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THOMAS N. TEMPLINWAYNE STATE UNIVERSITY
CENTER FOR HEALTH RESEARCH
MCUAAARMETHODOLOGY WORKSHOP
ISRMARCH 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 invariance—do 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 examplesLatent first difference modelCohort Sequential (accelerated longitudinal design)
Part ISome 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
Y1
Y2 Y3
Y4
e11
e21
e3
1
b2
b3c2
b1
c3
c1
Time is a proxy for unmeasured variables. Example: Cued recall of event memory
Autoregressive modelsGeneral Markov process (linear)
Stable process b1 = b2 = b3
Y1 Y2 Y3 Y4
e11
e21
e31
b1 b2 b3
y t bt 1yt 1 e t
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.
Y1 Y2 Y3 Y4
e11
e21
e31
a1 a2 a3
X1 X2 X3 X4
e4 e5 e6
1 1 1
b1 b2 b3
d1d2 d3
c1 c2 c3
Autoregressive models
Cross-lagged panel data model with covariate, Z, constant over time
Y1 Y2 Y3 Y4
e11
e21
e31
X1 X2 X3 X4
e4 e5 e6
1 1 1
Z
Autoregressive models
Cross-lagged panel data model with time dependent covariate
Y1 Y2 Y3 Y4
e11
e21
e31
X1 X2 X3 X4
e4 e5 e6
1 1 1
Z2 Z3 Z4
Autoregressive models
Cross-lagged panel data model: Example research question from Arab mother and child coping project
Child Cope
Mother-ChildRelationship
Child Cope
Mother-ChildRelationship
Child Cope
Mother-ChildRelationship
Our recursive model Predicts this
There are howevergood arguments in favor of this
Autoregressive models
Cross-lagged panel data model (incomplete without covariates)
Child Cope1
Child Cope2
Child Cope3
e11
e21
a1 a2
Mother-ChildRelationship 1
Mother-ChildRelationship 1
Mother-ChildRelationship 1
e3 e4
1 1
b1 b2
d1 d2
c1 c2
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
X1 X2 X3
Z2 Z3
Y1
1
1 1 1
Y2
1
1 1 1
Y3
1
1 1 1
GROUP 1
GROUP 2
X1 X2 X3
Z2 Z3
Y1
1
1 1 1
Y2
1
1 1 1
Y3
1
1 1 1
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
Z2 Z3
Y12
1
1 1 1
Y13.5
1
1 1 1
Y15
1
1 1 1
X12
1
111
X13.5
1
111
X15
1
111
1
Z2 Z3
Y13
1
1 1 1
Y14.5
1
1 1 1
Y16
1
1 1 1
X13
1
111
X14.5
1
111
X16
1
111
1
Cohort: 12 years old at baseline
Cohort: 13 years old at baseline
Z2 Z3
Y12
1
1 1 1
Y13.5
1
1 1 1
Y15
1
1 1 1
X12
1
111
X13.5
1
111
X15
1
111
1
Z2 Z3
Y13
1
1 1 1
Y14.5
1
1 1 1
Y16
1
1 1 1
X13
1
111
X14.5
1
111
X16
1
111
1
Z2 Z3
Y14
1
1 1 1
Y15.5
1
1 1 1
Y17
1
1 1 1
X141
111
X15.51
111
X171
111
1
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
Intercept
y1
1
1
y2
1
1
y3
1
1
Slope
y4 y5 y6
4
1
5
1
6
1
1
1
13
1
0
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 used—psychometrics, 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 years—regression 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
ExploratoryExploratory ConfirmatoryConfirmatory
All loadings are estimated
Pattern of loadings is fixed- others estimated
Evolution from EFA to CFA to random coefficients
F1
y11
y21
y31
F2
y4
y5
y6
1
1
1
0 00
000
F1
y11
y21
y31
F2
y4
y5
y6
1
1
1
Confirmator Confirmator Random coefficientsRandom coefficients
Pattern of loadings is fixed others estimated
All loadings are fixed in most models
Evolution from EFA to CFA to random coefficients
F1
y11
y21
y31
F2
y4
y5
y6
1
1
1
0 00
000
F1
y11
1
y21 1
y3
1
1
F2
y4
y5
y6
3
1
4 1
5
1
0 1 2
111
Confirmator Confirmator Random coefficientsRandom coefficients
Fixed mean & variance
Est. mean and variance
Evolution from EFA to CFA to random coefficients
Mean = 0, Var = 1
F1
y1
0,1
y2
0,1
y3
0,1
Mean = 0, Var = 1
F2
y4
0,
y5
0,
y6
0,
1
1
1
0 00
000
Mean = ?, Var. = ?
F1
y1
0,
1
1
y2
0,
1 1
y3
0,1
1
Mean = ?, Var = ?
F2
y4
0,
y5
0,
y6
0,
3
1
4 1
5
1
0 1 2
111
Confirmator Confirmator Random coefficientsRandom coefficients
Fixed mean & scaled variance
Est. mean and variance
Evolution from EFA to CFA to random coefficients
Mean = ?, Var. = ?
F1
y1
0,
1
1
y2
0,
1 1
y3
0,1
1
Mean = ?, Var = ?
F2
y4
0,
y5
0,
y6
0,
3
1
4 1
5
1
0 1 2
111
Mean = 0, Var scaled by y1
F1
y1
0,
11
y2
0,1
y3
0,1
Mean = 0, Var scaled by y6
F2
y4
0,
y5
0,
y6
0,
1
1
11
0 00
000
Unconditional random coefficients growth curve model
Intercept
y1
1
1
y2
1
1
y3
1
1
Slope
y4 y5 y6
4
1
5
1
6
1
1
1
13
1
0
Latent difference model (Steyer et al, 2000)
N=454Chi Square = 2.126, DF = 5Chi Square Probability = .831CFI = 1.000, RMSEA = .000
0
int
0, 36.95
e11 -2.51
ext
0, 79.79
e21 0
int2
0, 9.68
e3-1.15
ext2
0, 97.45
e41 1
0
int3
0, 27.13
e5-.26
ext3
0, 93.93
e61 1
21.76, 116.76E1
-2.39, 54.44E2-E1
-3.35, 56.41E3-E1
36.20
-29.01
.741.00
1.00 .74
.741.001.00 .74
.741.00
61.75 71.09
53.44
-27.82
Unconditional random coefficients latent contrast model
"Standard" Dummy Coding of Lambdakappa(i+1) = Time(i)mean - Time(4)mean
A change from endpoint modelGirls
Pottoff & Roy (1964) DataChi Square = .000, DF = 0
Chi Square Probability = \p, RMSEA = \rmsea, CFI = 1.000
0
y1
0, 4.10e11 0
y2
0, 3.29e21 0
y3
0, 5.08e31 0
y4
0, 5.40 e41
4.97
3.96
3.05
3.71
3.66
3.94
24.09, .00f1
-2.91, .00
y1-y4
-1.86, .00
y2-y4
-1.00, .00
y3-y4
1.001.00 1.00
1.00
1.001.001.00
These unconditional random coefficient models are just the starting point: Conditional random coefficient model
Level 2 specification of unconditional model
Conditional LCM specification
1 i 1
1x1i 1 i
2 i 2
2x1i 2i
Level 2 model:
Additional continuous or categorical predictors, (xqi) are added to the regression equations.
Conditional LCM standardized estimates
achprob achprob2 achprob3
e1 e2 e3
Intercept Slope
.88 .87.87
.00
.27.51
N=454Chi Square = 13.508, DF = 3Chi Square Probability = .004
CFI = .989, RMSEA = .088
d2d1
c_stress
-.28
.58
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
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
[t] s z 1]-Y[t f Y[t]
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).
0
ParentalMonitoring
True Score T1
0
mntr
0, e1
e11 1
0
ParentalMonitoring
True Score T2
0
mntr2
0, e1
e21
1
0
mntr3
0, e1
e31
0
ParentalMonitoring
True Score T3
1
Dual Change Score Model (McArdle & Hamagami, 2001)
0
T2 -T1
0
T3-T2
Slope
Status1
1
1
1
b
b
1
a
a
Additive slope parameter ‘a’ set to 1 for identification purposes
0
ParentalMonitoring
True Score T1
0
mntr
0, e1
e11 1
0
ParentalMonitoring
True Score T2
0
mntr2
0, e1
e21
1
0
mntr3
0, e1
e31
0
ParentalMonitoring
True Score T3
1
Dual Change Score Model (McArdle & Hamagami, 2001)
0
T2 -T1
0
T3-T2
Slope
Status1
1
1
1
b
b
1
1
1
Chi Square = 23.795, DF = 2Chi Square Probability = .000CFI = .963, RMSEA = .155
additive constant 'a' set to 1 for identification
0
ParentalMonitoring
True Score T1
0
mntr
0, 8.89
e11.00 1
0
ParentalMonitoring
True Score T2
0
mntr2
0, 8.89
e21.00
1
0
mntr3
0, 8.89
e31
0
ParentalMonitoring
True Score T3
1.00
Dual Change Score Model (McArdle & Hamagami, 2001)
0
T2 -T1
0
T3-T2
-9.27, 142.54
Slope
4.41, 29.73
Status
-65.04
1.00
1.00
1.00
1.00
2.08
2.08
1.00
1.00
1.00
Chi Square = 35.726, DF = 3Chi Square Probability = .000CFI = .944, RMSEA = .155
additive constant 'a' set to 1 for identification
0
ParentalMonitoring
True Score T1
0
mntr
0, 8.67
e11.00 1
0
ParentalMonitoring
True Score T2
0
mntr2
0, 8.67
e21.00
1
0
mntr3
0, 8.67
e31
0
ParentalMonitoring
True Score T3
1.00
Dual Change Score Model (McArdle & Hamagami, 2001)
0
T2 -T1
0
T3-T2
-.20, 3.20
Slope
4.42, 31.87
Status
-7.29
1.00
1.00
1.00
1.00
.00
.00
1.00
1.00
1.00
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
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 SocialSupport
Structural ModelMaternalStressors
Maternal Approach Coping
Child Avoidance Coping
Child Stress
MaternalDistress
Parent-ChildRelationship
ChildBehaviorProblems
Child Approach Coping
Family Socio-Demographic Risk
Maternal Avoidance Coping
Child SocialSupport
The reliability of the change score construct is evaluated statistically by the factor loadings of the change score variables
No PDS: Evaluate factor loadingsChi Square (307.886 ) / DF (210 ) = (1.466 )Chi Square Probability = .000RMSEA = .030, 90%CI = (.022, .037), CFI=.968, PCLOSE =1.000
MaternalStressors
Maternal ActiveCoping
Child Avoidance Coping
ChildStressors
MaternalDistress Mother-Child
Relationship
ChildBehaviorProblems
Child Active Coping
Maternal Avoidance Coping
pom_totdif cesd_totdif
Mother's SocialSupport
Child's SocialSupport
.09
.50
.14
-.36
-.11
-.41
.04.31
-.01
-.08
.64
.39
.07
.06.02
.20
.23 .28
.77.85
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 MODELSVery general class of models considered synonymous with
what I am calling CFA/random coefficientsSee 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
setup2
2This part was given as a handout for self study
LATENT CURVE MODELSVery general class of models considered synonymous with
what I am calling CFA/random coefficientsSee 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 IICFA / random coefficients: Basic equations and model
setup
CFA / random coefficients
Latent Curve Models (LCM) Example: Unconditional linear LCM
Y1 Y2 Y3
e11
e21
e31
Intercept Slope
1 11
0
12
CFA / random coefficients
Latent curve model (LCM) specification Level 1 equation:
yit 1i
2i t it
y a b x
CFA / random coefficients
Latent curve model (LCM) specification Level 1 equation:
yit 1i
2i t it
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
Y1 Y2 Y3
e11
e21
e31
1 11
0
12
12
22 , 0 2
3 , 021 , 0 i3i2i1
CFA / random coefficients
Latent curve model (LCM) specification Level 2 equations for the unconditional LCM:
μ1 is the population intercept and ζ1i is difference between an individuals intercept and the population intercept. μ2 is the population slope and ζ2i is 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.
11
2 12
21 222
CFA / random coefficients
Latent Curve Models (LCM) Example: Unconditional linear LCM
Y1 Y2 Y3
e11
e21
e31
1 11
0
12
12
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.
μ1, 12 μ2, 2
2
212
Degrees of freedom:Data points=3(3+1)/2 + 3 =9Free parameters =v1, v2,v3, i-m, s-m, i-v, s-v, cov = 6df= 9-6 = 3
0
achprob
0
achprob2
0
achprob3
0, v1
e11
0, v2
e21
0, v3
e31
i-m, i-v Intercepts-m, s-v
Slope
1 11
0
1.52.8
cov
0
achprob
0
achprob2
0
achprob3
0, 123.71
e11
0, 108.92
e21
0, 71.51
e31
34.96, 352.47 Intercept-1.24, 13.34
Slope
1.00 1.001.00
.00
1.502.80
-10.88
N=454Chi Square = 2.276, DF = 1
Chi Square Probability = .131CFI = .998, RMSEA = .053
Parameter estimates and standard errors
Parameter estimates and standard errors
0
achprob
0
achprob2
0
achprob3
0, 146.14
e11
0, 102.55
e21
0, 87.13
e31
34.91, 330.08 Intercept-1.23, 7.86
Slope
1.00 1.001.00
.00
1.502.80
N=454Chi Square = 3.211, DF = 2
Chi Square Probability = .201CFI = .998, RMSEA = .037
Model predicted and observed trajectory
y2(mean)=34.96+(-1.24)1.50=33.065
Conditional LCM specification
1 i 1
1x1i 1 i
2 i 2
2x1i 2i
Level 2 model:
Additional continuous or categorical predictors, (xqi) are added to the regression equations.
Conditional LCM specification
0
achprob
0
achprob2
0
achprob3
0, v1
e11
0, v2
e21
0, v3
e31
i-m Intercepts-m
Slope
1 11
0
1.52.8
0,d2
0,d1
c_stress
1 1
1 2
Conditional LCM example results
0
achprob
0
achprob2
0
achprob3
0, 107.75
e11
0, 118.21
e21
0, 63.41
e31
14.55 Intercept.82
Slope
1.00 1.001.00
.00
1.502.80
N=454Chi Square = 13.508, DF = 3Chi Square Probability = .004
CFI = .989, RMSEA = .088
0, 14.33d2
0, 237.52d1
10.74, 32.71
c_stress
-.19
1.90
11
Conditional LCM standardized estimates
achprob achprob2 achprob3
e1 e2 e3
Intercept Slope
.88 .87.87
.00
.27.51
N=454Chi Square = 13.508, DF = 3Chi Square Probability = .004
CFI = .989, RMSEA = .088
d2d1
c_stress
-.28
.58
LATENT DIFFERENCE MODELS
CFA / random coefficients
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
N=454Chi Square = 2.126, DF = 5Chi Square Probability = .831CFI = 1.000, RMSEA = .000
0
int
0, 36.95
e11 -2.51
ext
0, 79.79
e21 0
int2
0, 9.68
e3-1.15
ext2
0, 97.45
e41 1
0
int3
0, 27.13
e5-.26
ext3
0, 93.93
e61 1
21.76, 116.76E1
-2.39, 54.44E2-E1
-3.35, 56.41E3-E1
36.20
-29.01
.741.00
1.00 .74
.741.001.00 .74
.741.00
61.75 71.09
53.44
-27.82
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 caseThe inverse transform is used to generate SEM codingMany different error structures can be usedMore work on identification is needed.
CFA / random coefficients
Model fit with latent contrasts
0
ysr_int
0
ysr_int2
0
ysr_int3
0, 36.25
e11
0, 17.09
e21
0, 28.40
e31
10.88, 31.04 Time 1 1.66, .00
T1-T2
1.00 1.001.00
-1.00
N=454Chi Square = 1.967, DF = 2
Chi Square Probability = .374CFI = 1.000, RMSEA = .000
.10, .00
T2-T3
-1.00
-1.00
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
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