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Latent Class Analysis

Latent Class Analysis. Karen Bandeen-Roche October 27, 2016. Objectives For you to leave here knowing…. When is latent class analysis (LCA) model useful? What is the LCA model its underlying assumptions? How are LCA parameters interpreted? How are LCA parameters commonly estimated?

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Latent Class Analysis

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  1. Latent Class Analysis Karen Bandeen-Roche October 27, 2016

  2. ObjectivesFor you to leave here knowing… • When is latent class analysis (LCA) model useful? • What is the LCA model its underlying assumptions? • How are LCA parameters interpreted? • How are LCA parameters commonly estimated? • How is LCA fit adjudicated? • What are considerations for identifiability / estimability?

  3. Motivating ExampleFrailty of Older Adults “…the sixth age shifts into the lean and slipper’d pantaloon, with spectacles on nose and pouch on side, his youthful hose well sav’d, a world too wide, for his shrunk shank…” -- Shakespeare, “As You Like It”

  4. The Frailty Construct Fried et al., J Gerontol 2001; Bandeen-Roche et al., J Gerontol, 2006

  5. Frailty as a latent variable • “Underlying”:status or degree of syndrome • “Surrogates”: Fried et al. (2001) criteria • weight loss above threshold • low energy expenditure • low walking speed • weakness beyond threshold • exhaustion

  6. Part I: Model

  7. Latent class model ε1 Y1 Frailty Structural … η Ym εm Measurement

  8. Well-used latent variable models General software: MPlus, Latent Gold, WinBugs (Bayesian), NLMIXED (SAS)

  9. Analysis of underlying subpopulationsLatent class analysis POPULATION Ui … P1 PJ ∏11 ∏1M ∏J1 ∏JM … … Y1 YM Y1 YM Lazarsfeld & Henry, Latent Structure Analysis, 1968; Goodman, Biometrika, 1974

  10. Latent Variables: What?Integrands in a hierarchical model • Observed variables (i=1,…,n): Yi=M-variate; xi=P-variate • Focus: response (Y) distribution = GYx(y/x) ; x-dependence • Model: • Yi generated from latent (underlying) Ui: (Measurement) • Focus on distribution, regression re Ui: (Structural) • Overall, hierarchical model:

  11. Latent Variable ModelsLatent Class Regression (LCR) Model • Model: • Structural model: • Measurement model: = “conditional probabilities” > is MxJ • Compare to general form:

  12. Latent Variable ModelsLatent Class Regression (LCR) Model • Model: • Measurement assumptions: • Conditional independence • {Yi1,…,YiM} mutually independent conditional on Ui • Reporting heterogeneity unrelated to measured, unmeasured characteristics

  13. Latent Variable ModelsLatent Class Regression (LCR) Model • Model: • Measurement assumptions: • Conditional independence • {Yi1,…,YiM} mutually independent conditional on Ci • Reporting heterogeneity unrelated to measured, unmeasured characteristics

  14. Analysis of underlying subpopulations Method: Latent class analysis • Seeks homogeneous subpopulations • Features that characterize latent groups • Prevalence in overall population • Proportion reporting each symptom • Number of them = least to achieve homogeneity / conditional independence

  15. Latent class analysisPrediction • Of interest: Pr(C=j|Y=y) = posterior probability of class membership • Once model is fit, a straightforward calculation Pr(C=j|Y=y) = = =ij when evaluated at yi

  16. Part II: Fitting

  17. EstimationBroad Strokes • Maximum likelihood • EM Algorithm • Simplex method (Dayton & Macready, 1988) • Possibly with weighting, robust variance correction • ML software • Specialty: Mplus, Latent Gold • Stata: gllamm • SAS: macro • R: poLCA • Bayesian: winBugs

  18. EstimationMethods other than EM algorithm • Bayesian • MCMC methods (e.g. per Winbugs) • A challenge: label-switching • Reversible-jump methods • Advantages: feasibility, philosophy • Disadvantages • Prior choice (high-dimensional; avoiding illogic) • Burn-in, duration • May obscure identification problems

  19. EstimationLikelihood maximization: E-M algorithm A process of averaging over missing data – in this case, missing data is class membership.

  20. EstimationLikelihood maximization: E-M algorithm • Rationale: LVs as “missing” data • Brief review • “Complete” data • Complete data log likelihood taken as a function of ϕ • Iterate between • (K+1) E-Step: evaluate • (K+1) M-Step: maximize wrt ϕ • Convergence to a local likelihood maximum under regularity Dempster, Laird, and Rubin, JRSSB, 1977

  21. EstimationEM example: Latent Class Model

  22. EM-AlgorithmLatent class model A process of averaging over missing data – in this case, missing data is class membership. 1. Choose starting set of posterior probabilities • Use them to estimate P and π (M-step) • Calculate Log Likelihood • Use estimates of P and π to calculate posterior probabilities (E-step) • Repeat 2-4 until LL stops changing.

  23. Global and Local Maxima Multiple starting values very important!

  24. Example: Frailty Women’s Health & Aging Studies • Longitudinal cohort studies to investigate • Causes / course of physical and cognitive disability • Physiological determinants of frailty • Up to 7 rounds spanning 15 years • Companion studies in community, Baltimore, MD • ≥ moderately disabled women 65+ years: n=1002 • ≤ mildly disabled women 70-79 years: n=436 • This project: n=786 age 70-79 years at baseline • Probability-weighted analyses Guralnik et al., NIA, 1995; Fried et al., J Gerontol, 2001

  25. 2-Class Model Criterion 3-Class Model CL. 1 “NON-FRAIL” CL. 2 “FRAIL” CL. 2 “INTERMED.” CL. 3 “FRAIL” Weight Loss .073 .26 .072 .11 .54 Weakness .088 .51 .029 .26 .77 .15 .70 Slowness .004 .45 .85 CL. 1 “ROBUST” Low Physical Activity .000 .28 .70 .078 .51 Exhaustion .061 .34 .027 .16 .56 Class Prevalence (P) (%) 73.3 26.7 39.2 53.6 7.2 Example: Latent Frailty ClassesWomen’s Health and Aging Study Conditional Probabilities (π) Bandeen-Roche et al., J Gerontol, 2006

  26. 2-Class Model Criterion 3-Class Model CL. 1 “NON-FRAIL” CL. 2 “FRAIL” CL. 2 “INTERMED.” CL. 3 “FRAIL” Weight Loss .073 .26 .072 .11 .54 Weakness .088 .51 .029 .26 .77 .15 .70 Slowness .004 .45 .85 CL. 1 “ROBUST” Low Physical Activity .000 .28 .70 .078 .51 Exhaustion .061 .34 .027 .16 .56 Class Prevalence (P) (%) 73.3 26.7 39.2 53.6 7.2 Example: Latent Frailty ClassesWomen’s Health and Aging Study Conditional Probabilities (π) We estimate that 26% in the “frail” Subpopulation exhibit weight loss” Bandeen-Roche et al., J Gerontol, 2006

  27. Part III: Evaluating Fit

  28. Choosing the Number of Classes • a priori theory • Chi-Square goodness of fit • Entropy • Information Statistics • AIC, BIC, others • Lo-Mendell-Rubin (LMR) • Not recommended (designed for normal Y) • Bootstrapped Likelihood Ratio Test

  29. Entropy Measures classification error 0 – terrible 1 – perfect Ci=j Ci=j Dias & Vermunt (2006)

  30. Information Statistics • s = # of parameters • N= sample size • smaller values are better • AIC: -2LL+2s • BIC: -2LL + s*log(N) BIC is typically recommended - Theory: consistent for selection in model family - Nylund et al, Struct Eq Modeling, 2007

  31. Likelihood Ratio Tests • LCA models with different # of classes NOT nested appropriately for direct LRT. • Rather: LRT to compare a given model to the “saturated” model • LCA df (binary case): J-1 + J*M • Saturated df: 2M -1 • Goodness of fit df: 2M – J(M+1) P parameters (sum to 1) π parameters (M items*J classes)

  32. Bootstrapped Likelihood Ratio Test • In the absence of knowledge about theoretical distribution of difference in –2LL, can construct empirical distribution from data. • per Nylund (2006) simulation studies, performs “best”

  33. Example: Frailty Construct Validation Women’s Health & Aging Studies • Internal convergent validity • Criteriamanifestation is syndromic “a group of signs and symptoms that occur together and characterize a particular abnormality” - Merriam-Webster Medical Dictionary

  34. Validation: Frailty as a syndrome Method: Latent class analysis • If criteria characterize syndrome: • At least two groups (otherwise, no co-occurrence) • No subgrouping of symptoms (otherwise, more than one abnormality characterized)

  35. 2-Class Model Criterion 3-Class Model CL. 1 “NON-FRAIL” CL. 2 “FRAIL” CL. 2 “INTERMED.” CL. 3 “FRAIL” Weight Loss .073 .26 .072 .11 .54 Weakness .088 .51 .029 .26 .77 .15 .70 Slowness .004 .45 .85 CL. 1 “ROBUST” Low Physical Activity .000 .28 .70 .078 .51 Exhaustion .061 .34 .027 .16 .56 Class Prevalence (%) 73.3 26.7 39.2 53.6 7.2 Conditional Probabilities of Meeting Criteriain Latent Frailty Classes WHAS Bandeen-Roche et al., J Gerontol, 2006

  36. Results: Frailty Syndrome Validation • Data: Women’s Health and Aging Study • Single-population model fit: inadequate • Two-population model fit: good • Pearson χ2 p-value=.22; minimized AIC, BIC • Frailty criteria prevalence stepwise across classes—no subclustering • Syndromic manifestation well indicated

  37. ExampleResidual checking • Frailty construct

  38. Part IV: Identifiability / Estimability

  39. Identifiability • Rough idea for “non”-identifiability: More unknowns than there are (independent) equations to solve for them • Definition: Consider a family of distributions The parameter is (globally) identifiable iff

  40. IdentifiabilityRelated concepts • Local identifiability • Basic idea: ϕ identified within a neighborhood • Definition: F is locally identifiable at if there exists a neighborhood τ about for all τΦ. • Estimability, empirical identifiability: The information matrix for ϕ given y1,…,yn is non-singular.

  41. IdentifiabilityLatent class (binary Y) • Latent class analysis (measurement only) • Parameter dimension: 2M -1 • Unconstrained J-class model: J-1 + J*M • Need 2M ≥ J(M+1) (necessary, not sufficient) • Local identifiability: evaluate the Jacobian of the likelihood function (Goodman, 1974) • Estimability: Avoid fewer than 10 allocation per “cell” • n > 10*(2M) (rule of thumb)

  42. Identifiability / estimabilityFrailty example • Latent class analysis • Need 2M ≥ J(M+1) (necessary, not sufficient) • M=5; J=3; • 32 ≥ 3∙(5+1) – YES • By this criterion, could fit up to 9 classes • Local identifiability: evaluate the Jacobian of the likelihood function (Goodman, 1974) • Estimability: n > 10*(2M) • n > 10*(25) = 320 - YES

  43. ObjectivesFor you to leave here knowing… • When is latent class analysis (LCA) model useful? • What is the LCA model its underlying assumptions? • How are LCA parameters interpreted? • How are LCA parameters commonly estimated? • How is LCA fit adjudicated? • What are considerations for identifiability / estimability?

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