1. 1 Ordinal Models

2. 2 Estimating gender-specific LLCA with repeated ordinal data Examining the effect of time invariant covariates on class membership The effect of class membership on a later outcome

3. 3 The data 5 repeated measures of bedwetting 4½, 5½, 6½,7½ & 9½ yrs 3-level ordinal –Dry –Infrequent wetting (< 2 nights/week) –Frequent wetting (2+ nights/week)

4. 4 4 time point ordinal LLCA (Boys) title: 4 time point LLCA of ordinal bedwetting; data: file is 'C:\Work\bedwet_dsm4_llca\spss\llca_dsm4.txt'; listwise is ON; variable: names ID sex nwet_kk2 nwet_kk3 nwet_kk4 nwet_kk5 nwet_km2 nwet_km3 nwet_km4 nwet_km5 nwet_kp2 nwet_kp3 nwet_kp4 nwet_kp5 nwet_kr2 nwet_kr3 nwet_kr4 nwet_kr5 nwet_ku2 nwet_ku3 nwet_ku4 nwet_ku5; categorical = nwet_kk3 nwet_km3 nwet_kp3 nwet_kr3 nwet_ku3; usevariables nwet_kk3 nwet_km3 nwet_kp3 nwet_kr3 nwet_ku3; missing are nwet_kk3 nwet_km3 nwet_kp3 nwet_kr3 nwet_ku3 (999); classes = c (4); useobservations (sex==1);

5. 5 RESULTS IN PROBABILITY SCALE Latent Class 1 NWET_KK3 Category 1 0.190 0.030 6.430 0.000 Category 2 0.672 0.033 20.134 0.000 Category 3 0.138 0.026 5.409 0.000 NWET_KM3 Category 1 0.224 0.038 5.929 0.000 Category 2 0.727 0.036 20.254 0.000 Category 3 0.048 0.019 2.533 0.011 NWET_KP3 Category 1 0.160 0.045 3.540 0.000 Category 2 0.823 0.044 18.613 0.000 Category 3 0.017 0.011 1.473 0.141 NWET_KR3 Category 1 0.075 0.064 1.178 0.239 Category 2 0.903 0.061 14.686 0.000 Category 3 0.022 0.011 1.929 0.054 NWET_KU3 Category 1 0.456 0.054 8.486 0.000 Category 2 0.532 0.052 10.269 0.000 Category 3 0.012 0.008 1.501 0.133

6. 6 RESULTS IN PROBABILITY SCALE Latent Class 1 NWET_KK3 Category 1 0.190 0.030 6.430 0.000 Category 2 0.672 0.033 20.134 0.000 Category 3 0.138 0.026 5.409 0.000 NWET_KM3 Category 1 0.224 0.038 5.929 0.000 Category 2 0.727 0.036 20.254 0.000 Category 3 0.048 0.019 2.533 0.011 NWET_KP3 Category 1 0.160 0.045 3.540 0.000 Category 2 0.823 0.044 18.613 0.000 Category 3 0.017 0.011 1.473 0.141 NWET_KR3 Category 1 0.075 0.064 1.178 0.239 Category 2 0.903 0.061 14.686 0.000 Category 3 0.022 0.011 1.929 0.054 NWET_KU3 Category 1 0.456 0.054 8.486 0.000 Category 2 0.532 0.052 10.269 0.000 Category 3 0.012 0.008 1.501 0.133 Dry Infrequent wetting Frequent wetting

7. 7 Alternative 1 – three dimensions A 3D plot… or something made out of plasticine

8. 8 Alternative 2 – two figures Infrequent bedwettingFrequent bedwetting

9. 9 Alternative 3 – two figures Any bedwettingFrequent bedwetting

10. 10 Alternative 3 – two figures Any bedwettingFrequent bedwetting (1) A persistent wetting group who mostly wet to a frequent level (2) A persistent wetting group who mostly wet to an infrequent level (3) A delayed group comprising mainly infrequent wetters (4) Normative group

11. 11 Fit statistics - Boys BoysGirls # class# parmsBICBLRTEntropyBICBLRTEntropy 44315055.5< 0.0010.84110173.4< 0.0010.883 55415029< 0.0010.84910211.90.0320.893 665150740.3120.84210267.60.760.899 77615130.90.6460.83810344.110.91

12. 12 5-class model (boys) Any bedwettingFrequent bedwetting

13. 13 5-class model (boys) Normative (63.8%) –Mild risk of infrequent wetting at start which soon disappears Delayed-infrequent (18.2%) –Delayed attainment of nighttime bladder control but rarely attains frequent levels Persistent-infrequent (11.4%) –Persistent throughout period but rarely attains frequent levels Persistent-frequent (4.0%) –Persistently and frequently until late into period. Appears to be turning into lower frequency wetting however over 80% are still wetting to some degree at 9.5yr Delayed-frequent (2.7%) –Frequent wetting until half-way through time period, reducing to a lower level of wetting which appears to be clearing up by 9.5yr

14. 14 Fit statistics – Girls – Oh! BoysGirls # class# parmsBICBLRTEntropyBICBLRTEntropy 44315055.5< 0.0010.84110173.4< 0.0010.883 55415029< 0.0010.84910211.90.0320.893 665150740.3120.84210267.60.760.899 77615130.90.6460.83810344.110.91

15. 15 Fit statistics – Girls – Oh! BoysGirls # class# parmsBICBLRTEntropyBICBLRTEntropy 44315055.5< 0.0010.84110173.4< 0.0010.883 55415029< 0.0010.84910211.90.0320.893 665150740.3120.84210267.60.760.899 77615130.90.6460.83810344.110.91

16. 16 6-class model (girls) Any bedwettingFrequent bedwetting

17. 17 6-class model (girls) Normative (78.6%) Delayed-infrequent (11.7%) Persistent-infrequent (4.6%) Persistent-frequent (1.6%) Delayed-frequent (1.3%) Relapse (2.0%) –Initial period of dryness followed by a return to infrequent wetting

18. 18 Incorporating covariates 2-stage method Export class probabilities to another package – Stata Model class membership as a multinomial model with probability weighting Using classes derived from repeated BW measures with partially missing data (gloss over)

19. 19 Multinomial models (boys) label values class class_label label define class_label /// 1 "Pers INF [1]" /// 2 "DelayFRQ [2]" /// 3 "Normal [3]" /// 4 "Pers FRQ [4]" /// 5 "DelayINF [5]", add tab class foreach var of varlist bedwet_m bedwet_p […] toilet { tab `var' if class==1 xi: mlogit class `var' [iw = boy_weights], rrr test `var' }

20. 20 Typical output Multinomial logistic regression Number of obs = 5004 LR chi2(4) = 85.09 Prob > chi2 = 0.0000 Log likelihood = -5256.9295 Pseudo R2 = 0.0080 ------------------------------------------------------------------------------ class | RRR Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Pers INF [1] | bedwet_m | 2.456404.3459902 6.38 0.000 1.863829 3.237378 -------------+---------------------------------------------------------------- DelayFRQ [2] | bedwet_m | 3.019243.6958002 4.79 0.000 1.921915 4.743095 -------------+---------------------------------------------------------------- Pers FRQ [4] | bedwet_m | 3.795014.6783016 7.46 0.000 2.673461 5.387073 -------------+---------------------------------------------------------------- DelayINF [5] | bedwet_m | 1.540813.2046864 3.25 0.001 1.18761 1.999062 ------------------------------------------------------------------------------ (class==Normal [3] is the base outcome) ( 1) [Pers INF [1]]bedwet_m = 0 ( 2) [DelayDSM [2]]bedwet_m = 0 ( 3) [Pers DSM [4]]bedwet_m = 0 ( 4) [DelayINF [5]]bedwet_m = 0 chi2( 4) = 91.87, Prob > chi2 = 0.0000

21. 21 Selection of covariates (boys) MeasureDelay INFPers INFDelay DSMPers DSMChi, p Mum history of BW 1.54 [1.19, 2.00] 2.46 [1.86, 3.24] 3.02 [1.92, 4.74] 3.8 [2.67, 5.39] 91.9, < 0.001 Dad history of BW 1.32 [0.97, 1.79] 1.48 [1.04, 2.12] 2.14 [1.21, 3.78] 2.34 [1.47, 3.73] 20.7, < 0.001 Development at 18mn 1.06 [0.99, 1.13] 1.05 [0.96, 1.14] 1.27 [1.08, 1.48] 1.21 [1.07, 1.37] 17.9, 0.001 Development at 42mn 1.07 [0.99, 1.15] 1.17 [1.07, 1.28] 1.16 [0.98, 1.37] 1.27 [1.11, 1.46] 23.3, < 0.001 Birthweight 1 [0.93, 1.07] 0.99 [0.91, 1.08] 0.97 [0.84, 1.13] 0.98 [0.87, 1.11] 0.22, 0.995 Gestational age 1.01 [0.94, 1.09] 0.98 [0.89, 1.07] 0.89 [0.75, 1.05] 0.93 [0.81, 1.06] 3.52, 0.475 Maternal age 1.11 [1.02, 1.21] 1.1 [0.99, 1.23] 1.23 [1.00, 1.51] 1.02 [0.87, 1.19] 10.0, 0.041 Maternal education 1.1 [1.04, 1.16] 1.11 [1.04, 1.19] 1.12 [0.99, 1.27] 0.99 [0.90, 1.09] 21.4, < 0.001

22. 22 What if we had used modal-class? The further the posterior probabilities for class assignment are from 1 (i.e. the lower the entropy) the poorer the estimates from a model using the modal class In this example (partial missing data) –entropy = 0.788 1 2 3 4 5 1 0.800 0.011 0.017 0.020 0.153 2 0.063 0.804 0.000 0.069 0.064 3 0.008 0.001 0.923 0.000 0.068 4 0.049 0.104 0.004 0.813 0.030 5 0.110 0.009 0.170 0.006 0.706

23. 23 Estimates using mod class Delay INFPers INFDelay FRQPers FRQChi, p Weighted model Mum history of BW 1.54 [1.19, 2.00] 2.46 [1.86, 3.24] 3.02 [1.92, 4.74] 3.8 [2.67, 5.39] 91.9, < 0.001 Dad history of BW 1.32 [0.97, 1.79] 1.48 [1.04, 2.12] 2.14 [1.21, 3.78] 2.34 [1.47, 3.73] 20.7, < 0.001 Modal class Mum history of BW 1.78 [1.37, 2.32] 2.72 [2.06, 3.59] 2.63 [1.60, 4.32] 4.06 [2.91, 5.68] 104.6, < 0.001 Dad history of BW 1.32 [0.96, 1.80] 1.51 [1.05, 2.16] 2.11 [1.17, 3.80] 2.30 [1.46, 3.62] 20.6, < 0.001 Bias depends on class and also covariate

24. 24 A later outcome Boys’ data Outcomes: –Key Stage 3 at 13-14 yrs –Achieved level 5 or greater in English/Sci/Maths English failed = 1210 (27.9%) Science failed = 895 (20.6%) Maths failed = 845 (19.5%)

25. 25 2 stage procedure - Stata label values class class_label label define class_label /// 1 "Pers INF [1]" /// 2 "DelayFRQ [2]" /// 3 "Normal [3]" /// 4 "Pers FRQ [4]" /// 5 "DelayINF [5]", add recode class 3=0 foreach var of varlist maths science english { tab `var' if class==1 xi: logit `var' i.class [iw = b_par_p], or test _Iclass_2 _Iclass_3 _Iclass_4 _Iclass_5 }

26. 26 KS3 - English Logistic regression Number of obs = 4341 LR chi2(4) = 7.73 Prob > chi2 = 0.1018 Log likelihood = -2564.536 Pseudo R2 = 0.0015 ------------------------------------------------------------------------------ k3_lev5e | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Pers INF |.9896705.1131557 -0.09 0.928.7909827 1.238267 Delay FRQ |.8946308.1964416 -0.51 0.612.5817527 1.375781 Pers FRQ | 1.474409.2323757 2.46 0.014 1.082589 2.008041 Delay INF |.9135896.0852754 -0.97 0.333.76085 1.096991 ------------------------------------------------------------------------------ ( 1) _Iclass_1 = 0 ( 2) _Iclass_2 = 0 ( 3) _Iclass_4 = 0 ( 4) _Iclass_5 = 0 chi2( 4) = 7.97 Prob > chi2 = 0.0926

27. 27 KS3 - Maths Logistic regression Number of obs = 4341 LR chi2(4) = 11.29 Prob > chi2 = 0.0235 Log likelihood = -2133.6412 Pseudo R2 = 0.0026 ------------------------------------------------------------------------------ k3_lev5m | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Pers INF |.9460608.1241854 -0.42 0.673.7314512 1.223637 Delay FRQ |.9601245.2358923 -0.17 0.868.5931937 1.554027 Pers FRQ | 1.688991.2836716 3.12 0.002 1.215249 2.347413 Delay INF |.8976914.095965 -1.01 0.313.7280008 1.106935 ------------------------------------------------------------------------------ ( 1) _Iclass_1 = 0 ( 2) _Iclass_2 = 0 ( 3) _Iclass_4 = 0 ( 4) _Iclass_5 = 0 chi2( 4) = 12.05 Prob > chi2 = 0.0170

28. 28 KS3 - Science Logistic regression Number of obs = 4341 LR chi2(4) = 8.94 Prob > chi2 = 0.0626 Log likelihood = -2203.9946 Pseudo R2 = 0.0020 ------------------------------------------------------------------------------ k3_lev5s | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Pers INF |.8913295.1159146 -0.88 0.376.6907838 1.150097 Delay FRQ | 1.067237.2482602 0.28 0.780.6764799 1.683709 Pers FRQ | 1.525329.2566845 2.51 0.012 1.096786 2.121314 Delay INF |.8888344.092835 -1.13 0.259.7242967 1.09075 ------------------------------------------------------------------------------ ( 1) _Iclass_1 = 0 ( 2) _Iclass_2 = 0 ( 3) _Iclass_4 = 0 ( 4) _Iclass_5 = 0 chi2( 4) = 9.34 Prob > chi2 = 0.0531

29. 29 Summary Fitting ordinal models is similar to binary data however results (“trajectories”) are harder to interpret graphically Resulting classes can be used either as outcomes or categorical predictors using weighted regression in Stata Using variables derived from modal class assignment can often introduce very biased estimates