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Binary Models 1

Binary Models 1. A (Longitudinal) Latent Class Analysis of Bedwetting. How much fun can you have with 5 binary variables?. Croudace paper:. Gender-specific prevalence and levels of missing data. Latent Class Models. Deal with patterns of response e.g. 11111 = Yes at all five time points

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Binary Models 1

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

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

  3. Croudace paper:

  4. Gender-specific prevalence and levels of missing data

  5. 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.

  6. Latent Class Models

  7. 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)

  8. 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

  9. 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 | +--------------------------------+

  10. 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 | +--------------------------------+

  11. 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

  12. Thresholds Figure from Uebersax webpage

  13. 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

  14. 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

  15. 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

  16. 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????

  17. 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

  18. 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

  19. 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 Success Not there yet

  20. 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

  21. 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

  22. 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

  23. Model results 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 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

  24. 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

  25. These can be plotted • In Excel from the probabilities:

  26. … 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

  27. Yuck!

  28. Model fit stats

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. Model fit stats BLRT 4 class model is adequate fit to data and there is little improvement in fit when a 5th 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

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

  36. 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

  37. 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.

  38. Bring on the covariates

  39. 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

  40. Output from LCA: • Class distribution • Set of probabilities defining the likelihood that each observed pattern can be assigned to each class

  41. 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)

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

  43. Dataset:

  44. 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)

  45. 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

  46. 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 | +--------------------------------------------------------------------------------------------------+

  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 | +--------------------------------------------------------------------------------------------------+ Modal class covariates Posterior probs

  48. 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 ---------------------------------------------------------

  49. 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 Second kid Third kid

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