Presentation on theme: "MCUAAAR: Methods & Measurement Core Workshop: Structural Equation Models for Longitudinal Analysis of Health Disparities Data April 11th, 2007 11:00 to."— Presentation transcript:
1MCUAAAR: Methods & Measurement Core Workshop: Structural Equation Models for Longitudinal Analysis of Health Disparities DataApril 11th, 200711:00 to 1:00 ISR Thomas N. Templin, PhD Center for Health Research Wayne State University
2Many 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.
3SEM 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 structureContemporary ModelsAutoregressive models with means structures (Arbuckle, 1996)Growth curve modelsLatent means with no variance (Joreskog, 1989)Latent factors with means with variance (Tisak & Meridith,1990)Multigroup and Cohort Sequential DesignsLatent means and variance modeled separately (Random Effects Mixed Design) (Rovine & Molenaar,2001)Latent change and difference models (McArdle & Hamagami, 2001)
4SEM 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
5MG-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
6MG-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 constraintsBetween plus within-group coding.Test parameters directly.Covariance StructureModel error directly (replace error covariances with latent factors, etc)Model error indirectly (add latent structure to prediction equations)
13Contrast 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
14Design Matrix to Code Within and Between Effects
19Parameter Constraints Parameter constraints identified in “manage models”All intercepts are constrained to 0.ib1 = ib2 = ib3 = ib4 = ig1 = ig2 = ig3 = ig4=0Each of the p x q latent means is constrined to equality across group (boys = girls)mb1 = mg1mb2 = mg2mb3 = mg3mb4 = mg4mb5 = mg5mb6 = mg6mb7 = mg7mb8 = mg8
22ApplicationThis method is used to construct models for cohort sequential designs and for missing value treatments when there are distinct patterns of missingnessMay be useful for family models where the groups represent families of different sizes or composition
23Remember 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 interestYou 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 measurementInteractions are coded in the usual way as product design vectorsUsing the inverse transform of Y you can construct contrasts specific to your hypothesis if the standard ones are not adequate.
24Dummy 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.
25Amos Setup: Dummy Coding to Compare Each Follow-up Measure With the Baseline Measure
26Comments & 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
34Constraints on model parameters Constraints on Covariance Matrix: Homogeneity of Covariance Assumptionb12=g12b13 = g13b14 = g14b23 = g23b24 = g24b34 = g34Intercepts set to zero in both groupsm1 = m2 = m3 = m4=0Y variable variances are set equal within groupb1 = b2 = b3 = b4g1 = g2 = g3 = g4
36Compare toData in HandoutDo the slope and intercept estimates lookReasonable for each group?
37Part 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.
38Amos Setup: Growth Curve Model with Random Slope and Intercept
39Model Constraints Correlations among error terms are fixed to 0 b12=g12=0b13 = g13=0b14 = g14=0b23 = g23=0b24 = g24=0b34 = g34=0b3 = b4Intercepts fixed to 0.m1 = m2 = m3 = m4 = mg1 = mg2 = mg3 = mg4=0
40The covariance among the measures is now accounted for by the random effects
42The 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 literatureMixed Model (Rovine & Molenaar, 2001)GirlsLatent variable parameters constrained equal across groupsChi Square = , DF = 20Chi Square Probability = .039, RMSEA = .158, CFI = .806.00y10, 1.72e11y2e2y3e3y4e40, 2.91ICEPT0, .02Slope.0246-.0121.21, .00ICEPT-m.48, .00slope-m6.004.002.00
43If 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.
44b14b24b13b23b12b340, b20, b10, b30, b4e2e1e3e411m211m1m3m4y2y1y3y4Here individual differences in the intercept are modeled as a mediator of health outcome4121611Variable CorrelatedWith Race/EthnicityICEPT10,SlopeHealthOutcome11110,0,0,
45The 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.TimeYChange 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.
46For example, in this figure true change exists at the population level but is constant within groups.TimeYOnce 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.
47Pooled Interrupted Time Series Analyses Duncan & Duncan, 2004