mcuaaar: methods & measurement core workshop: structural equation models for longitudinal...
TRANSCRIPT
MCUAAAR: Methods & Measurement Core Workshop:Structural Equation Models for Longitudinal Analysis of Health Disparities Data
April 11th, 200711:00 to 1:00
ISR 6050
Thomas N. Templin, PhDCenter 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
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
Yi i
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1
2
3
4
i1 i1 i1 i1
Yi
y i1
y i2
y i3
y i4
E T 04x4
E T EcovY 4x4
EY E 4x1
Amos Setup: Within-Group Coding of Means Structure For Girls Group
ig1
y1
0, g1e1
1 ig2
y2
0, g2e2
1 ig3
y3
0, g3e3
1 ig4
y4
0, g4e4
1
g34
g14
g12
g24
g23
g13
,0m2
,0
m3
1 1
,0
m1
1
,0
m4
1
Amos Setup : Within-Group Coding For Boys Group
ib1
y1
0, b1e1
1 ib2
y2
0, b2e2
1 ib3
y3
0, b3e3
1 ib4
y4
0, b4e4
1
b34
b14
b12
b24
b23
b13
,0m2
,0
m3
1 1
,0
m1
1
,0
m4
1
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
MG-LBM Model (within group coding)Girls
Chi Square = .000, DF = 0Chi Square Probability = \p, RMSEA = \rmsea, CFI = 1.000
.00
y1
0, 4.10e1
1 .00
y2
0, 3.29e2
1 .00
y3
0, 5.08e3
1 .00
y4
0, 5.40e4
1
4.97
3.96
3.05
3.71
3.66
3.94
22.23, .00m2
23.09, .00
m3
1 1
21.18, .00
m1
1
24.09, .00
m4
1
Estimated Means Structure Model for Boys Group
MG-LBM Model (within group coding)Boys
Chi Square = .000, DF = 0 , Chi Square Probability = \p, RMSEA = \rmsea, CFI = 1.000
.00
y1
0, 5.64e1
1 .00
y2
0, 4.28e2
1 .00
y3
0, 6.59e3
1 .00
y4
0, 4.08e4
1
3.04
1.51
2.15
2.63
2.06
3.40
23.81, .00m2
25.72, .00
m3
1 1
22.88, .00
m1
1
27.47, .00
m4
1
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
Girls: Y1
y11
y12
y13
y14
Boys: Y2
y21
y22
y23
y24
1 2 3 4 5 6 7 8
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
Amos Coding for Means Structure: Girls Group
ig1
y1
0, g1e1
1 ig2
y2
0, g2e2
1 ig3
y3
0, g3e3
1 ig4
y4
0, g4e4
1
g34
g14
g12
g24
g23
g13
mg2, 0
m2
mg3, 0
m3
1 1
mg1, 0
m1
1
mg4, 0
m4
mg5, 0
m5
mg6, 0
m6
mg7, 0
m7
mg8, 0
m8
1
0 0 0 0
Amos Coding for Means Structure: Boys Group
ib1
y1
0, b1e1
1 ib2
y2
0, b2e2
1 ib3
y3
0, b3e3
1 ib4
y4
0, b4e4
1
b34
b14
b12
b24
b23
b13
mb2, 0
m2
mb3, 0
m3
0 0
mb1, 0
m1
0
mb4, 0
m4
mb5, 0
m5
mb6, 0
m6
mb7, 0
m7
mb8, 0
m8
0
1 1 1 1
Alternate Coding: Girls Group
ig1
y1
0, g1e1
1 ig2
y2
0, g2e2
1 ig3
y3
0, g3e3
1 ig4
y4
0, g4e4
1
g34
g14
g12
g24
g23
g13
mg2, 0
m2
mg3, 0
m3
1 1
mg1, 0
m1
1
mg4, 0
m4
mg5, 0
m5
mg6, 0
m6
mg7, 0
m7
mg8, 0
m8
1
Alternate Coding: Boys Group
ib1
y1
0, b1e1
1 ib2
y2
0, b2e2
1 ib3
y3
0, b3e3
1 ib4
y4
0, b4e4
1
b34
b14
b12
b24
b23
b13
mb2, 0
m2
mb3, 0
m3
mb1, 0
m1
mb4, 0
m4
mb5, 0
m5
mb6, 0
m6
mb7, 0
m7
mb8, 0
m8
1 1 1 1
Parameter ConstraintsParameter 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 = mg1mb2 = mg2mb3 = mg3mb4 = mg4
mb5 = mg5 mb6 = mg6mb7 = mg7mb8 = mg8
Estimated MeansMG-LBM Model
Girls
Chi Square = .000, DF = 0Chi Square Probability = \p, RMSEA = \rmsea, CFI = 1.000
.00
y1
0, 4.10e1
1 .00
y2
0, 3.29e2
1 .00
y3
0, 5.08e3
1 .00
y4
0, 5.40e4
1
4.97
3.96
3.05
3.71
3.66
3.94
22.23, .00
m2
23.09, .00
m3
1 1
21.18, .00
m1
1
24.09, .00
m4
22.88, .00
m5
23.81, .00
m6
25.72, .00
m7
27.47, .00
m8
1
Estimated Means
MG-LBM ModelBoyss
Chi Square = .000, DF = 0 , Chi Square Probability = \p, RMSEA = \rmsea, CFI = 1.000
.00
y1
0, 5.64e1
1 .00
y2
0, 4.28e2
1 .00
y3
0, 6.59e3
1 .00
y4
0, 4.08e4
1
3.04
1.51
2.15
2.63
2.06
3.40
22.23, .00
m2
23.09, .00
m3
21.18, .00
m1
24.09, .00
m4
22.88, .00
m5
23.81, .00
m6
25.72, .00
m7
27.47, .00
m8
1.00 1.00 1.00 1.00
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
1 2 3 4
y1 1 0 0 0
y2 1 1 0 0
y3 1 0 1 0
y4 1 0 0 1
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
0
y1
0, g1e1
1 0
y2
0, g2e2
1 0
y3
0, g3e3
1 0
y4
0, g4 e41
g34
g14
g12
g24
g23
g13
,0f1
,0
f2,0
f3,0
f4
1
1
11
11 1
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
Change From Baseline ModelDummy Coding of Lambda
kappa(i+1) = Time(i+1)mean - Time(1)meanGirls
Chi Square = .000, DF = 0Chi Square Probability = \p, RMSEA = \rmsea, CFI = 1.000
0
y1
0, 4.10e1
1 0
y2
0, 3.29e2
1 0
y3
0, 5.42e3
1 0
y4
0, 5.40 e41
5.14
3.96
3.05
3.71
3.82
3.88
21.18, .00f1
1.05, .00
f22.00, .00
f32.91, .00
f4
1.00
1.00
1.001.00
1.001.00 1.00
Statistical Tests of Change ContrastsAsymptotic 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 ContrastsBootstrapped Tests and 95% CI
Parameter Estimate Lower Upper P
f1 21.182 20.078 22.517 .010
f2 1.045 .331 1.666 .010
f3 2.000 1.145 2.743 .010
f4 2.909 2.068 3.602 .010
Novel Contrast Using Inverse Transform
A
y1 y2 y3 y4
1 1 1 1
1 1 0 0
0 0 1 1
0 0 0 1
, inverse:
1 2 3 4
y112
12
12
1
y212
12
12
1
y3 0 0 1 1
y4 0 0 0 1
Novel Coding Using Inverse TransformGirls
Chi Square = .000, DF = 0, Chi Square Probability = \p,RMSEA = \rmsea, CFI = 1.000
0
y1
0, 4.10e1
1 0
y2
0, 3.29e2
1 0
y3
0, 5.42e3
1 0
y4
0, 5.40 e41
5.14
3.96
3.05
3.71
3.82
3.88
-3.86, .00f1
-1.05, .00 f2-.91, .00
f324.09, .00
f4
1.00.50
1.00
1.001.00
.50
.50 -.50
.50.50
1.00
Growth Curve Model with Fixed Effects OnlyJöreskog, 1989
m1
y1
0, g1
e11 m2
y2
0, g2
e21 m3
y3
0, g3
e31 m4
y4
0, g4
e41
g34
g14
g12
g24
g23
g13
,0ICEPT
,0Slope
1 1
1 0
12
4
6
Girls
m1
y1
0, b1
e11 m2
y2
0, b2
e21 m3
y3
0, b3
e31 m4
y4
0, b4
e41
b34
b14
b12
b24
b23
b13
,0ICEPT
,0Slope
1 1
1 0
12
4
6
Boys
Constraints on model parameters
Constraints on Covariance Matrix: Homogeneity of Covariance Assumptionb12=g12b13 = g13b14 = g14b23 = g23b24 = g24b34 = g34
Intercepts set to zero in both groupsm1 = m2 = m3 = m4=0
Y variable variances are set equal within groupb1 = b2 = b3 = b4g1 = g2 = g3 = g4
Growth Curve ModelGirls
Joreskog & Sorbom (1989, LISREL 7 User Guide, 2nd Ed., p 261)
Chi Square = 11.454, DF = 16Chi Square Probability = .781, RMSEA = .000, CFI = 1.000
.00
y1
0, 3.70
e11 .00
y2
0, 3.70
e21 .00
y3
0, 3.70
e31 .00
y4
0, 3.70
e41
3.45
2.94
2.97
3.08
3.15
2.90
21.23, .00ICEPT
.48, .00Slope
1.00 1.00
1.00 .00
1.002.00
4.00
6.00
Growth Curve ModelBoys
(Pottoff & Roy, 1964)Joreskog & Sorbom (1989, LISREL 7 User Guide, 2nd Ed., p 261)
Chi Square = 11.454, DF = 16 , Chi Square Probability = .781, RMSEA = .000, CFI = 1.000
.00
y1
0, 5.84
e11 .00
y2
0, 5.84
e21 .00
y3
0, 5.84
e31 .00
y4
0, 5.84
e41
3.45
2.94
2.97
3.08
3.15
2.90
22.60, .00ICEPT
.79, .00Slope
1.00 1.00
1.00 .00
1.002.00
4.00
6.00
Compare toData 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.
mg1
y1
0, g1
e11
mg2
y2
0, g2
e21
mg3
y3
0, g3
e31
mg4
y4
0, g4
e41
g34
g14
g12
g24
g23
g13
ICEPT Slope
10
12
1
4
1
6
Amos Setup: Growth Curve Model with Random Slope and Intercept
Model Constraints
Correlations among error terms are fixed to 0
b12=g12=0b13 = g13=0b14 = g14=0b23 = g23=0b24 = g24=0b34 = 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
Growth Curve Model (Tisak & Meridith, 1990)Girls
Chi Square = 10.144, DF = 11, Probability = .517,RMSEA = .000, CFI = 1.000
.00
y1
0, .89
e11
.00
y2
0, .50
e21
.00
y3
0, .39
e31
.00
y4
0, .17
e41
.00
.00
.00
.00
.00
.00
21.24, 2.88ICEPT
.48, .02
Slope
.14
1.00.00
1.002.00
1.00
4.00
1.00
6.00
Growth Curve ModelBoys
Chi Square = 10.144, DF = 11 Probability = .517RMSEA = .000, CFI = 1.000
.00
y1
0, 3.78
e11
.00
y2
0, 2.36
e21
.00
y3
0, 2.55
e31
.00
y4
0, 2.55
e41
.00
.00
.00
.00
.00
.00
22.51, 2.11ICEPT
.81, -.01
Slope
.06
1.00.00
1.002.00
1.00
4.00
1.00
6.00
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 groupsChi Square = 32.467, DF = 20
Chi Square Probability = .039, RMSEA = .158, CFI = .806
.00
y1
0, 1.72
e11 .00
y2
0, 1.72
e21 .00
y3
0, 1.72
e31 .00
y4
0, 1.72
e41
.00
.00
.00
.00
.00
.00
0, 2.91ICEPT
0, .02
Slope
1 11
.012 4
6
-.01
21.21, .00
ICEPT-m
1 111
.48, .00
slope-m
6.004.00
2.00.00
m1
y1
0, b1
e11 m2
y2
0, b2
e21
m3
y3
0, b3
e31
m4
y4
0, b4
e41
b34
b14
b12
b24
b23
b13
ICEPT Slope0
1 214
1 6
0
HealthOutcome
0,
1
10,
10,
1
1
If the latent factors have sufficientvariance, 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
e11 m2
y2
0, b2
e21
m3
y3
0, b3
e31
m4
y4
0, b4
e41
b34
b14
b12
b24
b23
b13
ICEPT Slope0
1 21
4
16
0
HealthOutcome
0,
1
10,
10,
1
1Variable Correlated With Race/Ethnicity
0,1
Time
Y
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.
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
0
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.
Change From Baseline ModelDummy Coding of Lambda
kappa(i+1) = Time(i+1)mean - Time(1)meanGirls
Chi Square = 7.615, DF = 5 Chi Square Probability = .179,RMSEA = .229, CFI = .946
0y1
0e1
1.00
0y2
0e2
1.00
0y3
0
e31.00
0y4
0e4
1.00
21.18, .00f1
1.05, .00
f22.00, .00
f32.91, .00
f4
1.00
1.00
1.001.00
1.001.00 1.00
.74 1.16 .95
0, 2.56
r21
0, .75
r31
0, .75
r41
0, 2.56
r11