1. 1 Parallel Models

2. 2 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. 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 stiterations = 10; stscale = 15;

4. 4 4 class UNconstrained model model: %OVERALL% %c#1% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#3% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#4% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#2% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1];

5. 5 4 class UNconstrained model model: %OVERALL% %c#1% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#3% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#4% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#2% [dw_kk$1]; [dw_km$1]; [dw_kp$1]; [dw_kr$1]; [dw_ku$1]; [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; Red text = Not necessary, but useful for comparison

6. 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 NE_KM NE_KP NE_KR NE_KU Categorical latent variables C

7. 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 P-Value 0.0000 Likelihood Ratio Chi-Square Value 1468.348 Degrees of Freedom 979 P-Value 0.0000

8. 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. 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 Most Likely Latent Class Membership Latent Class

10. 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 Category 2 0.124 0.006 19.123 0.000 NE_KM Category 1 0.966 0.004 227.585 0.000 Category 2 0.034 0.004 8.085 0.000 NE_KP Category 1 0.978 0.003 337.963 0.000 Category 2 0.022 0.003 7.574 0.000 NE_KR Category 1 0.983 0.002 408.828 0.000 Category 2 0.017 0.002 7.155 0.000 NE_KU Category 1 0.990 0.002 521.300 0.000 Category 2 0.010 0.002 5.481 0.000

11. 11 Figure for 4-class Un-Con

12. 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. 13 4 class constrained model %c#1% [dw_kk$1] (1); [dw_km$1] (2); [dw_kp$1] (3); [dw_kr$1] (4); [dw_ku$1] (5); [ne_kk$1] (11); [ne_km$1] (12); [ne_kp$1] (13); [ne_kr$1] (14); [ne_ku$1] (15); %c#2% [dw_kk$1] (1); [dw_km$1] (2); [dw_kp$1] (3); [dw_kr$1] (4); [dw_ku$1] (5); [ne_kk$1] (16); [ne_km$1] (17); [ne_kp$1] (18); [ne_kr$1] (19); [ne_ku$1] (20); %c#3% [dw_kk$1] (6); [dw_km$1] (7); [dw_kp$1] (8); [dw_kr$1] (9); [dw_ku$1] (10); [ne_kk$1] (11); [ne_km$1] (12); [ne_kp$1] (13); [ne_kr$1] (14); [ne_ku$1] (15); %c#4% [dw_kk$1] (6); [dw_km$1] (7); [dw_kp$1] (8); [dw_kr$1] (9); [dw_ku$1] (10); [ne_kk$1] (16); [ne_km$1] (17); [ne_kp$1] (18); [ne_kr$1] (19); [ne_ku$1] (20);

14. 14 Daywetting constraints %c#1% [dw_kk$1] (1); [dw_km$1] (2); [dw_kp$1] (3); [dw_kr$1] (4); [dw_ku$1] (5); [ne_kk$1] (11); [ne_km$1] (12); [ne_kp$1] (13); [ne_kr$1] (14); [ne_ku$1] (15); %c#2% [dw_kk$1] (1); [dw_km$1] (2); [dw_kp$1] (3); [dw_kr$1] (4); [dw_ku$1] (5); [ne_kk$1] (16); [ne_km$1] (17); [ne_kp$1] (18); [ne_kr$1] (19); [ne_ku$1] (20); %c#3% [dw_kk$1] (6); [dw_km$1] (7); [dw_kp$1] (8); [dw_kr$1] (9); [dw_ku$1] (10); [ne_kk$1] (11); [ne_km$1] (12); [ne_kp$1] (13); [ne_kr$1] (14); [ne_ku$1] (15); %c#4% [dw_kk$1] (6); [dw_km$1] (7); [dw_kp$1] (8); [dw_kr$1] (9); [dw_ku$1] (10); [ne_kk$1] (16); [ne_km$1] (17); [ne_kp$1] (18); [ne_kr$1] (19); [ne_ku$1] (20);

15. 15 Bedwetting constraints %c#1% [dw_kk$1] (1); [dw_km$1] (2); [dw_kp$1] (3); [dw_kr$1] (4); [dw_ku$1] (5); [ne_kk$1] (11); [ne_km$1] (12); [ne_kp$1] (13); [ne_kr$1] (14); [ne_ku$1] (15); %c#2% [dw_kk$1] (1); [dw_km$1] (2); [dw_kp$1] (3); [dw_kr$1] (4); [dw_ku$1] (5); [ne_kk$1] (16); [ne_km$1] (17); [ne_kp$1] (18); [ne_kr$1] (19); [ne_ku$1] (20); %c#3% [dw_kk$1] (6); [dw_km$1] (7); [dw_kp$1] (8); [dw_kr$1] (9); [dw_ku$1] (10); [ne_kk$1] (11); [ne_km$1] (12); [ne_kp$1] (13); [ne_kr$1] (14); [ne_ku$1] (15); %c#4% [dw_kk$1] (6); [dw_km$1] (7); [dw_kp$1] (8); [dw_kr$1] (9); [dw_ku$1] (10); [ne_kk$1] (16); [ne_km$1] (17); [ne_kp$1] (18); [ne_kr$1] (19); [ne_ku$1] (20);

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 Likelihood Ratio Chi-Square Value 1797.171 Degrees of Freedom 1000 P-Value 0.0000

17. 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 3 4140 0.71097 4 1022 0.17551

18. 18 Figure for 4-class Con

19. 19 Association between classes ClassBedwettingDaywetting % Estimated model n (%) Modal assignment 3“Normal” 73.30%4140 (71.1%) 4Delayed“Normal”14.20%1022 (17.6%) 2Delayed 7.90%374 (6.4%) 1“Normal”Delayed4.50%287 (4.9%) 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. 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. 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. 22 Figure: 16 class constrained

23. 23 4-class Un-Con from earlier

24. 24 Crosstab: DW BWnormaldelayedchronicrelapse normal 4068498868 4273 (95.2%)(1.1%)(2.1%)(1.6%) (84.4%)(9.4%)(29.9%)(37.2%) relapse 10101986 206 (49.0%)(0.0%)(9.2%)(41.7%) (2.1%)(0.0%)(6.5%)(47.0%) delayed 374299370 710 (52.7%)(42.1%)(5.2%)(0.0%) (7.8%)(57.1%)(12.6%)(0.0%) chronic 27917615029 634 (44.0%)(27.8%)(23.7%)(4.6%) (5.8%)(33.6%)(51.0%)(15.8%) 48225242941835823

25. 25 Also possible with LCGA model: %OVERALL% i1 s1 q1 | ne_kk@0 ne_km@1 ne_kp@2 ne_kr@3 ne_ku@4; i2 s2 q2 | dw_kk@0 dw_km@1 dw_kp@2 dw_kr@3 dw_ku@4; %c#1% [i1] (1); [s1] (2); [q1] (3); [i2] (11); [s2] (12); [q2] (13); %c#2% [i1] (1); [s1] (2); [q1] (3); [i2] (14); [s2] (15); [q2] (16); %c#3% [i1] (1); [s1] (2); [q1] (3); [i2] (17); [s2] (18); [q2] (19); %c#4% [i1] (4); [s1] (5); [q1] (6); [i2] (11); [s2] (12); [q2] (13); %c#5% [i1] (4); [s1] (5); [q1] (6); [i2] (14); [s2] (15); [q2] (16); %c#6% [i1] (4); [s1] (5); [q1] (6); [i2] (17); [s2] (18); [q2] (19); %c#7% [i1] (7); [s1] (8); [q1] (9); [i2] (11); [s2] (12); [q2] (13); %c#8% [i1] (7); [s1] (8); [q1] (9); [i2] (14); [s2] (15); [q2] (16); %c#9% [i1] (7); [s1] (8); [q1] (9); [i2] (17); [s2] (18); [q2] (19);

26. 26 9-class constrained LCGA

27. 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. 28 How to examine residuals model: output: residual;

29. 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 Category 2 0.308 0.041 NE_KK Category 1 0.855 -0.028 Category 2 0.145 0.028 NE_KM Category 1 0.949 -0.028 Category 2 0.051 0.028 NE_KP Category 1 0.971 -0.025 Category 2 0.029 0.025 NE_KR Category 1 0.980 -0.001 Category 2 0.020 0.001 NE_KU Category 1 0.986 -0.013 Category 2 0.014 0.013

30. 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 Category 1 Category 1 0.195 -0.005 Category 1 Category 2 0.087 0.045 Category 2 Category 1 0.497 -0.036 Category 2 Category 2 0.221 -0.003 DW_KK NE_KK Category 1 Category 1 0.241 0.040 Category 1 Category 2 0.041 0.000 Category 2 Category 1 0.615 -0.068 Category 2 Category 2 0.104 0.028

31. 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 DW_KK DW_KR Category 1 Category 1 0.806 0.806 -0.035 Category 1 Category 2 0.031 0.031 0.081 Category 2 Category 1 0.120 0.120 0.043 Category 2 Category 2 0.043 0.043 -0.068 Bivariate Pearson Chi-Square 0.013 Bivariate Log-Likelihood Chi-Square 0.013

32. 32 Compare con/un-con 4 class uncon 5 class uncon 16 class con 25 class con # Parameters 43545574 DW_KK NE_KK 17.936.0566.253.11 DW_KM NE_KM 6.015.824.498.08 DW_KP NE_KP 2.171.994.271.41 DW_KR NE_KR 4.223.850.0010.38 DW_KU NE_KU 4.244.648.510.84 Overall307.45101.5649.9528.81 Bivariate Pearson Chi-Square

33. 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