Latent Growth Modeling Using Mplus Friday Harbor Psychometrics Workshop Richard N. Jones 1, 2, Frances M. Yang 1, 2, Douglas Tommet 1 1 Institute for Aging Research, Hebrew Senior. Life and Beth Israel Deaconess Medical Center, Division of Gerontology 2 Harvard Medical School jones@hrca. harvard. edu September 1, 2009 Corrected 9/2/2009 1
Acknowledgements • Funded in part by Grant R 13 AG 030995 -01 A 1 from the National Institute on Aging • The views expressed in written conference materials or publications and by speakers and moderators do not necessarily reflect the official policies of the Department of Health and Human Services; nor does mention by trade names, commercial practices, or organizations imply endorsement by the U. S. Government. 2 2
Session Overview • • Other Resources General Framework Comparison with Random Effects Modeling Framework Special Model Considerations Some Results from ROS Detailed Example from ROS Questions and Discussion 3
Other Resources • What is longitudinal data analysis? – Singer JD & Willett JB. Applied longitudinal data analysis: Modeling change and event occurrence. 2003, New York: Oxford University Press. (Also see worked examples at UCLA ATS) • How do I do latent growth curve modeling? – Duncan TE, Duncan SC, & Strycker LA. An introduction to latent variable growth curve modeling: concepts, issues and applications. Second ed. 2006, Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc. • Tell me more about the math behind latent curve methods – Bollen KA & Curran PJ. Latent curve models: a structural equation perspective. Wiley series in probability and statistics. 2006, Hoboken, N. J. : Wiley-Interscience. • Our workshop – 2009 workshop was LDA, come back in 2011 – http: //sites. google. com/site/lvmworkshop for slides, syntax, data 4
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Five+ Approaches to LDA in Mplus • • • Latent growth curve model Random effects model Multilevel model Latent change/Dual change score model Autoregressive/latent simplex model Latent Structural Models (Hyperbolic functions for learning data) 6
Random Effects and Latent Growth Curves: Same But Different • Reconceptualize random effects as latent variables • Use multivariate record layout (wide) • Main difference: – RE: time is data – LGC: time is a model parameter …unless it is data 7
Advantages HLM and Mixed Effect Modeling • Software for highly nested multilevel data better developed • Easier to get model fit and diagnostics • Use time-varying weights Note: HLM Hierarchical Linear Modeling LGC modeling • Embed in more complex models • Flexible curve shape (time is a parameter and/or data) • Modification Indices can help with misspecified models 8
Latent Growth Curve Models • Latent Growth Curve (LGC) modeling is just like CFA • Reconceptualize “factors” as “random effects” • Factor loadings are – (usually) not estimated but given by design or data, and – relate to the sequence of repeated observations • The action is in the mean structure part of the model (factor means, item means, factor variances) as opposed to factor loadings 9
Latent Growth Model (Linear Change) 10
Latent Growth Model (Linear Change) 11
Latent Growth Model (Linear Change) 12
Latent Growth Model (Linear Change) “*” Implies parameter freely estimated. All other parameters are held constant to the indicated value. 13
Latent Growth Model (Linear Change) 14
Latent Growth Model (Linear Change) 15
Latent Growth Model (Linear Change) 16
Change in Ordinal Outcome 17
Model a Retest Effect 18
Model a Retest Effect that is dependent on baseline 19
Regress Change on Baseline “*” Implies parameter freely estimated. All other parameters are held constant to the indicated value. 20
Multiple Indicator Growth Model 21
Growth Mixture Model 22
Alternative Time Bases 23
Model Building (LGC) † unless required to specify the time basis, e. g. , age group 24
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Post-Estimation Fit Evaluation • • • Save Factor Scores Import into stat package Compute expected scores Graph Residuals Empirical r-square 26
Example: PW 2008 • ROS • Change in global cognition (globcog) • Random effects growth mixture model – Time basis: Age centered at baseline mean within age group – Retest effect (occasion basis) – Mixture model part for growth parameters • Covariates: age group dummies (to define age metric) 27
Random Effects Mixture Model 28
TITLE: ROS GLOBCOG SINGLE CLASS 8/16/2009 DATA: FILE = __000001. dat ; VARIABLE: NAMES = y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 y 11 y 12 y 13 y 14 y 15 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 t 11 t 12 t 13 t 14 t 15 cagecat 1 cagecat 2 cagecat 3 cagecat 5 cagecat 6 projid ; MISSING ARE ALL (-9999) ; IDVARIABLE = projid ; TSCORES = t 1 -t 15 ; ANALYSIS: TYPE = random ; COVERAGE =. 02 ; MODEL: i s | y 1 -y 15 AT t 1 -t 15 ; r by y 2 -y 15 @1 ; [r] ; r@0 ; i s on cagecat 1 -cagecat 6* y 1 -y 15 *0. 1 (theta_1) ; ; 29
TITLE: ROS GLOBCOG Trajectories 8/26/2009 DATA: FILE = __000001. dat ; VARIABLE: NAMES = y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 y 11 y 12 y 13 y 14 y 15 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 t 11 t 12 t 13 t 14 t 15 cagecat 1 cagecat 2 cagecat 3 cagecat 5 cagecat 6 projid ; MISSING ARE ALL (-9999) ; IDVARIABLE = projid ; TSCORES = t 1 -t 15 ; CLASSES = c(3) ; ANALYSIS: TYPE = mixture random ; COVERAGE =. 02 ; STARTS = 0 ; PROCESSORS=2 ; ALGORITHM=integration ema ; INTEGRATION = montecarlo ; MCONVERGENCE = 0. 01 ; SAVEDATA: FILE = c: workrosposteddataderivedgmm 3 class 10 AUG 2009. dat ; SAVE = fscores cprob ; RESULTS = c: workrosposteddataderivedgmm 3 class 10 AUG 2009_results. dat ; 30
MODEL: %OVERALL% i s | y 1 -y 15 AT t 1 -t 15 ; r by y 2 -y 15 @1 ; [r] ; r@0 ; i on cagecat 1*. 714 cagecat 2*. 661 cagecat 3*. 224 cagecat 5*-. 397 cagecat 6*-. 764 ; s on cagecat 1*. 068 cagecat 2*. 06 cagecat 3*. 015 cagecat 5*-. 058 cagecat 6*-. 071 ; %c#1% [i*-. 31125 s*-. 27805 r*. 187] ; i*. 734 s*. 367 r@0 ; i with s *0 ; y 1 -y 15 *. 0375 (theta_1) ; %c#2% [i*-. 519 s*-. 622 r*. 187] ; i*. 734 s*. 734 r@0 ; i with s *0 ; y 1 -y 15 *. 075 (theta_2) ; %c#3% [i*-. 83 s*-1. 245 r*. 187] ; i*. 734 s*1. 468 r@0 ; i with s *0 ; y 1 -y 15 *. 15 (theta_3) ; 31
Figure 1. Cognitive Change Trajectories by Class and Age Group at First Observation as implied by Mixture Model Parameter Estimates 63% 27% 11% Education (and race/ethnicity, baseline mental status) associated with class Membership. But not age, not sex. 32
Figure 2. Burden of Amyloid and Tangle Neuropathology by Class Membership (N=326) 33
Trajectory Classes and Reserve • Neuropathology at autopsy does not perfectly account for membership in one of two population sub-groups experiencing substantial cognitive decline • Education, a proxy for cognitive reserve, may buffer the functional consequences of neuropathology: 34
A Different SEM model for Change The Latent Change Score Model 35 35
Latent Change Score Model TITLE: LCSM DATA: FILE = BLAH. dat ; VARIABLE: NAMES = y 1 y 2 ; MODEL: dy by y 2 @1 ; [y 1*] ; y 1* ; y 2 on y 1 @1 ; dy on y 1 * ; dy* ; [dy*] ; [y 2@0] ; y 2 @0 ; NB: As parameterized, just identified or saturated model = zero degrees of freedom. Just as many knowns as estimated parameters. 36 36
ex 03 -08. inp 37 37
Advantages Dual Change Score • More flexibility for estimating lagged and leading effects Latent GC modeling • Better fit (good for descriptive analysis) • Better for sequential patterns 38
Extensions to the LCSM • Dual Change Score Model • Bivariate CSM – Two outcomes are of interest • Multiple Indicator LCSM – change in a latent variable • Multiple Indicator Dual Change Score Model 39 39
Dual Change Score Model 40 40
Help Coding in Mplus … from Zhiyong Zhang, Ph. D. http: //www. psychstat. org/us/sort. php/7. htm%20 -%20 Programs 41 41
FHL 2009 Potential Topics • Replicate ROS analysis in MAP, and/or • Domain-specific REMM • Change-point model (LGCMM or REMM) – Reserve proxies associated with when cognitive decline starts and how fast decline occurs. 42
Questions Discussion 43
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