MCUAAAR: Methods

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MCUAAAR Methods Measurement Core
WorkshopStructural Equation Models for
Longitudinal Analysis of Health Disparities Data

• April 11th, 2007
• 1100 to 100ISR 6050Thomas N. Templin,
  PhDCenter for Health ResearchWayne State
  University

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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.
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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)

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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

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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

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MG-Latent Basis Model

• Two Parts
• Means 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)

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Means Structure Notation
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Amos Setup Within-Group Coding of Means
Structure For Girls Group
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Amos Setup Within-Group Coding For Boys Group
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Within-Group Coding
Parameter constraints identified in manage
models All intercepts are constrained to 0. ib1
ib2 ib3 ib4 ig1 ig2 ig3 ig40
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Estimated Means Structure Model for Girls Group
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Estimated Means Structure Model for Boys Group
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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

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Design Matrix to Code Within and Between Effects
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Amos Coding for Means Structure Girls Group
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Amos Coding for Means Structure Boys Group
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Alternate Coding Girls Group
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Alternate Coding Boys Group
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Parameter Constraints
Parameter constraints identified in manage
models All intercepts are constrained to
0. ib1 ib2 ib3 ib4 ig1 ig2 ig3
ig40 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
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Estimated Means
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Estimated Means
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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

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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.

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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 dont usually think
about it.
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Amos Setup Dummy Coding to Compare Each
Follow-up Measure With the Baseline Measure
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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

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Statistical Tests of Change ContrastsAsymptotic
Test
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Statistical Tests of Change ContrastsBootstrapped
Tests and 95 CI
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Novel Contrast Using Inverse Transform
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Growth Curve Model with Fixed Effects
OnlyJöreskog, 1989
Girls
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Boys
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Constraints on model parameters
Constraints on Covariance Matrix Homogeneity of
Covariance Assumption b12g12 b13 g13 b14
g14 b23 g23 b24 g24 b34 g34 Intercepts set
to zero in both groups m1 m2 m3 m40 Y
variable variances are set equal within group b1
b2 b3 b4 g1 g2 g3 g4
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Compare to Data in Handout Do the slope and
intercept estimates look Reasonable for each
group?
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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.

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Amos Setup Growth Curve Model with Random Slope
and Intercept
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Model Constraints
Correlations among error terms are fixed to
0 b12g120 b13 g130 b14 g140 b23
g230 b24 g240 b34 g340 b3
b4 Intercepts fixed to 0. m1 m2 m3 m4
mg1 mg2 mg3 mg40
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The covariance among the measures is now
accounted for by the random effects
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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
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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.
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b24
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b34
b12
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0, b4
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e2
e3
e4
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1
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y2
y3
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Here individual differences in the intercept are
modeled as a mediator of health outcome
4
2
1
6
1
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1
0
Variable Correlated With Race/Ethnicity
Slope
0,
ICEPT
1
0
Health
Outcome
1
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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.
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.
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For example, in this figure true change exists at
the population level but is constant within
groups.
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.
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Pooled Interrupted Time Series Analyses Duncan
Duncan, 2004
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Amos default Growth model