Latent Class Analysis Karen BandeenRoche October 27 2016

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? • How is LCA fit adjudicated? • What are considerations for identifiability / estimability?

Motivating Example Frailty 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”

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

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

Part I: Model

Latent class model ε 1 Y 1 … Frailty η Ym εm Measurement Structural

Well-used latent variable models Latent variable scale Observed variable scale Continuous Discrete Continuous Factor analysis Discrete FA LISREL IRT (item response) Discrete Latent profile Latent class Growth mixture analysis, regression General software: MPlus, Latent Gold, Win. Bugs (Bayesian), NLMIXED (SAS)

Analysis of underlying subpopulations Latent class analysis POPULATION … P 1 ∏ 11 Y 1 ∏ 1 M … YM Ui PJ ∏J 1 Y 1 ∏JM … YM Lazarsfeld & Henry, Latent Structure Analysis, 1968; Goodman, Biometrika, 1974

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:

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

Latent Variable Models Latent Class Regression (LCR) Model • Model: • Measurement assumptions: – Conditional independence Ø{Yi 1, …, Yi. M} mutually independent conditional on Ui ØReporting heterogeneity unrelated to measured, unmeasured characteristics

Latent Variable Models Latent Class Regression (LCR) Model • Model: • Measurement assumptions: – Conditional independence Ø{Yi 1, …, Yi. M} mutually independent conditional on Ci ØReporting heterogeneity unrelated to measured, unmeasured characteristics

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

Latent class analysis Prediction • 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

Part II: Fitting

Estimation Broad 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: po. LCA • Bayesian: win. Bugs

Estimation Methods 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

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

Estimation Likelihood 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

Estimation EM example: Latent Class Model

EM-Algorithm Latent class model 1. 2. 3. 4. 5. A process of averaging over missing data – in this case, missing data is class membership. 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.

Global and Local Maxima Multiple starting values very important!

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

Example: Latent Frailty Classes Women’s Health and Aging Study Conditional Probabilities (π) Criterion 2 -Class Model CL. 1 “NONFRAIL” CL. 2 “FRAIL” Weight Loss . 073 . 26 Weakness . 088 Slowness 3 -Class Model CL. 1 “ROBUST” CL. 2 “INTERMED. ” CL. 3 “FRAIL” . 072 . 11 . 54 . 51 . 029 . 26 . 77 . 15 . 70 . 004 . 45 . 85 Low Physical Activity . 078 . 51 . 000 . 28 . 70 Exhaustion . 061 . 34 . 027 . 16 . 56 Class Prevalence (P) (%) 73. 3 26. 7 39. 2 53. 6 7. 2 Bandeen-Roche et al. , J Gerontol, 2006

Example: Latent Frailty Classes Women’s Health and Aging Study Conditional Probabilities (π) Criterion 2 -Class Model 3 -Class Model CL. 1 “ROBUST” CL. 1 “NONFRAIL” CL. 2 “FRAIL” Weight Loss . 073 . 26 . 072 . 11 . 54 Weakness . 088 . 51 . 029 . 26 . 77 Slowness . 15 . 70 . 004 . 45 . 85 Low Physical Activity . 078 . 51 . 000 . 28 . 70 Exhaustion . 061 . 34 . 027 . 16 . 56 Class Prevalence (P) (%) 73. 3 26. 7 39. 2 53. 6 7. 2 CL. 2 “INTERMED. ” CL. 3 “FRAIL” We estimate that 26% in the “frail” Subpopulation exhibit weight loss” Bandeen-Roche et al. , J Gerontol, 2006

Part III: Evaluating Fit

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

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

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

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 + P parameters (sum to 1) J*M π parameters (M items*J classes) – Saturated df: 2 M -1 – Goodness of fit df: 2 M – J(M+1)

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

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

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

Conditional Probabilities of Meeting Criteria in Latent Frailty Classes WHAS Criterion 2 -Class Model CL. 1 “NONFRAIL” CL. 2 “FRAIL” Weight Loss . 073 . 26 Weakness . 088 Slowness 3 -Class Model CL. 1 “ROBUST” CL. 2 “INTERMED. ” CL. 3 “FRAIL” . 072 . 11 . 54 . 51 . 029 . 26 . 77 . 15 . 70 . 004 . 45 . 85 Low Physical Activity . 078 . 51 . 000 . 28 . 70 Exhaustion . 061 . 34 . 027 . 16 . 56 Class Prevalence (%) 73. 3 26. 7 39. 2 53. 6 7. 2 Bandeen-Roche et al. , J Gerontol, 2006

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

Example Residual checking • Frailty construct

Part IV: Identifiability / Estimability

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 identifiable iff is (globally)

Identifiability Related concepts • Local identifiability • Basic idea: ϕ identified within a neighborhood • Definition: F is locally identifiable at exists a neighborhood τ about for all τ Φ. if there • Estimability, empirical identifiability: The information matrix for ϕ given y 1, …, yn is non-singular.

Identifiability Latent class (binary Y) • Latent class analysis (measurement only) • Parameter dimension: 2 M -1 • Unconstrained J-class model: J-1 + J*M • Need 2 M ≥ 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*(2 M) (rule of thumb)

Identifiability / estimability Frailty example • Latent class analysis • Need 2 M ≥ 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*(2 M) • n > 10*(25) = 320 - YES

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? • How is LCA fit adjudicated? • What are considerations for identifiability / estimability?