Latent Growth Modeling Chongming Yang Research Support Center FHSS College.

Latent Growth Modeling

Chongming YangResearch Support Center

FHSS College

Objectives

• Understand the basics of LGM• Learn about some applications• Obtain some hands-on experience

Limitations of Traditional Repeated ANOVA / MANOVA / GLM

• Concern group-mean changes over time • Variances of changes not explicit parameters• List-wise deletion of cases with missing values• Can’t incorporate time-variant covariate

Recent Approaches Individual changes

• Multilevel/Mixed /HL modeling• Generalized Estimating Equations (GEE)• Structural equation modeling (latent growth

(curve) modeling)

Long Format Data Layout—Trajectory(T)(for Multilevel Modeling)

ID DV

Y

Time IVX

1 6.5 0 4.5

1 7.0 1 6.4

1 8.0 2 4.8

1 8.5 3 6.7

2 8.8 0 5.7

2 9.0 1 6.8

2 9.2 2 7.2

2 9.4 3 7.5

…

Run Linear Regression for each case

yit = i + iT + it

– i = individual– T = time variable

Intercept & Slope

Individual Level SummaryLinear Regression

id /class intercept slope …

1 7.72 2.50

2 8.51 3.26

3 7.64 4.07

4 16.25 0.92

5 13.17 1.27

6 11.21 3.85

7 9.05 4.21

8 17.11 1.32

9 15.32 2.11

•

Model Intercepts and Slopes

= i + i = s + s

IF variance of i = 0, Then = i , starting the same

IF variance of s = 0, Then = s, changing the same

Thus variances of i and s are important parameters

Unconditional Growth Model--Growth Model without Covariates

yt = + T + t

= i + i (i = intercept here) = s + s

Estimating Different Trajectories

ID Dependent

Variable

Linear Non-equidistant

Quadratic

curve

Logarithmic

curve

Exponential

curve

1 6.5 0 .0 0 0 0

1 7.0 1 .1 .1 .69 .172

1 8.0 2 .2 .4 1.10 .639

1 8.5 3 .3 .9 1.39 1.909

2 8.8 0 .0 0 0 0

2 9.0 1 .1 .1 .69 .172

2 9.2 2.5 .25 .4 1.10 .639

2 9.4 3 .35 .9 1.39 1.909

…

Conditional Growth Model--Growth Model with Covariates

• yt = i + iT + t3 + t

• i = i + i11 + i22 + i

• i = s + s11 + s22 + s

Note: i=individual, t = time, 1 and 2 = time-invariant covariates, 3 = time-variant covariate. i and I are functions of 1,2…n, yit is also a function of 3i.

Limitations of Multilevel/Mixed Modeling

• No latent variables• Growth pattern has to be specified• No indirect effect• No time-variant covariates

Latent Growth Curve Modeling within SEM Framework

• Data—wide format

id x1 x2 t1y1 t2y1 t3y1 …

1 2 5 1 2 3

2 3 4 3 4 5

3 4 3 6 7 8

Measurement Model of Y y = + +

Slope

y1

d1

1

y2

d2

1

y3

d3

1

y4

d4

1

Specific Measurement Models

• y1 = 1 + 1 + 1

• y2 = 2 + 2 + 2

• y3 = 3 + 3 + 3

• y4 = 4 + 4 + 4

= i + i

= s + s

Unconditional Latent Growth Model

y = + + y = 0 + 1*i + s +

Slope

y1

d1

1

y2

d2

1

y3

d3

1

y4

d4

1

Intercept

1 3

0

211 1

1

Five Parameters to Interpret

• Mean & Variance of Intercept Factor (2)

• Mean & Variance of Slope Factor (2)

• Covariance /correlation between Intercept and Slope factors (1)

Interchangeable Concepts

• Intercept = initial level = overall level• Slope = trajectory = trend = change rate• Time scores: factor loadings of the slope

factor

Growth Pattern Specification(slope-factor loadings)

• Linear: Time Scores = 0, 1, 2, 3 … (0, 1, 2.5, 3.5…)• Quadratic: Time Scores = 0, .1, .4, .9, 1.6• Logarithmic: Time Scores = 0, 0.69, 1.10, 1.39…• Exponential: Time Scores = 0, .172, .639, 1.909, • To be freely estimated: Time Scores = 0, 1, blank, blank…

Intercept/Level

Time2y

e21

Time3y

e3

Time4y

e4

Time1y

e1

1

1

Slope/Trend

1

Time5y

e5

Time-variantCovariate 1



Mediator

Time-invariantCovariate

DistalOutcome

Group

d41

d3

1

d1 1

d21

1 1 1

A latent Growth Model with Covariatesand A Outcome Variable

11

2

1

1

34

Intercept/Level

FactorTime2

FactorTime3

FactorTime4

Latent Growth Modeling of Factors

FactorTime1

1 1 1

Slope/Trend

3

1

21

FactorTime51

4

t1y1

e1

1

1

t1y2

e21

t2y1

e3

1

1

t2y2

e41

t3y1

e5

1

1

t3y2

e61

t4y1

e7

1

1

t4y2

e81

t5y1

e9

1

1

t5y2

e101

Parallel Growths

sy

y1

ey1

0

1

y2

ey2

1

1

y3

ey3

1

y4

ey4

1

iy

sz

z4

ez4

z3

ez3

x2

ez2

z1

ez1

11

1

11

iz

1

0

1

1 1 1

1 11

Cross-lagged Model

Frequency ofSubstance Use

(Baseline)


(3 Months)


(6 Months)


(12 Months)

Quality of Life(Baseline)

Quality of Life(3 Months)



d1 d2 d3

a2

c3

b1 b2

c1 c2

b3

a3a1

Parallel Growth with Covariates

y11 y12 y13 y14

Intercept1

slope1

1 1 1

1 2 3

e111

e121

e131

e141

1

y21 y22 y23 y24

Intercept2

slope2

e21 e22 e23 e24

11

12 3

1 1 1 1

X1

X2

X3

1

1

d1d2

d4d3

Antecedent and Subsequent (Sequential) Processes

s1

y1

e1

0

1

y2

e2

1

y3

e3

1

y4

e4

1

i1

s2

y8

e8

y7

e7

y6

e6

y5

e5

11

1

11

i2

0

1

1

d1 d2

11

1

11 1

1

yTime 1

yTime 2

yTime 3

yTime 4

yTime 5

yTime 6

Level1 Trend1 Level1 Trend1

AddedLevel

AddedTrend

20

1

e11

e21

e31

e4 e5 e6

1

2

1

0

1

11 1

Interrupted Time Series Latent Grwoth Model

1 1

1 1

1

1

01 21

Control Group

Experimental Group

Intercept/Level

Time2y

e21

Time3y

e3

Time4y

e4

Time1y

e1

1

1

Slope/Trend

Time5y

e5

34

Intercept/Level

Time2y

e2

Time3y

e3

Time4y

e4

Time1y

e1

Slope/Trend

Time5y

e5

1

1

1 1 1

34

111

1

1

2

1 1

2

11

1

AddedGrowth1

2 3 4

1 1 1

1

Cohort 1

Cohort 2

Cohort 3

yTime 1

yTime 2

yTime 3

Level1 Trend1

20

1

e11

e21

e31

1

1

1

yTime 2

yTime 3

yTime 4

Level1 ? Trend1 ?

e4 e5 e6

31

1

1 1 1

12

1

yTime 3

yTime 4

yTime 5

Level1 ? Trend1 ?

e7 e8 e9

42

1

1 1 1

13

1

Cohort-Sequential LGM

Piecewise Growth Model

Intercept

y1

e1

y2

e2

y3

e3

y4

e4

Slope1Slope2

1

1

20 1

2

11

1

Slope1

Slope2

Two-part Growth Model(for data with floor effect or lots of 0)

Dummy- Coding 0-1

y1 y2 y3 y4

Intercept1

slope1

1 1 1

1 2 3

e111

e121

e131

e141

1

u1 u2 u3 u4

Intercept2

slope2

e21 e22 e23 e24

11

12 3

1 1 1 1

X1

1

1

d1d2

d4d3

Original Rating 0-4

Continuous Indicators

Categorical Indicators

Mixture Growth Modeling

• Heterogeneous subgroups in one sample• Each subgroup has a unique growth pattern• Differences in means of intercept and slopes are

maximized across subgroups• Within-class variances of intercept and slopes

are minimized and typically held constant across all subgroups

• Covariance of intercept and slope equal or different across groups

Growth Mixtures

T-scores approach

• Use a variable that is different from the one that indicates measurement time to examine individual changes

• Example– Sample varies in age– Measurement was collected over time – Research question: How measurement changes

with age?

Advantage of SEM Approach

• Flexible curve shape via estimation• Multiple processes• Indirect effects • Time-variant and invariant covariates• Model indirect effects• Model growth of latent constructs • Multiple group analysis and test of parameter

equivalence• Identify heterogeneous subgroups with unique

trajectories

Model Specification growth of observed variable

ANALYSIS: MODEL: I S | y1@0 y2@1 y3 y4 ;

Specify Growth Model of Factorswith Continuous Indicators

MODEL: F1 BY y11 y12(1) y13(2); F2 BY y21 y22(1) y23(2); F3 BY y31 y32(1) y33(2); (invariant measurement over time)[Y11-Y13@0 Y21-Y23@0 Y31-Y33@0 F1-F3@0]; (intercepts fixed at 0) I S | F1@0 F2@1 F3 F4 ;

Why fix intercepts at 0 ?

• Y = 1 + F1

• F1 = 2 + Intercept

• Y = (1 = 2 =0) + Intercept

Y

F1

Intercept

Specify Growth Model of Factorswith Categorical Indicators

MODEL: F1 BY y11 y12(1) y13(2); F2 BY y21 y22(1) y23(2); F3 BY y31 y32(1) y33(2);

[Y11$1-Y13$1](3); [Y21$1-Y23$1](4); [Y31$1-Y33$1](5); (equal thresholds)[F1-F3@0]; (intercepts fixed at 0) [I@0]; (initial mean fixed 0, because no objective measurement for I) I S | F1@0 F2@1 F3 F4 ;

Practical Tip

• Specify a growth trajectory pattern to ensure the model runs

• Examine sample and model estimated trajectories to determine the best pattern

Practical Issues• Two measurement—ANCOVA or LGCM with variances

of intercept and slope factors fixed at 0• Three just identified growth (specify trajectory)• Four measurements are recommended for flexibility in • Test invariance of measurement over time when

estimating growth of factors• Mean of Intercept factor needs to be fixed at zero

when estimating growth of factors with categorical indicators

• Thresholds of categorical indicators need to be constrained to be equal over time

Unstandardized or StandardizedEstimates?

• Report unstandardized If the growth in observed variable is modeled,

• If latent construct measured with indicators are , report standardized

Resources• Bollen K. A., & Curren, P. J. (2006). Latent curve models: A structural

equation perspective. John Wiley & Sons: Hoboken, New Jersey• Duncan, T. E., Duncan, S. C., Strycker, L. A., Li, F., & Alpert A. (1999). An

introduction to latent variable growth curve modeling: Concepts, issues, and applications. Lawrence Erlbaum Associates, Publishers: Mahwah, New Jersey

• www.statmodel.com Search under paper and discussion for papers and answers to problems

http://www.statmodel.com/

Practice

1. Estimate an unconditional growth model 2. Compare various trajectories, linear, curve,

or unknown to determine which growth model fit the data best

3. Incorporate covariates4. Use sex or race as grouping variable and test

if the two groups have similar slopes.5. Explore mixture growth modeling

Latent Growth Modeling Chongming Yang Research Support Center FHSS College.

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