thomas n. templin wayne state university center for health research mcuaaar methodology workshop isr...

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)


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


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


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


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


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


Cross-lagged panel data model: Example research question from Arab mother and child coping project

Child Cope

Mother-ChildRelationship

Child Cope


Child Cope


Our recursive model Predicts this

There are howevergood arguments in favor of this


Cross-lagged panel data model (incomplete without covariates)

Child Cope1

Child Cope2

Child Cope3

e11

e21

a1 a2

Mother-ChildRelationship 1



e3 e4

1 1

b1 b2

d1 d2

c1 c2


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


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

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


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


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


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


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


Fixed mean & variance

Est. mean and variance


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


Fixed mean & scaled variance

Est. mean and variance


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


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


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


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


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


Latent Curve Models (LCM) Example: Unconditional linear LCM

Y1 Y2 Y3

e11

e21

e31

Intercept Slope

1 11

0

12


Latent curve model (LCM) specification Level 1 equation:

yit 1i

2i t it

y a b x


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


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


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.


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


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

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


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.


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


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


CFI = .989, RMSEA = .088

d2d1

c_stress

-.28

.58

LATENT DIFFERENCE MODELS



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


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.


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

thomas n. templin wayne state university center for health research mcuaaar methodology workshop isr...

Documents

wave data slide

baseline slide

autoregressive models

z slide

problem slide

b3 slide

addition models

unbalanced data