MCUAAAR: Methods & Measurement Core Workshop: Structural Equation Models for Longitudinal Analysis of Health Disparities Data April 11th, 2007 11:00 to 1:00 ISR 6050 Thomas N. Templin, PhD Center for Health Research Wayne State University

Many hypotheses concerning health disparities involve the comparison of longitudinal repeated measures data across one or more groups. A chief advantage of this type of design is that individuals act as their own control reducing confounding.

SEM Models for Balanced Continuous Longitudinal Data Early Models (Jöreskog, 1974, 1977) Autoregressive (2-wave or multi-wave) Covariance structure only (means were not modeled) Simplex , Markov, and other models for correlated error structure Contemporary Models Autoregressive models with means structures (Arbuckle, 1996) Growth curve models Latent means with no variance (Joreskog, 1989) Latent factors with means with variance (Tisak & Meridith,1990) Multigroup and Cohort Sequential Designs Latent means and variance modeled separately (Random Effects Mixed Design) (Rovine & Molenaar,2001) Latent change and difference models (McArdle & Hamagami, 2001)

SEM Models for Balanced Continuous Longitudinal Data Contemporary Models (cont.) Growth curve models (cont) Growth models for experimental designs (Muthen &Curran, 1997) Biometric Models (McArdle, et al,1998) Pooled interrupted time series model (Duncan & Duncan, 2004) Latent class GC models (Muthen, M-Plus) Multilevel GC models

MG-Latent Identity Basis Model Unlike the familiar two-wave autoregressive model, latent growth curve and change and difference models involve a different approach to SEM modeling. Many of these models appear to be variations of one another. I formulated what I am calling a multigroup latent identitly basis model (MG-LBM) that serves as a starting point for more specific longitudinal models. I will formulate this for model and then derive latent difference and growth, random effects, and other kinds of models that have appeared in the literature

MG-Latent Basis Model Two Parts Means Structure Covariance Structure Within group coding of within subject contrasts. Test parameters by comparing models with and without equality constraints Between plus within-group coding. Test parameters directly. Covariance Structure Model error directly (replace error covariances with latent factors, etc) Model error indirectly (add latent structure to prediction equations)

Means Structure Notation

Amos Setup: Within-Group Coding of Means Structure For Girls Group

Amos Setup : Within-Group Coding For Boys Group

Within-Group Coding Parameter constraints identified in “manage models” All intercepts are constrained to 0. ib1 = ib2 = ib3 = ib4 = ig1 = ig2 = ig3 = ig4=0

Estimated Means Structure Model for Girls Group

Estimated Means Structure Model for Boys Group

Contrast Coding Across Groups In order to explicitly estimate between group effects and interactions you need one design matrix for within and between effects. The more general coding described next will provide a foundation for this. With 4 repeated measures and 2 groups a total of 8 contrasts or identity vectors are needed. The same 8 means will be estimated but now there is one design matrix across both groups. This is achieved by constraining parameter estimates for each of the 8 identity vectors to be equal across groups

Design Matrix to Code Within and Between Effects

Amos Coding for Means Structure: Girls Group

Amos Coding for Means Structure: Boys Group

Alternate Coding: Girls Group

Alternate Coding: Boys Group

Parameter Constraints Parameter constraints identified in “manage models” All intercepts are constrained to 0. ib1 = ib2 = ib3 = ib4 = ig1 = ig2 = ig3 = ig4=0 Each of the p x q latent means is constrined to equality across group (boys = girls) mb1 = mg1 mb2 = mg2 mb3 = mg3 mb4 = mg4 mb5 = mg5 mb6 = mg6 mb7 = mg7 mb8 = mg8

Estimated Means

Estimated Means

Application This method is used to construct models for cohort sequential designs and for missing value treatments when there are distinct patterns of missingness May be useful for family models where the groups represent families of different sizes or composition

Remember Everything You Used to Know About Coding Regression With this mean structure basis you can now apply any of the familiar regression coding schemes to test contrasts of interest You can use dummy coding, contrast, or effects coding. Polynomial coding is used for growth curve models. Dummy coding will compare baseline to each follow-up measurement Interactions are coded in the usual way as product design vectors Using the inverse transform of Y you can construct contrasts specific to your hypothesis if the standard ones are not adequate.

Dummy Coding to Compare Each Follow-up Measure With the Baseline Measure Note that here we include the unit vector in the dummy coding. In regression, the unit vector is included automatically so you don’t usually think about it.

Amos Setup: Dummy Coding to Compare Each Follow-up Measure With the Baseline Measure

Comments & Interpretation There is nothing intuitive about the coding. It is based on the inverse transform. Here it looks like we are taking the average of all the measures to compare with each follow-up measure. In reality, we really are just comparing baseline (i.e, Y1) with each follow-up measure. The latent means estimate Y1, Y2-Y1, Y3-Y1, and Y4 –Y1. Check this out against the means in the handout

Statistical Tests of Change Contrasts Asymptotic Test Estimate S.E. C.R. P Label f1 21.182 .641 33.067 *** f2 1.045 .360 2.907 .004 f3 2.000 .421 4.750 f4 2.909 .398 7.312

Statistical Tests of Change Contrasts Bootstrapped Tests and 95% CI Parameter Estimate Lower Upper P f1 21.182 20.078 22.517 .010 f2 1.045 .331 1.666 f3 2.000 1.145 2.743 f4 2.909 2.068 3.602

Novel Contrast Using Inverse Transform

Growth Curve Model with Fixed Effects Only Jöreskog, 1989 Girls

Boys

Constraints on model parameters Constraints on Covariance Matrix: Homogeneity of Covariance Assumption b12=g12 b13 = g13 b14 = g14 b23 = g23 b24 = g24 b34 = g34 Intercepts set to zero in both groups m1 = m2 = m3 = m4=0 Y variable variances are set equal within group b1 = b2 = b3 = b4 g1 = g2 = g3 = g4

Compare to Data in Handout Do the slope and intercept estimates look Reasonable for each group?

Part II: Covariance Structure for Correlated Observations Standard techniques like we OLS regression, ANOVA, and MANOVA compare means and leave the correlated error unanalyzed. The SEM approach, and modern regression procedures like HLM, tap the information in the correlation structure. Latent structure can be brought out of the error side or the observed variable side of the model.

Amos Setup: Growth Curve Model with Random Slope and Intercept

Model Constraints Correlations among error terms are fixed to 0 b12=g12=0 b13 = g13=0 b14 = g14=0 b23 = g23=0 b24 = g24=0 b34 = g34=0 b3 = b4 Intercepts fixed to 0. m1 = m2 = m3 = m4 = mg1 = mg2 = mg3 = mg4=0

The covariance among the measures is now accounted for by the random effects

The fixed and random parts can be separated at the latent level The fixed and random parts can be separated at the latent level. The mathematical equivalence of this type of SEM and the hierarchical or mixed effects model with balanced data was shown by Rovine & Molenaar (2001) Extension to other kinds of multilevel or clustered data have appeared in the literature Mixed Model (Rovine & Molenaar, 2001) Girls Latent variable parameters constrained equal across groups Chi Square = 32.467, DF = 20 Chi Square Probability = .039, RMSEA = .158, CFI = .806 .00 y1 0, 1.72 e1 1 y2 e2 y3 e3 y4 e4 0, 2.91 ICEPT 0, .02 Slope .0 2 4 6 -.01 21.21, .00 ICEPT-m .48, .00 slope-m 6.00 4.00 2.00

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.

b14 b24 b13 b23 b12 b34 0, b2 0, b1 0, b3 0, b4 e2 e1 e3 e4 1 1 m2 1 1 m1 m3 m4 y2 y1 y3 y4 Here individual differences in the intercept are modeled as a mediator of health outcome 4 1 2 1 6 1 1 Variable Correlated With Race/Ethnicity ICEPT 1 0, Slope Health Outcome 1 1 1 1 0, 0, 0,

The longitudinal repeated measures advantage only applies for constructs that actually do change over time. In the example below, individual differences only exist in the average score or the intercept resulting in a between groups analysis subject to all the usual confounding. Time Y Change in Y would only be related to other variables by chance. In longitudinal analysis determining the variance in true change is critical but how to do it is somewhat of an issue.

For example, in this figure true change exists at the population level but is constant within groups. Time Y Once group is taken into account there are no individual differences in rate of change. Hence hypotheses concerning change in Y at the group level should be recognized as untestable.

Pooled Interrupted Time Series Analyses Duncan & Duncan, 2004

Amos default Growth model