1 Parallel Models

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 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 = stiterations = 10; stscale = 15;

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 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 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 Output for 4-class Un-Con TESTS OF MODEL FIT Loglikelihood H0 Value H0 Scaling Correction Factor for MLR Information Criteria Number of Free Parameters 43 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC Chi-Square Test of Model Fit Pearson Chi-Square Value Degrees of Freedom 979 P-Value Likelihood Ratio Chi-Square Value Degrees of Freedom 979 P-Value

8 Output for 4-class Un-Con FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES BASED ON THE ESTIMATED MODEL Latent classes CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP Class Counts and Proportions Latent classes

9 Output for 4-class Un-Con CLASSIFICATION QUALITY Entropy Average Latent Class Probabilities for Most Likely Latent Class Membership (Row) by Latent Class (Column) Most Likely Latent Class Membership Latent Class

10 Output for 4-class Un-Con RESULTS IN PROBABILITY SCALE: Latent Class 1 DW_KK Category Category DW_KM Category Category DW_KP Category Category DW_KR Category Category DW_KU Category Category NE_KK Category Category NE_KM Category Category NE_KP Category Category NE_KR Category Category NE_KU Category Category

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 %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 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 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 Output for 4-class Con TESTS OF MODEL FIT Loglikelihood H0 Value H0 Scaling Correction Factor for MLR Information Criteria Number of Free Parameters 23 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC Chi-Square Test of Model Fit Pearson Chi-Square Value Degrees of Freedom 1000 P-Value Likelihood Ratio Chi-Square Value Degrees of Freedom 1000 P-Value

17 Output for 4-class Con FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES BASED ON THE ESTIMATED MODEL Latent classes CLASSIFICATION QUALITY Entropy CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP Latent classes

18 Figure for 4-class Con

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 = Odds of delayed nighttime continence amongst delayed daywetters = 287 / 374 = 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 H0 Scaling Correction Factor Information Criteria Number of Free Parameters 43 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC Entropy class constrained TESTS OF MODEL FIT Loglikelihood H0 Value H0 Scaling Correction Factor Information Criteria Number of Free Parameters 55 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC Entropy 0.815

22 Figure: 16 class constrained

23 4-class Un-Con from earlier

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

25 Also possible with LCGA model: %OVERALL% i1 s1 q1 | i2 s2 q2 | %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 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: output: residual;

29 Residual output - univariate RESIDUAL OUTPUT UNIVARIATE DISTRIBUTION FIT FOR CLASS 1 Variable Estimated Residual (Observed-Estimated) DW_KK Category Category DW_KM Category Category DW_KP Category Category DW_KR Category Category DW_KU Category Category NE_KK Category Category NE_KM Category Category NE_KP Category Category NE_KR Category Category NE_KU Category Category

30 Residual output - bivariate BIVARIATE DISTRIBUTIONS FIT FOR CLASS 1 Variable Variable Estimated Residual (Observed-Estimated) DW_KK DW_KM Category 1 Category Category 1 Category Category 2 Category Category 2 Category DW_KK DW_KP Category 1 Category Category 1 Category Category 2 Category Category 2 Category DW_KK DW_KR Category 1 Category Category 1 Category Category 2 Category Category 2 Category DW_KK DW_KU Category 1 Category Category 1 Category Category 2 Category Category 2 Category DW_KK NE_KK Category 1 Category Category 1 Category Category 2 Category Category 2 Category

31 Tech10 BIVARIATE MODEL FIT INFORMATION Estimated Probabilities Standardized Variable Variable H1 H0 Residual (z-score) DW_KK DW_KM Category 1 Category Category 1 Category Category 2 Category Category 2 Category Bivariate Pearson Chi-Square Bivariate Log-Likelihood Chi-Square DW_KK DW_KP Category 1 Category Category 1 Category Category 2 Category Category 2 Category Bivariate Pearson Chi-Square Bivariate Log-Likelihood Chi-Square DW_KK DW_KR Category 1 Category Category 1 Category Category 2 Category Category 2 Category Bivariate Pearson Chi-Square Bivariate Log-Likelihood Chi-Square 0.013

32 Compare con/un-con 4 class uncon 5 class uncon 16 class con 25 class con # Parameters DW_KK NE_KK DW_KM NE_KM DW_KP NE_KP DW_KR NE_KR DW_KU NE_KU Overall 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