1. 1 Binary Models 1 A (Longitudinal) Latent Class Analysis of Bedwetting

2. 2 How much fun can you have with 5 binary variables?

3. 3 Croudace paper:

4. 4

5. 5 Gender-specific prevalence and levels of missing data 4.5 years5.5 years6.5 years7.5 years9.5 years Boys (n=7217) No bedwetting31393309329333703414 Bedwetting 1793 (36.4%) 1281 (27.9%) 1061 (24.4%) 844 (20.0%) 526 (13.4%) No response22852627286330033277 Girls (n=6756) No bedwetting35553635351635603587 Bedwetting 1074 (23.2%) 691 (16.0%) 586 (14.3%) 420 (10.6%) 225 (5.9%) No response212724302654)27762944

6. 6 Latent Class Models Deal with patterns of response e.g. 11111 = Yes at all five time points 00000 = No at all five time points 11000 = Yes early on, followed by no 10101 = Alternating pattern 11*** = Yes followed by missing 1*0*1 = etc.

7. 7 Latent Class Models

8. 8 Conditional independence manifest variables are independent given latent class, can be written as a product of all the probabilities for the individual items of that pattern. E.g. for a pattern ’01’, if pat(1) to be the first element of a pattern i.e. the first binary variable, then P( pattern = ’01’ | class = i) = P(pat(1) = ‘0’ | class = i)*P(pat(2) = ‘1’ | class = i)

9. 9 Huh? Basically, Bayes’ rule plus conditional independence assumption enable you to calculate the likelihood of each pattern being assigned to each class. By selecting starting values for the probabilities, ot alternatively a starting assignment for the patterns, one can iterate to convergence to find the best solution given your chosen number of classes

10. 10 Distribution of patterns (complete data) +--------------------------------+ | 4.5 5.5 6.5 7.5 9.5 N | |--------------------------------| 1. | 0 0 0 0 0 3535 | 2. | 1 0 0 0 0 529 | 3. | 1 1 1 1 1 303 | 4. | 1 1 0 0 0 241 | 5. | 1 1 1 1 0 199 | |--------------------------------| 6. | 1 1 1 0 0 176 | 7. | 0 1 0 0 0 138 | 8. | 0 0 1 0 0 99 | 9. | 1 0 1 0 0 82 | 10. | 0 0 0 1 0 69 | |--------------------------------| 11. | 1 0 1 1 0 48 | 12. | 0 0 0 0 1 43 | 13. | 1 1 0 1 0 40 | 14. | 0 1 1 0 0 39 | 15. | 1 1 1 0 1 29 | |--------------------------------| 16. | 1 0 0 1 0 28 | +--------------------------------+ | 4.5 5.5 6.5 7.5 9.5 N | |--------------------------------| 17. | 0 0 1 1 0 27 | 18. | 0 1 1 1 0 25 | 19. | 1 0 0 0 1 24 | 20. | 0 1 1 1 1 23 | |--------------------------------| 21. | 1 0 1 1 1 20 | 22. | 1 1 0 0 1 20 | 23. | 1 1 0 1 1 20 | 24. | 0 0 1 1 1 15 | 25. | 0 0 0 1 1 12 | |--------------------------------| 26. | 0 1 0 1 0 11 | 27. | 1 0 0 1 1 11 | 28. | 0 1 1 0 1 8 | 29. | 0 1 0 1 1 8 | 30. | 0 1 0 0 1 8 | |--------------------------------| 31. | 1 0 1 0 1 7 | 32. | 0 0 1 0 1 6 | +--------------------------------+

11. 11 Distribution of patterns with some missingness +--------------------------------+ | 4.5 5.5 6.5 7.5 9.5 N | |--------------------------------| 1. | 0 0 0 0. 458 | 2. | 0.... 398 | 3. | 0 0 0.. 260 | 4. | 0 0... 252 | 5. | 0 0 0. 0 248 | |--------------------------------| 6. | 1.... 150 | 7. | 0 0. 0 0 146 | 8. |. 0 0 0 0 138 | 9. | 0. 0 0 0 125 | 10. |... 0 0 124 | |--------------------------------| 11. |.... 0 123 | 12. | 0 0. 0. 109 | 13. |... 0. 107 | 14. | 0. 0.. 94 | 15. |. 0... 92 | |--------------------------------| +--------------------------------+ | 4.5 5.5 6.5 7.5 9.5 N | |--------------------------------| 16. | 1 0 0 0. 71 | 17. | 0 0.. 0 70 | 18. | 1 1 1 1. 66 | 19. | 0... 0 65 | 20. | 1 1... 62 | |--------------------------------| 21. | 0.. 0 0 57 | 22. |.. 0.. 55 | 23. | 1 0 0. 0 54 | 24. | 0. 0 0. 53 | 25. | 1 0... 50 | |--------------------------------| Etc. 182. |.. 0 1 0 1 | 183. | 0 0. 1 1 1 | +--------------------------------+

12. 12 Thresholds Mplus thinks of binary variables as being a dichotomised continuous latent variable The point at which a continuous N(0,1) variable must be cut to create a binary variable is called a threshold A binary variable with 50% cases corresponds to a threshold of zero A binary variable with 2.5% cases corresponds to a threshold of 1.96

13. 13 Thresholds Figure from Uebersax webpage

14. 14 Categorical variables A categorical variable with n-levels requires n-1 thresholds i.e. you need to make n-1 cuts in a continuous N(0,1) variable to make your observed n-level variable

15. 15 Degrees of freedom 32 possible patterns (missing data patterns don’t count) Each additional class requires –5 df to estimate the 5 prevalence of wetting within that class (i.e. 5 thresholds) –1 df for an additional cut of the latent variable defining the class distribution Hence a 5-class model uses up 5*5 + 4 = 29 degrees of freedom leaving 3 df to test the model

16. 16 Procedure Fit 1-5 class models in turn Output = within class thresholds (prevalences) and latent class distribution Select preferred model using various criteria –Statistical – measures of fit –Ease of interpretation of results –Parsimony –Face validity –Astrology Celebrate

17. 17 How to fit in Mplus – black box way data: file is ‘bedwetting_5tp.txt'; listwise is on; variable: names sex bwt marr m_age parity educ tenure ne_kk ne_km ne_kp ne_kr ne_ku; categorical = ne_kk ne_km ne_kp ne_kr ne_ku; usevariables ne_kk ne_km ne_kp ne_kr ne_ku; missing are ne_kk ne_km ne_kp ne_kr ne_ku (-9); classes = c (4); analysis: type = mixture; starts = 1000 500; stiterations = 10; stscale = 20; model: %OVERALL% Is that it????

18. 18 What you’re actually doing: model: %OVERALL% [c#1 c#2 c#3]; %c#1% [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#2% [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#3% [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; %c#4% [ne_kk$1]; [ne_km$1]; [ne_kp$1]; [ne_kr$1]; [ne_ku$1]; 4 class model => 3 estimated threshold for latent class variable 5 more thresholds for each class

19. 19 How many random starts? Depends on –Sample size –Complexity of model Number of manifest variables Number of classes Aim to find consistently the model with the lowest likelihood, within each run

20. 20 Success Not there yet Loglikelihood values at local maxima, seeds, and initial stage start numbers: -10148.718 987174 1689 -10148.718 777300 2522 -10148.718 406118 3827 -10148.718 51296 3485 -10148.718 997836 1208 -10148.718 119680 4434 -10148.718 338892 1432 -10148.718 765744 4617 -10148.718 636396 168 -10148.718 189568 3651 -10148.718 469158 1145 -10148.718 90078 4008 -10148.718 373592 4396 -10148.718 73484 4058 -10148.718 154192 3972 -10148.718 203018 3813 -10148.718 785278 1603 -10148.718 235356 2878 -10148.718 681680 3557 -10148.718 92764 2064 Loglikelihood values at local maxima, seeds, and initial stage start numbers -10153.627 23688 4596 -10153.678 150818 1050 -10154.388 584226 4481 -10155.122 735928 916 -10155.373 309852 2802 -10155.437 925994 1386 -10155.482 370560 3292 -10155.482 662718 460 -10155.630 320864 2078 -10155.833 873488 2965 -10156.017 212934 568 -10156.231 98352 3636 -10156.339 12814 4104 -10156.497 557806 4321 -10156.644 134830 780 -10156.741 80226 3041 -10156.793 276392 2927 -10156.819 304762 4712 -10156.950 468300 4176 -10157.011 83306 2432

21. 21 What the output looks like TESTS OF MODEL FIT Loglikelihood H0 Value -10153.129 H0 Scaling Correction Factor 1.007 for MLR Information Criteria Number of Free Parameters 23 Akaike (AIC) 20352.258 Bayesian (BIC) 20505.737 Sample-Size Adjusted BIC 20432.649 (n* = (n + 2) / 24) Chi-Square Test of Model Fit for the Binary outcomes Pearson Chi-Square 11.543 Degrees of Freedom 8 P-Value 0.1728 Likelihood Ratio Chi-Square 11.210 Degrees of Freedom 8 P-Value 0.1901

22. 22 Measures of entropy Average Latent Class Probabilities for Most Likely Latent Class Membership (Row) by Latent Class (Column) 1 2 3 4 1 0.846 0.067 0.040 0.047 2 0.110 0.808 0.037 0.046 3 0.030 0.095 0.875 0.000 4 0.041 0.007 0.000 0.952 Entropy (global) 0.844

23. 23 Model based + modal class latent class distribution FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES BASED ON THE ESTIMATED MODEL Latent classes 1 790.04734 0.13521 2 297.76475 0.05096 3 511.59879 0.08756 4 4243.58913 0.72627 CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP Class Counts and Proportions Latent classes 1 674 0.11535 2 211 0.03611 3 545 0.09327 4 4413 0.75526

24. 24 Model results Estimates S.E. Est./S.E. Latent Class 1 Thresholds NE_KK$1 -1.540 0.238 -6.475 NE_KM$1 -0.891 0.199 -4.471 NE_KP$1 0.346 0.121 2.864 NE_KR$1 2.500 0.472 5.294 NE_KU$1 2.359 0.248 9.527 Latent Class 2 Thresholds NE_KK$1 -0.294 0.208 -1.415 NE_KM$1 0.482 0.429 1.123 NE_KP$1 -0.632 0.230 -2.754 NE_KR$1 -1.539 0.750 -2.053 NE_KU$1 0.501 0.197 2.540 Estimates S.E. Est./S.E. Latent Class 3 Thresholds NE_KK$1 -3.122 0.544 -5.739 NE_KM$1 -15.000 0.000 0.000 NE_KP$1 -3.172 0.481 -6.593 NE_KR$1 -3.224 0.609 -5.294 NE_KU$1 -0.563 0.117 -4.800 Latent Class 4 Thresholds NE_KK$1 2.093 0.073 28.484 NE_KM$1 3.698 0.198 18.651 NE_KP$1 3.796 0.140 27.200 NE_KR$1 4.213 0.180 23.383 NE_KU$1 4.420 0.169 26.172 Categorical Latent Variables Means C#1 -1.681 0.114 -14.746 C#2 -2.657 0.240 -11.092 C#3 -2.116 0.090 -23.551

25. 25 RESULTS IN PROBABILITY SCALE Latent Class 1 NE_KK Category 1 0.177 0.035 5.106 Category 2 0.823 0.035 23.816 NE_KM Category 1 0.291 0.041 7.078 Category 2 0.709 0.041 17.250 NE_KP Category 1 0.586 0.029 19.981 Category 2 0.414 0.029 14.138 NE_KR Category 1 0.924 0.033 27.917 Category 2 0.076 0.033 2.292 NE_KU Category 1 0.914 0.020 46.773 Category 2 0.086 0.020 4.420 Latent Class 2 NE_KK Category 1 0.427 0.051 8.387 Category 2 0.573 0.051 11.258 NE_KM Category 1 0.618 0.101 6.103 Category 2 0.382 0.101 3.769 NE_KP Category 1 0.347 0.052 6.673 Category 2 0.653 0.052 12.555 NE_KR Category 1 0.177 0.109 1.620 Category 2 0.823 0.109 7.551 NE_KU Category 1 0.623 0.046 13.445 Category 2 0.377 0.046 8.150

26. 26 These can be plotted In Excel from the probabilities:

27. 27 … or in Mplus plot: type is plot2; series is anybw_t1 (4.5) anybw_t2 (5.5) anybw_t3 (6.5) anybw_t4 (7.5) anybw_t5 (9.5); Then Graph> view graphs> estimated probabilities

28. 28 Yuck!

29. 29 Model fit stats 1 class2 class3 class4 class5 class Estimated params 511172329 Likelihood -13784.5-10432.6-10200.2-10153.1-10148.7 BLRT <0.0001 0.17 BIC 27612.420960.62054820505.720548.9 Entropy -0.8890.8240.8440.804

30. 30 Model fit stats - BIC Bayesian Information Criterion = -2*Log-likelihood + (# params)*ln(sample size) Function of likelihood which rewards a more parsimonius model Decrease followed by an increase as extra classes are added

31. 31 Model fit stats - Entropy Measure of the certainty with which patterns (and therefore subjects) are assigned to latent classes. Higher values indicate a greater delineation of classes – ideally approaching 1 (Celeux & Soromenho) 0.62 would indicate 'fuzzyness' (Ramaswamy) Thorough job: –Examine global entropy, –Examine class-specific entropy –Examine the assignment probabilities for each individual pattern

32. 32 Model fit stats - BLRT Bootstrap Likelihood Ratio Test Traditional Δ-likelihood test cannot be used to test nested latent class models as the difference in likelihoods is not chi-square distributed The BLRT empirically estimates the difference distribution providing a p-value for the observed difference which can be used in tests of model fit

33. 33 How to obtain BLRT Fit k-class model Select an optseed which replicates the solution with the lowest likelihood for k-classes Re-fit k-class model using optseed, specifying tech14 in the output section and selecting an appropriate number of runs for bootstrapping k-1starts = 100 20; lrtstarts 0 0 250 50; lrtbootstrap 100; Ensure that optimal k-1 class model has been replicated

34. 34 BLRT Output PARAMETRIC BOOTSTRAPPED LIKELIHOOD RATIO TEST FOR 4 (H0) VERSUS 5 CLASSES H0 Loglikelihood Value -2097.721 2 Times the Loglikelihood Difference 5.513 Difference in the Number of Parameters 7 Approximate P-Value 0.7600 Successful Bootstrap Draws 100 Does this agree with what you obtained for 4 class run? B LRT p-value implies no improvement in fit when moving from 4 to 5 classes

35. 35 Model fit stats 1 class2 class3 class4 class5 class Estimated params 511172329 Likelihood -13784.5-10432.6-10200.2-10153.1-10148.7 BLRT <0.0001 0.17 BIC 27612.420960.62054820505.720548.9 Entropy -0.8890.8240.8440.804 BLRT 4 class model is adequate fit to data and there is little improvement in fit when a 5 th class is added BIC attains it’s minimum value at the 4 class model (penalising 5-class for it’s lack of parsimony) Entropy 2-class model has highest entropy, however all values are reasonably high

36. 36 Other considerations - interpretation 4 class model 5 class model

37. 37 Final decision 4 and 5 class model fit the data, parsimony favours 4 class Both 4 and 5 class models make intuitive sense –4 classes: ‘normative’, ‘delayed’, ‘persistent’, ‘relapse’ –5 class brings us ‘severe delay’ 5 class model is in good agreement with Croudace et al (2004) Job’s a good ‘un

38. 38 Conclusions Like EFA, LCA is an exploratory tool with the aim of summarising the variability in the dataset in a simple/interpretable way These results do not prove that there are 4 or 5 groups of children in mere life. LCA will find groupings in the data even if there is no reason to think such groups might exist.

39. 39 Bring on the covariates

40. 40 Predicting class membership One can strengthen the assertion that subjects can be neatly packaged in little groups if one can show that these groups differ with respect to –Co-morbid conditions –Aetiogical factors –Later outcomes Even better if such findings support what is seen in (a) other epidemiological studies, (b) clinical settings

41. 41 Output from LCA: Class distribution Set of probabilities defining the likelihood that each observed pattern can be assigned to each class PatternNormativeDelayedSev delayPersistRelapse 000000.9890.0080.001- 11111-0.00.0260.9690.005 110000.0380.7610.010.0020.189 010100.1050.0880.4730.0140.320

42. 42 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)

43. 43 Save data from Mplus savedata: file is “boys_5class_output_completecase.txt"; save cprob; Don’t forget to add the ID variable: variable: idvariable is ID;

44. 44 Dataset: t1t2t3t4t5IDp(c1)p(c2)p(c3)p(c4)p(c5) Modal class 10000300040.22400.00100.7755 00000300310.0170000.9835 00011300330.17700.4800.3423 00000300420.0170000.9835 00000300500.0170000.9835 2222030051000104 00000300640.0170000.9835 10000300680.22400.00100.7755 00000300700.0170000.9835 00000300720.0170000.9835 00000300950.0170000.9835 22200301060.0010.98900.0102 10000301100.22400.00100.7755 10000301160.22400.00100.7755 00000301190.0170000.9835 22100301400.0530.9430.0040.00102 00000301470.0170000.9835

45. 45 Then what? Merge Mplus output with covariates using ID Mplus will only permit one ID variable. –ALSPAC has one ID to identify parents but two ID’s to identify the kids – parent ID plus another one –Create a composite ID e.g. 1000.1, 1000.2 and then rederive proper ID before matching Read into Stata (or similar)

46. 46 Reshaping the dataset Weighted model requires a reshaping of the dataset so that each child has n-rows (for an n- class model) rather than just 1

47. 47 Pre-shaped – first 20 kids | ID sex dev_18 dev_42 pclass1 pclass2 pclass3 pclass4 pclass5 modclass | |--------------------------------------------------------------------------------------------------| | 30004 male 3..001 0.803 0.197 3 | | 30008 male 2 1.908 0 0.007.085 1 | | 30010 male 2 2.053.001.052 0.894 5 | | 30023 male 1 3.115 0.596.001.288 3 | | 30031 male 3 4 0 0.983 0.016 3 | |--------------------------------------------------------------------------------------------------| | 30033 male 4 4.392 0.397 0.211 3 | | 30042 male 1 3 0 0.983 0.016 3 | | 30050 male 3 2 0 0.983 0.016 3 | | 30051 male 2 2 0 0 0 1 0 4 | | 30057 male 1 3.135 0.002 0.864 5 | |--------------------------------------------------------------------------------------------------| | 30058 male 1 4 0 0.958 0.041 3 | | 30064 male 2 4 0 0.983 0.016 3 | | 30068 male 4 3.001 0.803 0.197 3 | | 30070 male 3 4 0 0.983 0.016 3 | | 30072 male 1 1 0 0.983 0.016 3 | |--------------------------------------------------------------------------------------------------| | 30075 male 3 3 0 0.982 0.018 3 | | 30088 male 3 4.03.002.889.003.076 3 | | 30095 male 3. 0 0.983 0.016 3 | | 30098 male 3..068.158.173.018.583 5 | | 30104 male 4 1.008 0.775 0.217 3 | +--------------------------------------------------------------------------------------------------+

48. 48 Pre-shaped – first 20 kids | ID sex dev_18 dev_42 pclass1 pclass2 pclass3 pclass4 pclass5 modclass | |--------------------------------------------------------------------------------------------------| | 30004 male 3..001 0.803 0.197 3 | | 30008 male 2 1.908 0 0.007.085 1 | | 30010 male 2 2.053.001.052 0.894 5 | | 30023 male 1 3.115 0.596.001.288 3 | | 30031 male 3 4 0 0.983 0.016 3 | |--------------------------------------------------------------------------------------------------| | 30033 male 4 4.392 0.397 0.211 3 | | 30042 male 1 3 0 0.983 0.016 3 | | 30050 male 3 2 0 0.983 0.016 3 | | 30051 male 2 2 0 0 0 1 0 4 | | 30057 male 1 3.135 0.002 0.864 5 | |--------------------------------------------------------------------------------------------------| | 30058 male 1 4 0 0.958 0.041 3 | | 30064 male 2 4 0 0.983 0.016 3 | | 30068 male 4 3.001 0.803 0.197 3 | | 30070 male 3 4 0 0.983 0.016 3 | | 30072 male 1 1 0 0.983 0.016 3 | |--------------------------------------------------------------------------------------------------| | 30075 male 3 3 0 0.982 0.018 3 | | 30088 male 3 4.03.002.889.003.076 3 | | 30095 male 3. 0 0.983 0.016 3 | | 30098 male 3..068.158.173.018.583 5 | | 30104 male 4 1.008 0.775 0.217 3 | +--------------------------------------------------------------------------------------------------+ covariatesPosterior probs Modal class

49. 49 The reshaping reshape long pclass, i(id) j(class) (note: j = 1 2 3 4 5) Data wide -> long --------------------------------------------------------- Number of obs. 5584 -> 27920 Number of variables 66 -> 63 j variable (5 values) -> class xij variables: pclass1 pclass2... pclass5 -> pclass ---------------------------------------------------------

50. 50 Re-shaped – first 3 kids +--------------------------------------------------+ | id sex dev_18 dev_42 pclass class | |--------------------------------------------------| 1. | 30004 male 3..001 1 | 2. | 30004 male 3. 0 2 | 3. | 30004 male 3..803 3 | 4. | 30004 male 3. 0 4 | 5. | 30004 male 3..197 5 | |--------------------------------------------------| 6. | 30008 male 2 1.908 1 | 7. | 30008 male 2 1 0 2 | 8. | 30008 male 2 1 0 3 | 9. | 30008 male 2 1.007 4 | 10. | 30008 male 2 1.085 5 | |--------------------------------------------------| 11. | 30010 male 2 2.053 1 | 12. | 30010 male 2 2.001 2 | 13. | 30010 male 2 2.052 3 | 14. | 30010 male 2 2 0 4 | 15. | 30010 male 2 2.894 5 | +--------------------------------------------------+ First kid Third kid Second kid

51. 51 Multinomial model log using "temperament.log", replace foreach var of varlist activity rhythmicity approach /// adaptability intensity mood persistence distractibility /// threshold { xi: mlogit class `var' [iw = pclass], rrr test `var' xi: mlogit class `var' kz021 [iw = pclass], rrr test `var' xi: mlogit class `var' kz021 b1 b2 b5 b6 b10 b16 b17 i.wisc_70 [iw = pclass], rrr } log close

52. 52 Multinomial model log using "temperament.log", replace foreach var of varlist activity rhythmicity approach /// adaptability intensity mood persistence distractibility /// threshold { xi: mlogit class `var' [iw = pclass], rrr test `var' xi: mlogit class `var' kz021 [iw = pclass], rrr test `var' xi: mlogit class `var' kz021 b1 b2 b5 b6 b10 b16 b17 i.wisc_70 [iw = pclass], rrr } log close Weight by class membership probabilities

53. 53 Typical Output Multinomial logistic regression Number of obs = 9535 LR chi2(4) = 57.66 Prob > chi2 = 0.0000 Log likelihood = -9613.0706 Pseudo R2 = 0.0030 ------------------------------------------------------------------------------ class | RRR Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1 | adaptability | 1.146521.0431225 3.64 0.000 1.065042 1.234232 -------------+---------------------------------------------------------------- 2 | adaptability | 1.180821.0643773 3.05 0.002 1.061151 1.313986 -------------+---------------------------------------------------------------- 3 | adaptability | 1.230596.0459751 5.55 0.000 1.143706 1.324087 -------------+---------------------------------------------------------------- 5 | adaptability | 1.172061.0421842 4.41 0.000 1.092231 1.257727 ------------------------------------------------------------------------------ (class==4 is the base outcome) ( 1) [1]adaptability = 0 ( 2) [2]adaptability = 0 ( 3) [3]adaptability = 0 ( 4) [5]adaptability = 0 chi2( 4) = 57.12 Prob > chi2 = 0.0000

54. 54 Summary Can use Latent Class Analysis to summarise the variability in patterns of response to binary longitudinal repeated measures ~ LLCA [Longitudinal LCA] Can use time invariant covariates to predict class membership in a 2-stage model in Stata using posterior probabilities