Latent Growth Modeling
Using Mplus
Friday Harbor Psychometrics Workshop
Richard N. Jones1,2, Frances M. Yang1,2, Douglas Tommet1
1Institute for Aging Research, Hebrew SeniorLife and Beth Israel Deaconess
Medical Center, Division of Gerontology
2Harvard Medical School
[email protected]
September 1, 2009
Corrected 9/2/2009
1
Acknowledgements
• Funded in part by Grant R13AG030995-01A1
from the National Institute on Aging
• The views expressed in written conference
materials or publications and by speakers and
moderators do not necessarily reflect the official
policies of the Department of Health and Human
Services; nor does mention by trade names,
commercial practices, or organizations imply
endorsement by the U.S. Government.
2
2
Session Overview
•
•
•
•
•
•
•
•
Other Resources
General Framework
Comparison with Random Effects
Modeling Framework
Special Model Considerations
Some Results from ROS
Detailed Example from ROS
Questions and Discussion
3
Other Resources
• What is longitudinal data analysis?
– Singer JD & Willett JB. Applied longitudinal data analysis:
Modeling change and event occurrence. 2003, New York: Oxford
University Press. (Also see worked examples at UCLA ATS)
• How do I do latent growth curve modeling?
– Duncan TE, Duncan SC, & Strycker LA. An introduction to latent
variable growth curve modeling: concepts, issues and
applications. Second ed. 2006, Mahwah, New Jersey: Lawrence
Erlbaum Associates, Inc.
• Tell me more about the math behind latent curve methods
– Bollen KA & Curran PJ. Latent curve models: a structural
equation perspective. Wiley series in probability and statistics.
2006, Hoboken, N.J.: Wiley-Interscience.
• Our workshop
– 2009 workshop was LDA, come back in 2011
– http://sites.google.com/site/lvmworkshop for slides, syntax, data
4
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ut
f
c
c
x
5
Five+ Approaches to LDA in
Mplus
•
•
•
•
•
•
Latent growth curve model
Random effects model
Multilevel model
Latent change/Dual change score model
Autoregressive/latent simplex model
Latent Structural Models (Hyperbolic
functions for learning data)
6
Random Effects and Latent Growth
Curves: Same But Different
• Reconceptualize random effects as latent
variables
• Use multivariate record layout (wide)
• Main difference:
– RE: time is data
– LGC: time is a model parameter
…unless it is data
7
Advantages
HLM and Mixed Effect
Modeling
• Software for highly
nested multilevel data
better developed
• Easier to get model fit
and diagnostics
• Use time-varying weights
Note: HLM Hierarchical Linear Modeling
LGC modeling
• Embed in more complex
models
• Flexible curve shape
(time is a parameter
and/or data)
• Modification Indices can
help with misspecified
models
8
Latent Growth Curve Models
• Latent Growth Curve (LGC) modeling is just like
CFA
• Reconceptualize “factors” as “random effects”
• Factor loadings are
– (usually) not estimated but given by design or data,
and
– relate to the sequence of repeated observations
• The action is in the mean structure part of the
model (factor means, item means, factor
variances) as opposed to factor loadings
9
Latent Growth Model (Linear Change)




y1 y2 y3 y4

 [ ]
1

 [ ]
2
x1
10
Latent Growth Model (Linear Change)




y1 y2 y3 y4

1
1
2



x1
11
Latent Growth Model (Linear Change)




y1 y2 y3 y4
10
Performance

5
 [ ]
1

 [ ]
2
0
x1
1
2
3
4
time
12
Latent Growth Model (Linear Change)
1  2 3  4
Typical parameterization for
[1]
[2]
[3]
linear change and equallyspaced time steps
[4]
 [ ]
1
31
=

1=1
2
1
11

x1
00
 = 0 
0 
 42=3
 [ ]
2
=1
 21
=1
 32=

 41
 22
 11=1
y1 y2 y3 y4
2
21
11 01
 = 1 2 
1 3 

 = VAR() = 

*
 = *
* 000
0* 00
00* 0
000*



*
 =*
*
 = VAR() =  * *
“*” Implies parameter freely estimated. All
other parameters are held constant to the
indicated value.
13
Latent Growth Model (Linear Change)
1  2 3  4
[1]
[2]
[3]
[4]
LEVEL 1:
1
31
=

1=1
2

yit = t + 1×i1 + t×i2 + tqxiq + it
yit =
i1 + t×i2 + tqxiq + it
LEVEL 2:
 [ ]
1
11
y =  +  + x + 
 42=3
 [ ]
2
=1
 21
=1
 32=

 41
 22
 11=1
y1 y2 y3 y4
2
21
 =  + x + 
=+
x + 
i1 = 1 + 11×xi1 + i1
i2 = 2 + 21×xi2 + i2
x1
14
Latent Growth Model (Linear Change)
1  2 3  4
[1]
[2]
[3]
[4]
y1 y2 y3 y4
TITLE:
Latent Growth Curve
(Short hand notation)
DATA:
File = blah.dat;
1
31
=

1=1
2

eta1 eta2 | y1@0 y2@1
y3@2 y4@3;
eta1 eta2 on x1 ;
 [ ]
1
11
MODEL:
 42=3
 [ ]
2
=1
 21
 32=

=1
 41
 22
 11=1
VARIABLE: Names = y1-y4 x1;
2
21
x1
15
Latent Growth Model (Linear Change)
1  2 3  4
[1]
[2]
[3]
[4]
y1 y2 y3 y4
TITLE:
Latent Growth Curve
(Long hand notion)
DATA:
File = blah.dat;
 [ ]
1
31
=

1=1
2
1
11

MODEL:
 42=3
 [ ]
2
=1
 21
 32=

=1
 41
 22
 11=1
VARIABLE: Names = y1-y4 x1;
eta1 by y1-y4@1;
eta2 by y1@0 y2@1 y3@2 y4@3;
[y1-y4@0] ;
[eta1 eta2] ;
eta1 with eta2 ;
eta1 eta2 on x1 ;
2
21
x1
16
Change in Ordinal Outcome
1  2 3  4
[1]
[2]
[3]
[4]
1
31
=

1=1
2
11

DATA:
File = blah.dat;
MODEL:
 [ ]
1
Latent Growth Curve
VARIABLE: Names = u1-u4 x1;
Categorical = u1-u4;
 42=3
 [ ]
2
=1
 21
1
 32=

=
 41
 22
 11=1
u1 u2 u3 u4
TITLE:
eta1 eta2 | u1@0 u2@1 u3@2 u4@3;
eta1 eta2 on x1 ;
! constrain the scale parameters
! see Mplus Web note 4 for more info
{u1-u4@1;}
2
21
x1
17
Model a Retest Effect

[ 3]
1
1
2
3
4
TITLE:
Latent Growth Curve
With retest effect
DATA:
File = blah.dat;
VARIABLE: Names = y1-y4 x1;
y1 y2 y3 y4
1

1
1
1
1 2
 [ ]

eta1 eta2 | y1@0 y2@1 y3@1 y4@3;
eta3 by y2-y3@1
[eta3] ; eta3@0;
eta1 eta2 eta3 on x1 ;
3
 [ ]
1
11
MODEL:
2
21
x1
18
Model a Retest Effect
that is dependent on baseline
1
2
3
4



0
0
= 0
0
y1 y2 y3 y4

 [ ]
1
11

x1
21
 [ ]
1
1
= 
1

1

 = VAR() = 


[ 3]
* 000
0* 00
00* 0
000*
0 0
1 1 
2 1

3 1



 0 0 0
 =  0 0 0
 * 0 0
2
 *
 =  *
 *
 *
 =  *
 *
*

*
*
 = VAR() = 

? ? ?
19
Regress Change on Baseline
Typical parameterization for
1  2 3  4
linear change and equally[1]
[2]
[3]
spaced time steps
[4]
1= 1
2

=1
41
 [ ]
1
11

x1
2
=1
 [ ]

 42=3
1
 32=


=
31
 22
 11=1
y1 y2 y3 y4
00
 = 0
0
2
21
11 01
 = 1 2
1 3

 = VAR() = 

*
 = *

* 000
0* 00
00* 0
000*



*
 =*

*
 = VAR() = 0 *
0
 = * 0
“*” Implies parameter freely estimated. All
other parameters are held constant to the
indicated value.
20
Multiple Indicator Growth Model
y11 y21 y31 y12 y22 y32 y13 y23 y33 y14 y24 y34
f1

f2
 [ ]
1

x1
f3
f4
 [ ]
2
21
Growth Mixture Model
1
2
[1]
3
[3]
[2]

31

=1
31
y3
 32=1.2
=1
y2
=0.
 22
 11=1
y1
6

 21
 [ ]
 [ ]
1
2
21

23
c
x1 x2 x3
22
Alternative Time Bases
•
Observation number
•
Time of observation
•
Time of observation relative to some anchor (death,
last scheduled follow-up)
•
Age at observation
•
Age at observation centered at baseline mean
•
Age at observation centered at baseline within age
group (and include age group dummies)
•
Completely general
23
Model Building (LGC)
Step 0: Descriptive analysis, graphs
Step 1: Start with simple model (unconditional
†
model, i.e. no covariates )
Step 2: Add covariates, regress intercept and
slope on covariates.
Step 3: Possible model modifications
† unless
required to specify the time basis, e.g., age group
24
Study Data
SAS, SPSS,
R/S-Plus
STATA
Preprocess
ASCII Data
Command
File
Also LISREL,EQS,
WinBUGS
Mplus
Output &
Inferences
clean data, handle missingness
select cases, variables
transformations
descriptives
A text file with selected data
elements. Comma delimited works
best for Mplus
Also a raw text (ASCII) file
Instructions for a single analysis
STATA modules are
available to automate
this process
Write the
Paper
25
Post-Estimation Fit Evaluation
•
•
•
•
•
Save Factor Scores
Import into stat package
Compute expected scores
Graph Residuals
Empirical r-square
26
Example: PW 2008
• ROS
• Change in global cognition (globcog)
• Random effects growth mixture model
– Time basis: Age centered at baseline mean
within age group
– Retest effect (occasion basis)
– Mixture model part for growth parameters
• Covariates: age group dummies (to define
age metric)
27
Random
Effects
Mixture
Model

[ 3]
1

2
 15
y1 y2
y15
 [ ]
 [ ]
1

2
age
50-64
age
90-102
exclude age 75-79 as
reference group
c
28
TITLE:
ROS GLOBCOG SINGLE CLASS 8/16/2009
DATA:
FILE = __000001.dat ;
VARIABLE:
NAMES = y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
cagecat1 cagecat2 cagecat3 cagecat5 cagecat6 projid ;
MISSING ARE ALL (-9999) ;
IDVARIABLE = projid ;

TSCORES = t1-t15 ;
[ 3]
ANALYSIS:
TYPE = random ;
COVERAGE = .02 ;
MODEL:
i s | y1-y15 AT t1-t15 ;
r by y2-y15 @1 ;
[r] ;
r@0 ;
i s on cagecat1-cagecat6*
y1-y15 *0.1 (theta_1) ;
1
;

2
 15
y1 y2
y15
 [ ]
 [ ]
1

age
50-64
2
age
90-102
exclude age 75-79 as
reference group
29
TITLE: ROS GLOBCOG Trajectories 8/26/2009
DATA:
FILE = __000001.dat ;
VARIABLE:
NAMES =
y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
cagecat1 cagecat2 cagecat3 cagecat5 cagecat6 projid ;
MISSING ARE ALL (-9999) ;
IDVARIABLE = projid ;
TSCORES = t1-t15 ;
CLASSES = c(3) ;
ANALYSIS:
TYPE = mixture random ;
COVERAGE = .02 ;
STARTS = 0 ;
PROCESSORS=2 ;
ALGORITHM=integration ema ;
INTEGRATION = montecarlo ;
MCONVERGENCE = 0.01 ;
SAVEDATA:
FILE = c:\work\ros\posted\data\derived\gmm3class10AUG2009.dat ;
SAVE = fscores cprob ;
RESULTS = c:\work\ros\posted\data\derived\gmm3class10AUG2009_results.dat ;
30
MODEL:
%OVERALL%
i s | y1-y15 AT t1-t15 ;
r by y2-y15 @1 ;
[r] ;
r@0 ;
i on cagecat1*.714 cagecat2*.661 cagecat3*.224
cagecat5*-.397 cagecat6*-.764 ;
s on cagecat1*.068 cagecat2*.06 cagecat3*.015
cagecat5*-.058 cagecat6*-.071 ;
%c#1%
[i*-.31125 s*-.27805 r*.187] ;
i*.734 s*.367 r@0 ;
i with s *0 ;
y1-y15 *.0375 (theta_1) ;
%c#2%
[i*-.519 s*-.622 r*.187] ;
i*.734 s*.734 r@0 ;
i with s *0 ;
y1-y15 *.075 (theta_2) ;
%c#3%
[i*-.83 s*-1.245 r*.187] ;
i*.734 s*1.468 r@0 ;
i with s *0 ;
y1-y15 *.15 (theta_3) ;

[ 3]
1

2
 15
y1 y2
y15
 [ ]
 [ ]
1

2
age
50-64
age
90-102
exclude age 75-79 as
reference group
c
31
Figure 1. Cognitive Change Trajectories by Class and
Age Group at First Observation as implied by Mixture
Model Parameter Estimates
2
Slow
Moderate
Fast
1
63%
27%
11%
(z-Score)
0
-1
-2
-3
-4
-5
-6
60
70
80
90
100
110
Age
Education (and race/ethnicity, baseline mental status) associated with class
Membership. But not age, not sex.
32
Figure 2. Burden of Amyloid and Tangle Neuropathology by Class Membership
(N=326)
Amyloid
Tangles
80
10
Pathology Score
60
5
40
20
0
0
Slow
Moderate
Class Membership
Fast
Slow
Moderate
Fast
Class Membership
33
Trajectory Classes and Reserve
• Neuropathology at autopsy does not perfectly account
for membership in one of two population sub-groups
experiencing substantial cognitive decline
• Education, a proxy for cognitive reserve, may buffer the
functional consequences of neuropathology:
34
A Different SEM model for
Change
The Latent Change Score Model
35
35
Resources
General Framework Comparison with RE Modeling Framework Special Considerations Some ROS Results Detailed Example
Latent Change Change Score Model
*
0
y1
y2
 y=*

1
1  =*

y
y
*
TITLE:
LCSM
DATA:
FILE = BLAH.dat ;
VARIABLE:NAMES = y1 y2 ;
MODEL:
dy by y2 @1 ;
[y1*] ;
y1* ;
y2 on y1 @1 ;
dy on y1 * ;
dy* ;
[dy*] ;
[y2@0] ;
y2 @0 ;
NB: As parameterized, just identified or
saturated model = zero degrees of freedom.
Just as many knowns as estimated parameters.
36
36
0
344.506
N0
N1
1


-0.241
0.199
254.206
1
N
108.300

0.129
98.608
0.186
-0.273
F

1
0
420.510
1
F0

ex03-08.inp
F1

37
37
Advantages
Dual Change Score
• More flexibility for
estimating lagged and
leading effects
Latent GC modeling
• Better fit (good for
descriptive analysis)
• Better for sequential
patterns
38
Extensions to the LCSM
• Dual Change Score Model
• Bivariate CSM
– Two outcomes are of interest
• Multiple Indicator LCSM
– change in a latent variable
• Multiple Indicator Dual Change Score
Model
39
39
Dual Change Change Score Model
y1
y1
y2
[=0]
1

0
1

0
y3
0
[=0]
1

y4

y43
y32
0
0
0
[=0]
1
1
y21
0
0
y4


1
y0
y
y3
[=0]

1
*
y2
(1)
(1)
(1)
(1)
4*
3*
2*
1*
=1
=1
=1
 y=*
 s=*
1
s
s

 s,y=*
*
40
40
Help Coding in Mplus
… from Zhiyong Zhang, Ph.D.
http://www.psychstat.org/us/sort.php/7.htm%20-%20Programs
41
41
FHL 2009 Potential Topics
• Replicate ROS analysis in MAP, and/or
• Domain-specific REMM
• Change-point model (LGCMM or REMM)
– Reserve proxies associated with when
cognitive decline starts and how fast decline
occurs.
42
Questions
Discussion
43