un-Con

• INPUT READING TERMINATED NORMALLY
• 4 class unconstrained parallel model
• SUMMARY OF ANALYSIS
• Number of groups
  1
• Number of observations
  5823
• Number of dependent variables
  10
• Number of independent variables
  0
• Number of continuous latent variables
  0
• Number of categorical latent variables
  1
• Observed dependent variables
• Binary and ordered categorical (ordinal)
• DW_KK DW_KM DW_KP DW_KR
  DW_KU NE_KK

7
Output for 4-class Un-Con

• TESTS OF MODEL FIT
• Loglikelihood
• H0 Value
  -17302.499
• H0 Scaling Correction Factor
  1.067
• for MLR
• Information Criteria
• Number of Free Parameters
  43
• Akaike (AIC)
  34690.998
• Bayesian (BIC)
  34977.790
• Sample-Size Adjusted BIC
  34841.148
• Chi-Square Test of Model Fit
• Pearson Chi-Square
• Value
  2149.662
• Degrees of Freedom
  979

8
Output for 4-class Un-Con

• FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT
  CLASSES
• BASED ON THE ESTIMATED MODEL
• Latent classes
• 1 4127.12486 0.70876
• 2 363.58862 0.06244
• 3 260.72357 0.04477
• 4 1071.56295 0.18402
• CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST
  LIKELY LATENT CLASS MEMBERSHIP
• Class Counts and Proportions
• Latent classes
• 1 4118 0.70720
• 2 346 0.05942
• 3 246 0.04225
• 4 1113 0.19114

9
Output for 4-class Un-Con

• CLASSIFICATION QUALITY
• Entropy 0.894
• Average Latent Class Probabilities for Most
  Likely Latent Class Membership (Row) by Latent
  Class (Column)
• 1 2 3 4
• 1 0.977 0.010 0.000 0.013
• 2 0.079 0.870 0.020 0.031
• 3 0.000 0.024 0.909 0.067
• 4 0.070 0.014 0.027 0.890

Latent Class
Most Likely Latent Class Membership
10
Output for 4-class Un-Con

• RESULTS IN PROBABILITY SCALE Latent Class 1
• DW_KK
• Category 1 0.936 0.004
  208.189 0.000
• Category 2 0.064 0.004
  14.265 0.000
• DW_KM
• Category 1 0.983 0.002
  401.303 0.000
• Category 2 0.017 0.002
  7.117 0.000
• DW_KP
• Category 1 0.979 0.003
  352.602 0.000
• Category 2 0.021 0.003
  7.487 0.000
• DW_KR
• Category 1 0.986 0.002
  441.507 0.000
• Category 2 0.014 0.002
  6.389 0.000
• DW_KU
• Category 1 0.992 0.002
  637.980 0.000
• Category 2 0.008 0.002
  4.928 0.000
• NE_KK
• Category 1 0.876 0.006
  135.029 0.000

11
Figure for 4-class Un-Con
12
Why should we constrain this?

• Although the age at attainment of daytime
  continence is related to that for nighttime
  continence, there is considerable variability
• We might like to know
• the odds of late nighttime development for a
  child with normal daytime development
• Whether a relapse in bedwetting is more likely if
  a child is late in its daytime development

13
4 class constrained model

• c1
• dw_kk1 (1)
• dw_km1 (2)
• dw_kp1 (3)
• dw_kr1 (4)
• dw_ku1 (5)
• ne_kk1 (11)
• ne_km1 (12)
• ne_kp1 (13)
• ne_kr1 (14)
• ne_ku1 (15)
• c2
• dw_kk1 (1)
• dw_km1 (2)
• dw_kp1 (3)
• dw_kr1 (4)
• dw_ku1 (5)
• ne_kk1 (16)

• c3
• dw_kk1 (6)
• dw_km1 (7)
• dw_kp1 (8)
• dw_kr1 (9)
• dw_ku1 (10)
• ne_kk1 (11)
• ne_km1 (12)
• ne_kp1 (13)
• ne_kr1 (14)
• ne_ku1 (15)
• c4
• dw_kk1 (6)
• dw_km1 (7)
• dw_kp1 (8)
• dw_kr1 (9)
• dw_ku1 (10)
• ne_kk1 (16)

14
Daywetting constraints

• c1
• dw_kk1 (1)
• dw_km1 (2)
• dw_kp1 (3)
• dw_kr1 (4)
• dw_ku1 (5)
• ne_kk1 (11)
• ne_km1 (12)
• ne_kp1 (13)
• ne_kr1 (14)
• ne_ku1 (15)
• c2
• dw_kk1 (1)
• dw_km1 (2)
• dw_kp1 (3)
• dw_kr1 (4)
• dw_ku1 (5)
• ne_kk1 (16)

• c3
• dw_kk1 (6)
• dw_km1 (7)
• dw_kp1 (8)
• dw_kr1 (9)
• dw_ku1 (10)
• ne_kk1 (11)
• ne_km1 (12)
• ne_kp1 (13)
• ne_kr1 (14)
• ne_ku1 (15)
• c4
• dw_kk1 (6)
• dw_km1 (7)
• dw_kp1 (8)
• dw_kr1 (9)
• dw_ku1 (10)
• ne_kk1 (16)

15
Bedwetting constraints

• c1
• dw_kk1 (1)
• dw_km1 (2)
• dw_kp1 (3)
• dw_kr1 (4)
• dw_ku1 (5)
• ne_kk1 (11)
• ne_km1 (12)
• ne_kp1 (13)
• ne_kr1 (14)
• ne_ku1 (15)
• c2
• dw_kk1 (1)
• dw_km1 (2)
• dw_kp1 (3)
• dw_kr1 (4)
• dw_ku1 (5)
• ne_kk1 (16)

• c3
• dw_kk1 (6)
• dw_km1 (7)
• dw_kp1 (8)
• dw_kr1 (9)
• dw_ku1 (10)
• ne_kk1 (11)
• ne_km1 (12)
• ne_kp1 (13)
• ne_kr1 (14)
• ne_ku1 (15)
• c4
• dw_kk1 (6)
• dw_km1 (7)
• dw_kp1 (8)
• dw_kr1 (9)
• dw_ku1 (10)
• ne_kk1 (16)

16
Output for 4-class Con

• TESTS OF MODEL FIT
• Loglikelihood
• H0 Value
  -17462.086
• H0 Scaling Correction Factor
  1.072
• for MLR
• Information Criteria
• Number of Free Parameters
  23
• Akaike (AIC)
  34970.172
• Bayesian (BIC)
  35123.572
• Sample-Size Adjusted BIC
  35050.485
• Chi-Square Test of Model Fit
• Pearson Chi-Square
• Value
  2748.584
• Degrees of Freedom
  1000
• P-Value
  0.0000

17
Output for 4-class Con

• FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT
  CLASSES
• BASED ON THE ESTIMATED MODEL
• Latent classes
• 1 264.25426 0.04538
• 2 459.37184 0.07889
• 3 4269.80907 0.73327
• 4 829.56483 0.14246
• CLASSIFICATION QUALITY
• Entropy 0.892
• CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST
  LIKELY LATENT CLASS MEMBERSHIP
• Latent classes
• 1 287 0.04929
• 2 374 0.06423

18
Figure for 4-class Con
19
Association between classes
Odds of delayed nighttime continence amongst
normal daywetters 1022 / 4140 0.247 Odds of
delayed nighttime continence amongst delayed
daywetters 287 / 374 0.767
Odds Ratio 3.11
20
Extension to larger models

• Interest in association between 4 classes of
  bedwetting and 4 classes of daywetting
• Fit this with a constrained 16 class model in
  same way
• Should recreate the groups/curves found with
  separate models for BW and DW

21
Compare with 4-class Un-Con

• 4 class unconstrained
• TESTS OF MODEL FIT
• Loglikelihood
• H0 Value -17302.499
• H0 Scaling Correction Factor 1.067
• Information Criteria
• Number of Free Parameters 43
• Akaike (AIC) 34690.998
• Bayesian (BIC) 34977.790
• Sample-Size Adjusted BIC 34841.148
• Entropy 0.894

• 16 class constrained
• TESTS OF MODEL FIT
• Loglikelihood
• H0 Value -16973.255
• H0 Scaling Correction Factor 1.098
• Information Criteria
• Number of Free Parameters 55
• Akaike (AIC) 34056.510
• Bayesian (BIC) 34423.336
• Sample-Size Adjusted BIC 34248.562
• Entropy 0.815

22
Figure 16 class constrained
23
4-class Un-Con from earlier
24
Crosstab
25
Also possible with LCGA

• model
• OVERALL
• i1 s1 q1 ne_kk_at_0 ne_km_at_1 ne_kp_at_2 ne_kr_at_3
  ne_ku_at_4
• i2 s2 q2 dw_kk_at_0 dw_km_at_1 dw_kp_at_2 dw_kr_at_3
  dw_ku_at_4

c1 i1 (1) s1 (2) q1
(3) i2 (11) s2 (12)
q2 (13) c2 i1 (1) s1
(2) q1 (3) i2 (14)
s2 (15) q2 (16) c3 i1
(1) s1 (2) q1 (3)
i2 (17) s2 (18) q2
(19)
c4 i1 (4) s1 (5) q1
(6) i2 (11) s2 (12)
q2 (13) c5 i1 (4) s1
(5) q1 (6) i2 (14)
s2 (15) q2 (16) c6 i1
(4) s1 (5) q1 (6)
i2 (17) s2 (18) q2
(19)
c7 i1 (7) s1 (8) q1
(9) i2 (11) s2 (12)
q2 (13) c8 i1 (7) s1
(8) q1 (9) i2 (14)
s2 (15) q2 (16) c9 i1
(7) s1 (8) q1 (9)
i2 (17) s2 (18) q2
(19)
26
9-class constrained LCGA
27
Correlations within class

• One assumption of LCA is that the latent class
  variable totally accounts for the observed
  correlations between the manifest variables
  (local independence)
• Not assessed by fit statistics so should be
  checked by examining within class residuals
• The more variables you model, particularly if
  they are not simply repeated measures, the more
  you run the risk of there being a residual
  bivariate correlation

28
How to examine residuals

• model
• ltsnipgt
• output
• residual

29
Residual output - univariate

• RESIDUAL OUTPUT
• UNIVARIATE DISTRIBUTION FIT FOR CLASS 1
• Variable Estimated Residual
  (Observed-Estimated)
• DW_KK
• Category 1 0.281
  0.039
• Category 2 0.719
  -0.039
• DW_KM
• Category 1 0.424
  0.020
• Category 2 0.576
  -0.020
• DW_KP
• Category 1 0.341
  -0.022
• Category 2 0.659
  0.022
• DW_KR
• Category 1 0.527
  -0.060
• Category 2 0.473
  0.060
• DW_KU
• Category 1 0.692
  -0.041

30
Residual output - bivariate

• BIVARIATE DISTRIBUTIONS FIT FOR CLASS 1
• Variable Variable Estimated
  Residual (Observed-Estimated)
• DW_KK DW_KM
• Category 1 Category 1 0.119
  0.069
• Category 1 Category 2 0.162
  -0.029
• Category 2 Category 1 0.304
  -0.049
• Category 2 Category 2 0.414
  0.010
• DW_KK DW_KP
• Category 1 Category 1 0.096
  -0.012
• Category 1 Category 2 0.185
  0.051
• Category 2 Category 1 0.245
  -0.010
• Category 2 Category 2 0.473
  -0.030
• DW_KK DW_KR
• Category 1 Category 1 0.148
  -0.014
• Category 1 Category 2 0.133
  0.053
• Category 2 Category 1 0.378
  -0.046
• Category 2 Category 2 0.340
  0.007
• DW_KK DW_KU

31
Tech10

• BIVARIATE MODEL FIT INFORMATION
• 
  Estimated Probabilities
• 
  Standardized
• Variable Variable H1
  H0 Residual
• 
  (z-score)
• DW_KK DW_KM
• Category 1 Category 1 0.806
  0.797 1.683
• Category 1 Category 2 0.031
  0.040 -3.451
• Category 2 Category 1 0.103
  0.112 -2.146
• Category 2 Category 2 0.060
  0.051 3.082
• Bivariate Pearson Chi-Square
  25.115
• Bivariate Log-Likelihood Chi-Square
  25.693
• DW_KK DW_KP
• Category 1 Category 1 0.793
  0.792 0.105
• Category 1 Category 2 0.044
  0.045 -0.207
• Category 2 Category 1 0.103
  0.104 -0.140
• Category 2 Category 2 0.060
  0.059 0.181
• Bivariate Pearson Chi-Square
  0.092
• Bivariate Log-Likelihood Chi-Square
  0.092

32
Compare con/un-con
Bivariate Pearson Chi-Square
33
Summary

• This approach makes it possible to model two
  longitudinal processes in parallel
• One can examine the association between the
  classes obtained from two n-class models
• The more manifests you have, the less likely
  local independence is to hold
• One can use the n² classes as the
  outcome/predictor in a further (2-stage) analysis