Latent Class Analysis - Harvard University

1 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 1.

2 Y1. frailty Structural . Ym m Measurement Well-used Latent variable models Latent Observed variable scale 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, WinBugs (Bayesian), NLMIXED (SAS). Analysis of underlying subpopulations Latent Class Analysis POPULATION. Ui P1 PJ. 11 1M J1 JM. Y1 YM Y1 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 = GYYxx(y/x). ( y x ) ; x-dependence Model: Yi generated from Latent (underlying) Ui: F (y U ) = u (Measurement). , x; ). Y U ,x Focus on distribution, regression re Ui: F (u x; ) (Structural).

3 Ux Overall, hierarchical model: FY x ( y x ) = FY U ,x ( y U = u, x )dFU x (u x ). Latent Variable Models Latent Class Regression (LCR) Model J M. Model: fY x ( y x ) = Pj mj ym (1 mj )1 ym j =1 m =1. Structural model: [U x ] = Pr{U. i i i = j } = Pr{ = j } = Pj , j = 1,.., J. Measurement model: [Yi Ui ]. mj = Pr{Yim = 1Ui = j } = Pr{Yim = 1 i = j }. = conditional probabilities . > is MxJ. Compare to general form: FY x ( y x ) = FY U ,x ( y U = u, x )dFU x (u x ). Latent Variable Models Latent Class Regression (LCR) Model Model: J M 1 y m fY x ( y x ) = Pj mj ym (1 ). mj j =1 m =1. Measurement assumptions: [Yi Ui ]. Conditional independence {Yi1, ,YiM} mutually independent conditional on Ui Reporting heterogeneity unrelated to measured, unmeasured characteristics Latent Variable Models Latent Class Regression (LCR) Model Model: J M 1 y m fY x ( y x ) = Pj mj ym (1 ).

4 Mj j =1 m =1. Measurement assumptions: [Yi Ci ]. Conditional independence {Yi1, ,YiM} mutually independent conditional on Ci Reporting 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) = Pr (C = j, Y = y ). Pr (Y = y ). M. Pj mjym (1 mj ). 1 ym = m =1. J m ym 1 ym k mk (1 mk ). P. k =1.. m =1. = ij when evaluated at yi part II: Fitting Estimation Broad Strokes Maximum likelihood EM Algorithm Simplex method (Dayton & Macready, 1988).

5 Possibly with weighting, robust variance correction ML software Specialty: Mplus, Latent Gold Stata: gllamm SAS: macro R: poLCA. Bayesian: winBugs Estimation Methods other than EM algorithm Bayesian MCMC methods ( 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 W = {Y , x, u}. Complete data log likelihood = w ( | w). = log Fy ,u|x ( y , u | x, ) taken as a function of . Iterate between [. (K+1) E-Step: evaluate Q( | ( k ) ) = Eu| y , x w ( | W ) | y, x; ( k ) ].

6 (K+1) M-Step: maximize Q( | (k ) ) wrt . Convergence to a local likelihood maximum under regularity Dempster, Laird, and Rubin, JRSSB, 1977. Estimation EM example: Latent Class Model m . J 1 y im . J. max L = log Pj mj (1 mj ) + Pj y im i =1 j =1 m =1 j =1. L n ij (yim mj ) n yim ij : = 0 mj = n mj i =1 mj (1 mj ) i =1.. h =1. hj L n 1. : { ij Pj n}= 0 Pj = ij Pj i =1 n EM-Algorithm Latent Class model A process of averaging over missing data in this case, missing data is Class membership. 1. Choose starting set of posterior probabilities 2. Use them to estimate P and (M-step). 3. Calculate Log Likelihood 4. Use estimates of P and to calculate posterior probabilities (E-step). 5. 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.

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

8 FRAIL . FRAIL We estimate that 26% in the frail . Subpopulation exhibit weight loss . Weight Loss .073 .26 .072 .11 .54. Weakness .088 .51 .029 .26 .77. Slowness .15 .70 .004 .45 .85. Low Physical .078 .51 .000 .28 .70. Activity Exhaustion .061 .34 .027 .16 .56. Class Prevalence (P). (%). 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 N J % % . Pr( SCi i=j = j | Yi ) *log Pr( SCi i=j = j | Yi ) . E = 1 i =1 j =1 . N *log( J ). Dias & Vermunt (2006). Information Statistics s = # of parameters N= sample size smaller values are better AIC: -2LL+2s BIC: -2LL + s*log(N).

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

10 - Merriam-Webster Medical Dictionary 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). Conditional Probabilities of Meeting Criteria in Latent frailty Classes WHAS. Criterion 2- Class Model 3- Class Model CL. 1 CL. 2 CL. 1 CL. 2 CL. 3. NON- FRAIL ROBUST INTERMED. FRAIL . FRAIL . Weight Loss .073 .26 .072 .11 .54. Weakness .088 .51 .029 .26 .77. Slowness .15 .70 .004 .45 .85. Low Physical .078 .51 .000 .28 .70. Activity Exhaustion .061 .34 .027 .16 .56. Class Prevalence (%). 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 2 p-value=.