Growth Mixture Model

Transcription

1 Growth Mixture Model Latent Variable Modeling and Measurement Biostatistics Program Harvard Catalyst The Harvard Clinical & Translational Science Center Short course, October 28, 2016 Slides contributed by Jeannie-Marie Leoutsakos, Assistant Professor of Psychiatry & Mental Health, Johns Hopkins University

2 Well-used latent variable models Latent variable scale Observed variable scale Continuous Discrete Continuous Factor analysis LISREL Discrete FA IRT (item response) Discrete Latent profile Growth mixture Latent class analysis, regression

3 Outline Motivating Example: The ADAPT trial Two Options for Modeling Heterogeneity 1. Mixed Effects/Growth Curve Models 2. Growth Mixture Models Comparison of Options & Final Thoughts

4 AD and Inflammation AD characterized by β-amyloid plaques and neurofibrillary tangles AD is progressive, with a long preclinical period. Inflammatory processes have been linked to plaque and tangle formation Inflammatory processes also linked to clearance of β-amyloid.

5 AD and NSAIDs In observational studies, NSAID use associated with reduced risk of AD AD treatment trials show no effect of NSAIDs In an MCI prevention trial, NSAID increased risk.

6 The ADAPT Trial Multi-site prevention trial N=2528 Participants 70+, family history of AD 200 mg of celecoxib bid, 220 mg of naproxen sodium bid, or matching placebo (1:1:1.5) Enrollment began in 2001, halted December 2004 due to safety concerns. Study cohort is still being followed.

7 ADAPT 3MS Total

8 Option 1. LGCM i indexes individuals j indexes timepoints Each individual i has her own personal intercept η 0i and personal slope η 1i Y = η + η time + ε ij 0i 1i j ij η = γ + γ age + ζ 0i i η = γ + γ age + γ I( drug ) + γ I( drug ) + ζ 1i i 8

9 Option 1. Mixed Effects Model β 0 +b 0i β 1 t ij +β 5 nap t ij β 0 β 1 t ij + b 1i t ij 3MS β 1 t ij time β 1 t ij + β 3 cel t ij Y ij =γ 00 +ζ 0i + γ 10 t ij +ζ +b 1i t ij i + γ 12 cel t ij + γ 13 nap t ij +ε ij

10 SAS Syntax Random Intercept + Random Slope Model proc mixed data = adapt method=ml covtest; model h=time agec drug1 drug2 time*drug1 time*drug2 /s; random int time/ type = un sub = id; run; Model Random Effects 10

11 Mixed Effects with Quadratic Term β 0 +β 4 β 1 t ij +β 5 nap t ij + β 8 nap t 2 ij β 0 +b 0i β 1 t ij +b 1i t ij + β 6 t ij 2 β 0 β 0 +β 2 β 1 t ij + β 6 t ij 2 3MS time β 1 t ij + β 3 cel t ij + β 7 cel t ij 2 Y ij =β 0 +b 0i + β 1 t ij +b 1i t ij +β 2 cel+ β 3 cel t ij + β 4 nap+ β 5 nap t ij + β 6 t ij2 + β 7 cel t ij2 + β 8 nap t ij2 + ε ij

12 Parameter Estimates

13 Expected Change Over 4 Years

14 Observed and Predicted Trajectories Model I Placebo Celecoxib 3MS Total Naproxen Model II Placebo 3MS Total Celecoxib 3MS Total Naproxen

15 Rates of Decline in Pre-Clinical AD Prior to clinical dx, decline rate not constant Can consider trajectories as being in classes : no, slow, & fast decline.

16 The Timing Hypothesis Contradiction between observational and clinical trials due to differences in timing of exposure to NSAIDs Early/little or no decline: NSAIDs good Later/ substantial decline: NSAIDs bad Observational trials: most individuals in no/slow decline class when exposed Clinical trials: larger proportion of individuals in the fast decline class

17 Testing the Timing Hypothesis Ideal: Stratify individuals by decline class, fit mixed effects models with NSAID effects separately for each class. Problem: we don t know for sure how many classes there are, or who is in each class. Class is a latent variable.

18 Mixed Effects Models as Growth Models Term used in developmental research Growth getting taller, smarter, etc. Fixed effects (intercept, slope, quadratic) referred to as Growth Factors

19 Mixture Models Useful when you believe your population is actually a mixture of subpopulations. Mixture here has nothing to do with Mixed Effects

20 Option 2. Growth Mixture Models Allows for the estimation of a prespecified number of latent classes of trajectories Determined via a combination of substantive theory, fit indices, and bootstrapped likelihood ratio tests. Estimates mixed effects model (growth model) parameters for each latent class

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22 Option 2. LGMM i indexes individuals j indexes timepoints K indexes latent classes Each class k has its own class-specific intercept η 0k and slope η 1k, and classspecific drug-effects Y = η + η time + ε ij 0i 1i j ij η = γ + ζ 0i 0k 0i η = γ + γ Idrug ( ) + γ Idrug ( ) + ζ 1i 10k 11k 1 12k 2 1i 22

23 Growth Mixture Model Parameters For each class (indexed by k), we now have Y ij =γ 0k +ζ 0i + γ 10k t ij +ζ 1i t ij + γ 11k cel t ij + γ 12k nap t ij +ε ij Simultaneously, model probability of membership in each class via multinomial logistic regression - this allows for inclusion of predictors of class membership (e.g., age, such that older individuals have greater probability of membership in the fast-decline class.

24 Step 1. for LGMM Single-class LGCM (no covariates) Assess variability of intercepts and slopes (graphically) Determine whether a quadratic term is needed. Assess the correlation structure among outcomes across time. (eg, is it OK to hold residual variance of Ys constant over time?). 24

25 Step 2. For LGMM Fit Latent Class Growth Analyses Unlike GMM, var(i) and var(s) is fixed at 0. Using the same methods as with LCA (BIC and BLRT) determine the appropriate number of classes Do this with and without covariates 25

26 MPLUS Input for LCGA Specify latent classes Fix I and S variances Estimate I and S separately for each class 26

27 MPLUS Input for BLRT one set of start values for parameter estimation Specify # of starts for BLRT Ask for BLRT 27

28 Choosing Number of Classes Without covariates (and with, not shown) a two-class model does not fit the data as well as a three-class model 28

29 Step 3. Fit GMM First with class invariant I, S variances Also possible to have class-varying I,S variances Add covariates Ponder meaning 29

30 Class-Invariant I S Variances 30

31 Adding Covariates Age as a predictor of class membership Drug1 and Drug 2 modify class-specific* slopes. 31

32 GMM with Covariates Input 32

33 Growth Mixture Model Parameter Estimates

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35 Expected Change Over Time A model where drug effects are forced to be the same across classes fits the data significantly worse (-2LLD: ; p<0.001)

36 Comparison of Options Mixed Effects Assumes one population Simpler interpretation More parsimonious Standard software Results can be more definitive Growth Mixture Model Models subpopulations Complex interpretation More parameters Need larger sample Need $pecial $oftware Results not definitive; post-hoc subgroup analysis

37 Final Thoughts on Growth Mixture Models What does it all mean? possible to get fit indices, etc which support a multi-class mixture when really there are no underlying subgroups. Entails a number of assumptions about the within-person correlation and random effects, results can be highly sensitive to those assumptions Assumptions/model fit difficult to check Hypothesis generating/refining rather than confirming.