THOMAS N. TEMPLIN WAYNE STATE UNIVERSITY CENTER FOR HEALTH RESEARCH MCUAAAR METHODOLOGY WORKSHOP ISR MARCH 14, 2009 Overview of Latent Variable Longitudinal SEM

Intro Classification of models for continuous outcomes for longitudinal panel data (short series) Autoregressive CFA based/random coefficients Autoregressive and CFA together Other models for continuous outcomes Biometric model Family and twin data Latent first difference model Cohort Sequential (accelerated longitudinal design) Latent class growth curve analysis (Muthen)

Intro Other models Generalized SEM models Dichotomous, count, and censored outcomes Time series Latent Markov modeling Survival analysis Latent class analysis

Intro Advantages of SEM for longitudinal analyses Causal analysis not limited to control by partial correlation Mediation Reciprocal causation Instrumental variables Moderator analysis is facilitated using multigroup SEM Time varying covariates Psychometric analysis can determine quality of measurement measurement invariancedo measures change their meaning over time? Complex and correlated error structures not problem

Intro Disadvantages Traditional ANOVA and MANOVA approaches are more powerful, simpler to perform, more trusted, and most of all easier to explain in research reports for publication. Variation in trajectories can be explored by including individual characteristics as factors in the design. Unbalanced data handled better within a regression framework like HLM. Example: unbalanced data M-Plus claims to handle unbalanced data so stay tuned to new developments The tradition approaches and the SEM approaches are both useful for the same kind of balanced data.

Autoregressive CFA based/random coefficients Autoregressive and CFA together Other examples Latent first difference model Cohort Sequential (accelerated longitudinal design) Part I Some models in more detail

Autoregressive models Heise (1969) Reliability of a single item can be determined even when true score changes if you have three waves of data Joreskog (1977, 1979) Autoregressive models with many complex error structures- simplex, factor, etc. Popular stability of alienation example Wothke (2000) Time dependent process specification Cross-lagged panel data model (2-wave, multi-wave data) Reciprocal causation Cross-lagged panel data model with first differencing (3-wave data)

Autoregressive models Time dependent process specification Time is a proxy for unmeasured variables. Example: Cued recall of event memory

Autoregressive models General Markov process (linear) Stable process b1 = b2 = b3

Autoregressive models Reciprocal-causation Parental monitoring and child behavior problems (r =.29 to.35) Blood pressure self-care behavior and systolic blood pressure (-.10) Cross-lagged panel data model

Autoregressive models Cross-lagged panel data model A series of chi-square difference tests enables selection of parsimonious model, for example, c1 = c2 = c3, or d1 = d2 = d3 = 0. Y1Y2Y3Y4 e1 1 e2 1 e3 1 a1a2a3 X1X2X3X4 e4 e5 e b1b2b3 d1 d2 d3 c1 c2 c3

Autoregressive models Cross-lagged panel data model with covariate, Z, constant over time Y1Y2Y3Y4 e1 1 e2 1 e3 1 X1X2X3X4 e4 e5 e Z

Autoregressive models Cross-lagged panel data model with time dependent covariate Y1Y2Y3Y4 e1 1 e2 1 e3 1 X1X2X3X4 e4 e5 e Z2 Z3 Z4

Autoregressive models Cross-lagged panel data model: Example research question from Arab mother and child coping project Our recursive model Predicts this There are however good arguments in favor of this

Autoregressive models Cross-lagged panel data model (incomplete without covariates)

Autoregressive models The models described were single group and used observed variables as outcomes. In addition models can include: Multiple group analysis Interaction effects Different models for different racial/ethnic groups Multiple indicators at each wave of measurement Allows estimation of reliability and appropriate path coefficient adjustment for unreliability Psychometric assessment of measurement invariance Multiple Covariates Time invariant covariates, gender, or personal characteristics Time varying covariates, household income. Complex error structures

GROUP 1 GROUP 2

Autoregressive models: Example relevant to development and aging research Age related changes in relationship between quality of mother-child relationship and child coping Measurement invariance time cohort Convergence

Cohort: 12 years old at baseline Cohort: 13 years old at baseline

Cohort: 12 years old at baseline Cohort: 13 years old at baseline Cohort: 14 years old at baseline

Growth curve models (Meredith & Tisak, 1984, 1990) Latent difference/true score model (Steyer, et al, 2000,1997) Latent contrast analysis (general contrast specification, not described in literature yet) CFA based/random coefficients

Unconditional random coefficients growth curve model

CFA based/random coefficient models Motivating example Prenatal cocaine exposure and school aged outcomes study Many outcomes collected over several years, 10, 11, 12, 13, 15 Achenbach CBCL, YSR, self esteem, school grades Many covariates (mothers age, income, marital status, exposure to violence in home, in neighborhood, educational level, perinatal care and nutrition, other substance abuse history) Analysis at each time point would have generated a lot of correlated results Solution: slope and status analysis reduces the repeated measures to two meaning statistics Statisticians have been doing this for years

CFA based/random coefficient models Advantage of CFA approach to growth curve modeling The CFA approach provides appropriate standard errors, and allows all the other powerful SEM modeling techniques to be usedpsychometrics, mediation, multigroup analysis, constant and time varying covariates, and most recently latent classes analysis (Muthen). Also appealing because you can model data at the group level and the individual level (group trajectory and individual predictors of variance in the group trajectory)-multilevel interpretation Analysis is based on a simple idea that statisticians have used for many yearsregression analysis of summary statistics

CFA based/random coefficient models Disadvantages Routine applications require data uniformly spaced across individuals This may be changing (see Mplus). Practical limitation on number of longitudinal variables in one analysis

Exploratory Confirmatory All loadings are estimated Pattern of loadings is fixed- others estimated Evolution from EFA to CFA to random coefficients

Confirmator Random coefficients Pattern of loadings is fixed others estimated All loadings are fixed in most models Evolution from EFA to CFA to random coefficients

Confirmator Random coefficients Fixed mean & variance Est. mean and variance Evolution from EFA to CFA to random coefficients

Confirmator Random coefficients Fixed mean & scaled variance Est. mean and variance Evolution from EFA to CFA to random coefficients

Unconditional random coefficients growth curve model

Latent difference model (Steyer et al, 2000)

Unconditional random coefficients latent contrast model

These unconditional random coefficient models are just the starting point: Conditional random coefficient model Level 2 specification of unconditional model

Conditional LCM specification Level 2 model: Additional continuous or categorical predictors, (x qi ) are added to the regression equations.

Conditional LCM standardized estimates

If the latent factors have sufficient variance, they can be used as variables in a more comprehensive model. Here the intercept has substantial variance but the slope does not. Individual differences in the intercept could be an important predictor of health outcome.

Here individual differences in the intercept are modeled as a mediator of health outcome m1 y1 0, b1 e1 1 m2 y2 0, b2 e2 1 m3 y3 0, b3 e3 1 m4 y4 0, b4 e4 1 b34 b14 b12 b24 b23 b13 ICEPT Slope Health Outcome 0, Variable Correlated With Race/Ethnicit y 0, 1

Dual change score (DCS) model ( McArdle & Hamagami, 2001) Second order latent growth curve and dual change score models (Emilio Ferrer, et al, 2008) Bollen & Curran - Autoregression and CFA combined

Autoregressive and CFA combined Dual change score model McArdle & Hamagami, 2001 Status and slope factor can be interpreted as random coefficient as in growth curve models

Dual change score model (McArdle & Hamagami, 2001) set up to estimate all parameters (mean and variance of status and slope, a, b, and common error variance (8 parameters, 9 moments including means).

Additive slope parameter a set to 1 for identification purposes

General strategy common to random coefficient CFA methods Define latent measures of status and change The latent measures are a function of the repeated observed variables One measure of status and one or more measures of change Status What is the mean level of the outcome overall or at some predetermined point (baseline, end of treatment) Change What kind of change is meaningful? Polynomial trends –latent curve analysis (Meredith & Tisak) Change scores or contrasts (McArdle, Steyer, et al, latent contrast analysis?) Piecewise trends Are the measures of change significant? Do status and change vary in relation to individual characteristics or other variables?

CFA / random coefficients Modeling extensions Piecewise coding SEM features Multiple variables Multiple groups Multiple indicators Bootstrapping Missing value imputation Full information maximum likelihood Multiple imputation Designs that span time by bridging longitudinal and cross sectional data (accelerated longitudinal designs) Missing data by design Models that combine autoregression and CFA

CFA / random coefficients Limitations Data limitations Consistent time intervals Need a manageable number of distinct patterns See M-plus advancements (say tuned) Difference coding and contrast coding Many kinds of research questions are better posed in terms of differences and contrasts rather than curves

Biometric models (McArdle, 1986) Latent first difference model Cohort-sequential model (described by Duncan et al.,1999) Specially designed models

Latent first difference model 2-wave SEM developed for complex model of stress, coping and adjustment (in preparation) Latent change is defined by multiple difference score indicators Causal model controls for confounders that are constant. Models A -> B Allows many variables and mediated relations Similar to econometric models using first differences except that change is defined at the construct level Takes unreliability of the difference scores into account.

Maternal Social Support Structural Model Maternal Stressors Maternal Approach Coping Child Avoidance Coping Child Stress Maternal Distress Parent-Child Relationship Child Behavior Problems Child Approach Coping Family Socio- Demogra phic Risk Maternal Avoidance Coping Child Social Support

The reliability of the change score construct is evaluated statistically by the factor loadings of the change score variables

DISCUSSION AND CONCLUSIONS How to choose a longitudinal SEM?

Many model choices Two basic types --autoregressive and random coefficients and some creative adaptations Each type can be expanded for the purpose of the analysis to include Multiple repeated measures variables with, for example, the slope of one affecting the intercept of the other and visa versa. Multiple indicators of the repeated measures construct can be included or constructed for multi-item scales for psychometric evaluation of measurement invariance Multiple groups can be used to examine interaction Causal analysis can be facilitated by including appropriate mediating variables and covariates Covariates may be constant or time varying Many creative applications appear in the literature

LATENT CURVE MODELS Very general class of models considered synonymous with what I am calling CFA/random coefficients See Bollen & Curran (2006) for an excellent description of the model, how to code it, and interesting extensions. See Duncan et al (1999) for introduction and many interesting applications Handout Part II CFA / random coefficients: Basic equations and model setup 2 2 This part was given as a handout for self study

LATENT CURVE MODELS Very general class of models considered synonymous with what I am calling CFA/random coefficients See Bollen & Curran (2006) for an excellent description of the model, how to code it, and interesting extensions. See Duncan et al (1999) for introduction and many interesting applications Part II CFA / random coefficients: Basic equations and model setup

CFA / random coefficients Latent Curve Models (LCM) Example: Unconditional linear LCM

CFA / random coefficients Latent curve model (LCM) specification Level 1 equation:

CFA / random coefficients Latent curve model (LCM) specification Level 1 equation: While this equation gives a regression line for each case, it is a theoretical equation and the factor scores, 1i, and, 2i, are not really estimated as part of the model. They are latent variables and can be estimated after the fact using factor score estimation equations. The variances of the, terms are estimated parameters. The other model parameters of interest are in the level two equations. The random coefficients are latent variables

CFA / random coefficients Latent Curve Models (LCM) Unconditional linear LCM: Level 1 error , 0 2 3, 0 2 1, 0 i3 i2 i1

CFA / random coefficients Latent curve model (LCM) specification Level 2 equations for the unconditional LCM: μ 1 is the population intercept and ζ 1i is difference between an individuals intercept and the population intercept. μ 2 is the population slope and ζ 2i i s the difference between the individual slope and the population slope. If there is a significant amount of variance in these intercept and/or slope differences, additional variables, like, age, gender, education could be added to the equations in order to account for this variance individual trajectories.

CFA / random coefficients Latent curve model (LCM) specification Level 2 equation errors: variance covariance matrix of level 2 equation errors 1i and 2i These terms, factor variance and covariance, give the variance of individual difference in status and slope.

CFA / random coefficients Latent Curve Models (LCM) Example: Unconditional linear LCM 1 2 In the unconditional model, the factor variance and the Level 2 error variance are the same so no terms are needed to graph the model. μ 1, 1 2 μ 2,

Degrees of freedom: Data points= 3(3+1)/2 + 3 =9 Free parameters = v1, v2,v3, i-m, s-m, i-v, s-v, cov = 6 df= 9-6 = 3

Parameter estimates and standard errors

Model predicted and observed trajectory y2(mean)=34.96+(-1.24)1.50=33.065

Conditional LCM specification Level 2 model: Additional continuous or categorical predictors, (x qi ) are added to the regression equations.

Conditional LCM specification 1 2

Conditional LCM example results

Conditional LCM standardized estimates

LATENT DIFFERENCE MODELS CFA / random coefficients

Latent difference models Latent differences vs. Latent curves Intervention studies Pre vs. Post treatment Intercept may be at beginning or end of an interval Naturalistic designs Start of season vs. End of season During the school year vs. Over the summer Years pre vs. Year post event- 5 years post retirement, 10 years post retirement Nonlinear curves Separate trend components may be harder to interpret than change scores over specific intervals

CFA / random coefficients Latent difference models (Steyer, et al, 2000) Two types presented Change from baseline Change in successive intervals

LATENT CONTRAST ANALYSIS (NOT IN LITERATURE YET) Any meaningful contrasts of the type used in MANOVA can be constructed to address specific research questions The Steyer et al approach is a special case The inverse transform is used to generate SEM coding Many different error structures can be used More work on identification is needed. CFA / random coefficients

Model fit with latent contrasts

Example true score trajectories No individual differences in rate of change

Dual change score (DCS) model ( McArdle & Hamagami, 2001) Second order latent growth curve and dual change score models (Emilio Ferrer, et al, 2008) Bollen & Curran - Autoregression and CFA combined