Latent Growth Modeling

Chongming Yang Research 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 2 2 2 1 1 1 1 2 DV Y 6.5

7.0

8.0

8.5

8.8

9.0

9.2

9.4

Time 1 2 3 0 1 2 3 0 IV X 4.5

6.4

4.8

6.7

5.7

6.8

7.2

7.5



y

it

=



i

Run Linear Regression for each case +



i

T +



it

– i = individual – T = time variable

Intercept & Slope

• id /class 1 2 3 4 5 6 7 8 9

Individual Level Summary

Linear Regression … intercept 7.72

8.51

7.64

16.25

13.17

11.21

9.05

17.11

15.32

slope 2.50

3.26

4.07

0.92

1.27

3.85

4.21

1.32

2.11

Model Intercepts and Slopes

 = 

i

+ 

i

 = 

s

+ 

s

IF variance of 

i

= 0, Then  IF variance of 

s

= 0, Then  Thus variances of 

i

and 

s

= = 

i

, starting the same 

s

, changing the same are important parameters

Unconditional Growth Model--

Growth Model without Covariates

y t

=  +  T + 

t

 = 

i

+ 

i

(i = intercept here)  = 

s

+ 

s

Estimating Different Trajectories … ID 2 2 2 1 1 1 1 2 Dependent Variable 6.5

7.0

8.0

8.5

8.8

9.0

9.2

9.4

Linear 1 2.5

3 0 1 2 3 0 Non equidistant .0

.1

.2

.3

.0

.1

.25

.35

Quadratic curve 0 .1

.4

.9

0 .1

.4

.9

Logarithmic curve 0 Exponential curve 0 .69

1.10

.172

.639

1.39 1.909

0 0 .69

1.10

.172

.639

1.39 1.909

Conditional Growth Model--

Growth Model with Covariates • • •

y t

= 

i

 i = 

i

 i = 

s

+ 

i

T + 

t



3

+ 

i1

+ 

s1

 

1 1 +



i2 +



s2

+ 

t

 

2 2 +



i

+ 

s

Note: i=individual, t = time, variant covariate.  i

and

 

1

I and 

2

= time-invariant covariates, are functions of 

1,2…n , y it



3

= time-

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 2 3

id

1 3 4

x1

2 4 3

x2

5 3 6

t1y1

1 4 7

t2y1

2 5 8

t3y1

3

…

d1 1 y1 Measurement Model of Y y =  +  +  d2 1 y2 d3 1 y3 d4 1 y4 Slope

Specific Measurement Models

• • • • y 1 y 2 y 3 y 4 =  1 =  2 =  3 =  4 +  1  +  2  +  3  +  4  +  1 +  2 +  3 +  4  = 

i

+ 

i

 = 

s

+ 

s

Unconditional Latent Growth Model

y =  +  +   y = 0 + 1*  i +  s +  d1 1 y1 d2 1 y2 d3 1 y3 d4 1 y4 1 1 Intercept 1 1 2 0 1 Slope 3

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…

1 e1 Time1 y Group 1 d1 1 Time2 y Intercept /Level e2 1 1 Time-variant Covariate 1 Time-variant Covariate 2 Time-variant Covariate 3 1 e3 1 Time3 y 1 1 e4 Time4 y 1 1 e5 Time5 y d4 1 Distal Outcome 1 3 2 Slope /Trend 4 1 d3 Mediator 1 d2 Time-invariant Covariate A latent Growth Model with Covariates and A Outcome Variable

e1 1 t1y1 1 e2 1 t1y2 Factor Time1 1 e3 1 t2y1 1 e4 1 t2y2 Factor Time2 1 e5 1 t3y1 1 e6 1 t3y2 1 Factor Time3 1 e7 1 t4y1 1 e8 1 t4y2 Factor Time4 1 e9 1 t5y1 1 e10 1 t5y2 Factor Time5 Intercept /Level 3 1 2 Slope /Trend 4 Latent Growth Modeling of Factors

Parallel Growths

ey1 1 y1 0 1 1 iy 1 1 ey2 1 y2 1 sy ey3 1 y3 ey4 1 y4 1 iz 1 1 1 0 z1 1 ez1 1 x2 1 ez2 sz z3 1 ez3 z4 1 ez4

Cross-lagged Model

Frequency of Substance Use (Baseline) a1 Frequency of Substance Use (3 Months) a2 Frequency of Substance Use (6 Months) a3 Frequency of Substance Use (12 Months) b1 b2 b3 c1 Quality of Life (Baseline) d1 c2 c3 Quality of Life (3 Months) d2 Quality of Life (6 Months) d3 Quality of Life (12 Months)

Parallel Growth with Covariates

X1 X2 X3 e11 1 y11 e12 1 y12 e13 1 y13 e14 1 y14 1 1 Intercept1 1 1 1 2 3 slope1 1 Intercept2 1 1 1 y21 1 e21 y22 1 e22 d1 d3 1 slope2 2 3 y23 1 e23 y24 1 e24 d2 d4

Antecedent and Subsequent (Sequential) Processes e1 1 y1 0 1 1 1 1 i1 e2 1 y2 1 e3 1 y3 e4 1 y4 s1 d1 i2 11 1 1 y5 1 0 1 y6 1 e5 e6 y7 1 e7 s2 d2 y8 1 e8

e1 1 y Time 1 1 e2 1 1 y Time 2 1 e3 1 y Time 3 Level1 0 1 Trend1 2 Added Level 1 1 1 e4 1 y Time 4 1 0 1 Added Trend 2 e5 1 1 y Time 5 e6 1 1 y Time 6 Level1 0 1 Trend1 2 Interrupted Time Series Latent Grwoth Model

1 e1 Time1 y 1 Intercept /Level

Control Group



Experimental Group

 Intercept /Level e2 1 Time2 y 1 1 e3 1 Time3 y 1 e4 1 Time4 y 1 3 2 Slope /Trend 4 1 e5 1 Time5 y 1 Time1 y 1 e1 1 Time2 y 1 e2 1 Slope /Trend 3 2 4 1 Added Growth 2 3 4 1 Time3 y 1 e3 1 Time4 1 y e4 1 Time5 y 1 e5

Cohort 1 Cohort 2 Cohort 3 1 e1 1 y Time 1 Level1 e2 1 1 y Time 2 1 1 0 Trend1 e3 1 y Time 3 2 e4 1 y Time 2 1 Level1 ?

e5 1 1 y Time 3 1 1 2 3 Trend1 ?

e6 1 y Time 4 Cohort-Sequential LGM e7 1 y Time 3 1 Level1 ?

e8 1 1 y Time 4 1 2 3 4 Trend1 ?

e9 1 y Time 5

Piecewise Growth Model

e1 y1 1 Intercept 1 e2 y2 1 e3 y3 1 e4 y4 1 Slope1 2 2 0 1 Slope2 Slope2 Slope1

Original Rating 0-4 Dummy Coding 0-1

Two-part Growth Model

(for data with floor effect or lots of 0) e11 1 y1 e12 1 y2 e13 1 y3 e14 1 y4 X1 1 1 Intercept1 1 1 1 2 3 slope1 d1 d2 u1 1 e21 1 Intercept2 1 1 1 u2 1 e22 d3 1 slope2 2 3 u3 1 e23 u4 1 e24 d4 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 Factors

with 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 • Y =  1 + F1 • F1 =  2 + Intercept F1 • Y = (  1 =  2 =0) + Intercept Intercept

Specify Growth Model of Factors

with 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 Standardized Estimates?

• 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

answers to problems Search under paper and discussion for papers and

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 covariates 4. Use sex or race as grouping variable and test if the two groups have similar slopes.

5. Explore mixture growth modeling