Ronald H. Heck
EDEP 768 (S2011): Seminar in Structural Equation Modeling
University of Hawai‘i at Mānoa
1
April 4, 2011
Handout #20
Specifying Latent Curve and Other Growth Models Using Mplus
(Revised 12-1-2014)
The SEM approach offers a contrasting framework for use in analyzing various kinds of
longitudinal data. In this framework, the distinction between level 1 (within individuals) and
level 2 (between individuals) in a two-level random coefficients growth model is not made
because the outcomes are considered as multivariate data in a T-dimensional vector
(y1,y2,…,yT)′. Because of this difference in conceptualizing the data structure in SEM, the
repeated measures over time can be expressed as a type of single-level confirmatory factor
analysis (or measurement model), where the intercept and growth latent factors are measured by
the multiple indicators of yT. The structural part of the SEM analysis can then be used to
investigate the effects of covariates or other latent variables on the latent growth curve factors.
Once the overall latent curve model has been specified through relating the observed repeated
measures variables to the latent factors that represent the change process, it can be further
divided into its within- and between-group components, similar to other types of multilevel
models. This implies that the three-level multilevel univariate growth model (with repeated
measures nested within individuals and individuals nested within groups) can be expressed two
level latent growth model using the SEM specification.
Until recently, full information ML for multilevel latent curve models was limited by the
complexity of specifying models within existing software and the computational demands in
trying to estimate the models. The availability of full information ML for missing data analysis
has enhanced the usefulness of SEM methods (e.g., alleviating the need for listwise deletion,
allowing unequal intervals between measurement occasions) for examining individual change
(Mehta & Neale, 2005). As Mehta and Neal note, this has allowed the fitting of SEM to
individual data vectors by making available model estimation methods based on individual
likelihood that could accommodate unbalanced data structures (e.g., individuals within clusters).
Consider an example where we were interested in examining individual development in math
over four years (y1- y4) in a random sample of elementary school students. In conducting a latent
growth curve analysis, the researcher’s first concern is typically to establish the level of initial
status (or end status) and the rate of change. Fitting a two-factor latent growth curve model is
accomplished first by fixing the intercept factor loadings to 1.0. Because the repeated measures
are considered as factor loadings, there are several choices regarding how to parameterize the
growth rate factor. In this case, we can specify a linear growth trajectory by using 0, 1, 2, 3 to
represent fixed successive linear measurements. Since the first factor loading is fixed at 0, it is
shown as a dotted line in Figure 1. Alternatively, researchers may decide not indicate the loading
from the factor to the observed indicator in their path diagrams since it is 0.
Ronald H. Heck
2
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
Figure 1. Proposed latent curve model.
The typical two-level, random-components growth model with a vector of repeated measures at
level 1 nested within individuals at level 2 can be expressed in SEM terms as a single-level
multivariate model by utilizing the general measurement and structural models (Muthén, 2002).
The measurement model specifies a vector of repeated observed measures which define a smaller
set of latent growth factors (often an initial status factor and a linear growth factor) as follows:
yit   t  i  xi   it ,
(1)
where yit is a vector of math outcomes for individual i at time t ( yi1 , yi 2 ,..., yiT ) ,  t is a vector of
measurement intercepts set to 0 at the individual level,  t is a p x m design matrix representing
the change process, i is an m-dimensional vector of latent variables (0 ,1 ) , Κ is a p x q
parameter matrix of regression slopes relating xi ( x1i , x2i ,..., x pi ) to the latent factors, and εit
represents time-specific errors, which are contained in a covariance matrix designated as  .
The factor loadings for the latent factors (i.e., in this case, η0i and η1i) are contained in the
 t factor-loading matrix, as in the following:
1
1
t 
1

1
0
1 
2

3
(2)
Fixing the loadings of the four measurement occasions on the first factor (η0i) to a value of 1.0
ensures that it is interpreted as a true (i.e., error-free) initial status factor, that is, representing a
baseline point of the underlying developmental process under investigation. Because the repeated
measures are specified as parameters (i.e., factor loadings) in a latent curve model (i.e., instead
of data points in a multilevel model), the researcher can hypothesize that they follow particular
shapes or patterns and test the hypothesized growth pattern against the data. The slope factor
(η1i) can be defined as linear with successive time points of 0, 1, 2, 3. This scaling indicates the
Ronald H. Heck
3
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
slope coefficient will represent successive yearly intervals, starting with initial status as time
score 0 (i.e., the loading from the growth rate factor to y1 is shown as a dotted line in Figure 1).
All parameters are fixed, therefore, in the Λt matrix in this type of specification (which in some
cases may be an undesirable restriction).i Below is the lambda matarix for specifying the latent
intercept (I) and growth factors (S) as specified in the Mplus TECH1 output. Because the
loadings are fixed (at 1.0 for the intercept factor, and from 0-3 for the growth factor), in this
growth model specification there are no “freely-estimated” growth parameters specified.
LAMBDA
_______________________________________
I
S
______________________________________
GRADE1
0
0
GRADE2
0
0
GRADE3
0
0
GRADE4
0
0
__________________________________
As noted, defining the shape of the change ahead of testing has been referred to as an intercept
and slope (IS) approach (Raykov & Marcoulides, 2006). By fixing the factor loadings to a
particular pattern (t = 0, 1, 2, 3), the hypothesized growth shape may then be tested against the
actual data and its fit determined by examining various SEM fit indices. If the linear model does
not fit the data well, one might add a quadratic factor to the factor loading matrix (i.e., which
requires an intercept factor, a linear slope factor, and a quadratic slope factor with loadings 0, 1,
4, 9 as a third column). The psi covariance matrix (Ψ) contains latent factor variances and
covariances.ii There are also other possible ways to define the factor loadings in the Λt matrix to
indicate different patterns of individual change. The residuals are specified in the theta
covariance matrix (Θ).
THETA
___________________________________________________________________
GRADE1
GRADE2
GRADE3
GRADE4
___________________________________________________________________
GRADE1
1
GRADE2
0
2
GRADE3
0
0
3
GRADE4
0
0
0
4
__________________________________________________________________
Alternatively, we may actually know little about the shape of the individual trajectories. Various
types of nonlinear growth can also be captured within the basic two-factor LCA formulation by
using different patterns of free (estimated) and fixed factor loadings within the basic level and
shape (LS) formulation. In Mplus model statements, the asterisk (*) option is used to designate a
free parameter, such as a factor loading. For the LS formulation, we fix the first time score to
zero and the last time score to 1.0, while allowing the middle time scores to be freely estimated
(0, *, *, 1). This can allow for valleys and peaks within the overall growth interval being
measured (e.g., 0, -0.5. 1.2, 1). The factor-loading matrix would be the following:
Ronald H. Heck
4
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
1
1
t 
1

1
0
* 
*

1
(3)
We could also capture other types of nonlinear growth, using combinations of free and fixed
factor loadings within the factor-loading matrix. For example, we could allow the last time score,
or the last two time scores, to be freely estimated [(0, 1, 2,*) or (0, 1, *, *)]. In addition to
describing possible peaks or valleys, this type of formulation would allow for the estimation of
various trajectories where the change might accelerate (e.g., 0, 1, 3, 6) or decelerate (e.g., 0, 1,
1.5, 1.75) over time.
As these models suggest, the basic two-factor LCA formulation is quite flexible in allowing the
researcher to consider both the levels of the growth or decline over time as well as the shape of
the trend, which summarizes the growth rate over a specified time interval. The analyst can
observe whether the change is positive or negative and whether there is heterogeneity in the level
and rate of change across the individuals in the sample (Willett & Sayer, 1996).
The Structural Model
After the measurement model is used to relate the successive observed measures to the initial
status and growth rate factors (similar to factor analysis), the second SEM model (i.e., the
structural model) is used to relate one or more time-invariant covariates to the random effects
(i.e., initial status and growth trend factors). The general structural model for individual i is then
i    i  xi   i ,
(4)
where  is vector of intercepts for the equations,  is an m x m parameter matrix of slopes for
the regressions among the latent variables, Γ is an m x q slope parameter matrix relating x
covariates to latent variables, and  i is a vector of residuals with zero means and are contained
in covariance matrix Ψ. Consistent with the two-level growth modeling approach, i contains
intercept ( 0i ) and slope ( 1i ) latent factors whose variability can be explained by one or more
covariates. For example, here would be a simple model to explain latent growth in math using
two covariates (e.g., gender, motivation) to explain differences in the initial status and linear
growth factors over time.
0i   0   01 x1i   02 x2i   0i ,
1i  1   11 x1i   12 x2i   1i ,
(5)
(6)
where  0 and 1 are equation intercepts and  coefficients are structural parameters describing
Ronald H. Heck
5
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
the regressions of latent variables on the covariates. As noted, the equation intercepts are
contained in alpha.
ALPHA
__________________________________________________________________
I
S
FEMALE
MOTIVATION
__________________________________________________________________
5
6
0
0
________________________________________________________
In practice, it is convenient in Mplus to specify the model such that the regression of the latent
intercept (I) and slope (S) factors on the covariates are contained in the B structural matrix
(instead of a gamma matrix) .iii
BETA
_______________________________________________________________________
I
S
FEMALE
MOTIVATION
_______________________________________________________________________
I
0
0
7
8
S
0
0
9
10
FEMALE
0
0
0
0
MOTIVATION
0
0
0
0
_______________________________________________________________________
Each growth equation has its own residual (  0i and  1i ) which permits the quality of
measurement associated with each individual’s growth trajectory to differ from those of other
individuals. Residual variances and covariances for the latent factors are contained in Ψ.
PSI
______________________________________________________________________
I
S
GENDER
MOTIVATION
_____________________________________________________________________
I
11
S
12
13
FEMALE
0
0
0
MOTIVATION
0
0
0
0
______________________________________________________________________
It is important to keep in mind that the random-coefficients and the latent-curve approaches
describe individual development in a slightly different manner. First, as noted, the SEM
formulation treats the individual’s change trajectory as single-level, multivariate data. The
approach accounts for the correlation across time by the same random effects influencing each of
the variables in the outcome vector (Muthén & Asparouhov, 2003). In contrast, the randomcoefficients approach treats the trajectories as univariate data and accounts for correlation across
time by having two levels in the model. A second difference is the manner in which the time
scores (i.e., the scaling of the observations) are considered. For random-coefficients growth
modeling, the time scores are data; that is, they are pieces of information entered in the level-1
(within individual) data set to represent what the individual’s math score was at each particular
point in time. For LCA, however, the time scores are considered as model parameters; therefore,
they are defined as factor loadings on the growth factors. The individual scores at each time
interval are therefore not represented as data points within the data set. Finally, latent growth
Ronald H. Heck
6
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
curve modeling typically proceeds from an analysis of means and covariance structures. In
contrast, random coefficients (multilevel) growth modeling creates a separate growth trajectory
with intercept and slope for each individual. Random-coefficients multilevel models are typically
not described in terms of means and covariance structures (Raudenbush, 2001).
The SEM submodels for the single-level, multivariate specification of latent change can be easily
expanded to facilitate specifying multilevel latent change models that decompose observed and
latent variables into their uncorrelated within- and between-cluster components. This type of
two-level formulation is consistent with a three-level univariate random coefficients growth
model, except that the scores are assumed to vary across individuals and clusters in this
formulation and there are not time-varying covariates present (Muthen & Asparouhov, 2011).
The general measurement model for individual i in group j at time t can be expressed as
Ytij   tWij   tij ,
(7)
where  tW is the within-group factor loading matrix for the latent growth factors and  tij are timerelated residuals are assumed to have no means and are contained in ΘW. The measurement
intercepts for the repeated measures defining the growth factors are considered to be random
intercepts that are referred to in the between part of the model, where in Mplus diagrams they are
shown in circles since they are continuous latent variables that vary across clusters (i.e., they
have between-cluster variances).
For example, for a simple multilevel latent curve model defining three repeated math measures
within groups, we have the following specification for the time-related lambda matrix within
groups.
LAMBDA
_____________________________________________________________________
LW
SW
LB
SB
_____________________________________________________________________
MATH1
0
0
0
0
MATH2
0
0
0
0
MATH3
0
1
0
0
_____________________________________________________________________
Once again, within groups the latent level factor (LW) has factor loadings fixed to 1.0, so there
are no parameters estimated. The latent shape factor (SW) has the first two occasions fixed (0.1)
and the third occasion is freely estimated. The between-group latent factors are not specified
within groups (with parameters fixed to 0). The time-related residuals are defined in theta as
follows:
THETA
_______________________________________________________
MATH1
MATH2
MATH3
_______________________________________________________
MATH1
0
MATH2
0
2
MATH3
0
0
3
_______________________________________________________
Ronald H. Heck
7
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
The matrix specification indicates that the residual for MATH1 was fixed to 0 (which was
necessary for model convergence.
It is useful to note that in the multilevel specification parameter matrices such as Λ, Β, Γ, Θ, and
Ψ can be specified both within clusters and between clusters. The corresponding within-groups
structural model can be specified as
ij   j   jij   j xij   ij ,
(8)
where the cluster (j) subscripts on the parameter matrices suggest the possibility that some
elements of these matrices vary over clusters. The residuals in equations are assumed to have
zero means and are contained in ΨW.
Between groups, a corresponding measurement model can be defined specifying the
relationships of the repeated measures to the intercept and growth latent factors:
 j     tB j   tj ,
(9)
where ν is a vector measurement intercepts (which are generally fixed to zero because the latent
factor means are instead estimated) and  tj are time-related residuals that are contained in ΘB. It
is possible to specify a different latent growth model between groups from the within-school
model describing individual change. Once again, the growth factors are specified in lambda
(Between) as follows:
LAMBDA
___________________________________________________________________
LW
SW
LB
SB
___________________________________________________________________
MATH1
0
0
0
0
MATH2
0
0
0
0
MATH3
0
0
0
7
____________________________________________________________
Time related residuals are contained in Theta (Between), where the first time-related error was
fixed to 0:
THETA
____________________________________________________
MATH1
MATH2
MATH3
____________________________________________________
MATH1
0
MATH2
0
8
MATH3
0
0
9
_______________________________________________
Factor intercepts, which vary across groups, are contained in alpha (and consequently are not
estimated within groups.)
ALPHA
_______________________________________________________________
LW
SW
LB
SB
_______________________________________________________________
0
0
10
11
_______________________________________________________________
Ronald H. Heck
8
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
Finally, the factor variances and covariances between groups are contained in PSI (between):
PSI
____________________________________________________________________
SL
SW
LB
SB
____________________________________________________________________
SL
0
SW
0
0
LB
0
0
12
SB
0
0
13
14
____________________________________________________________________
It is also possible to specify randomly-varying intercepts and slopes across clusters for other
parts of the growth model. For example, in the example we are developing, we might specify that
the effect of student SES on the latent growth factors within groups varies across schools. In this
latter formulation, multilevel measurement and structural models with random effects can be
investigated by permitting elements of some coefficient matrices (e.g., Β and Γ) to vary at the
cluster level, as in Eq. 8 (Preacher et al., 2010). Because the slopes vary randomly across groups,
they are not estimated in the matrix of structural relationships within groups. In this case,
between groups, the intercepts for the random slopes are specified in the alpha vector:
ALPHA
________________________________________________________________________
LW
SW
S1
S2
LB
SB
________________________________________________________________________
0
0
10
11
12
13
________________________________________________________________________
Between schools, we can then specify a structural model to examine variability in the withingroup random intercept and slope effects. A general between-groups structural model can be
defined as
 j      j  x j   j ,
(10)
where  j is a vector of the between-group latent growth factors and random effects (e.g., random
slopes) from the within-groups model,  is a vector of intercepts for the between-groups
equations, x j represent between-groups covariates, Β and Γ represent structural relationships
regarding latent and observed variables, and  j are errors in equations that have zero means and
are contained in ΨB . We might now add a school-level predictor (school quality) to explain
variation in the two latent growth parameters (LB and SB) and the random within-groups slopes.
We can specify the final model in Figure 2. The freely-estimated (*) math3 parameters within
and between groups suggest there might be acceleration or deceleration over the last time period.
Ronald H. Heck
9
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
Figure 2. Proposed multilevel latent curve model.
The specification of the Beta matrix suggests that the four latent-variable outcomes are regressed
on the quality of the school’s educational processes.
BETA
_____________________________________________________________________________
LW
SW
S1
S2
LB
SB
LowSES
Quality
_____________________________________________________________________________
LW
0
0
0
0
0
0
0
0
SW
0
0
0
0
0
0
0
0
S1
0
0
0
0
0
0
0
14
S2
0
0
0
0
0
0
0
15
LB
0
0
0
0
0
0
0
16
SB
0
0
0
0
0
0
0
17
LOWSES
0
0
0
0
0
0
0
0
QUALITY
0
0
0
0
0
0
0
0
______________________________________________________________________________
Errors in the between-group equations are once again contained in ΨB. Of course, we can
introduce more complex relationships between groups including direct and indirect effects as
well as school-level change processes that might explain growth in student leaning in math. The
Mplus input file for the two-level latent curve model is next summarized.
Ronald H. Heck
10
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
______________________________________________________________________
TITLE:
DATA:
VARIABLE:
ANALYSIS:
Two-level latent curve model of math growth (M1);
FILE IS C:\program files\mplus\ch7 ex2.txt;
Format is 10f8.0,2f8.2;
Names are schcode math1 math2 math3
female lowses slep sped middle notrans cses
quality;
Usevariables schcode math1 math2 math3 lowses
quality;
within = lowses;
between = quality;
CLUSTER IS schcode;
TYPE = twolevel random;
Estimator is MLR;
Model:
OUTPUT:
%BETWEEN%
lb sb | math1@0 math2@1 math3*;
math1@0;
lb sb S1 S2 on quality;
%WITHIN%
lw sw |math1@0 math2@1 math3*;
math1@0;
S1 | lw on lowses;
S2 | sw on lowses;
Sampstat TECH1;
______________________________________________________________________
Defining A Random-Coefficients Growth Model in Mplus
We can also specify a random-coefficients model in Mplus, where the repeated observations on
YT nested within individuals at level 1 and differences between individuals are examined at level
2. Consider, for example, where we wish to look at difference in initial status graduation rates for
a sample of four-year institutions and their change in graduation rates over time. We have four
years of graduation rates nested within institutions (with four data lines per individual institution
within the data set). We also have covariates related to the type of institution (public = 1, private
= 0) and prestige (1 = selective student admissions standards, 0 = else). Since there are no latent
growth parameters, at level 1 we can write a simple growth model as follows:
Yti  i  i X ti   ti
(11)
where i is a vector of intercepts for the equations between individuals, Βi is a regression matrix
of relationships between growth parameters and covariates, X ti is a vector of time-related
variables (e.g., linear and possible quadratic components, time-varying covariates), and  ti is a
vector of time-related errors which are contained in ΨW. We will specify the initial status
intercept and the growth rate to vary randomly across individual institutions at level 2. In this
Ronald H. Heck
11
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
simple growth model, there is only one parameter specified within groups (i.e., the residual
variance for the initial status graduation rate).
PSI
_________________________________________________________________________________
S
GRADUATE
GROWRATE
PRIVATE
PRESTIGE
_________________________________________________________________________________
S
0
GRADUATE
0
1
GROWRATE
0
0
0
PRIVATE
0
0
0
0
PRESTIGE
0
0
0
0
0
_________________________________________________________________________
Between-individuals, we can specify a structural model to explain the random effects (i.e., the
initial status intercept and random growth rate) as follows:
i    i  X i   i
(12)
where  is a vector of intercepts for the initial status and growth rate random effects,  is a
matrix of regression coefficients specifying relationships between the random components, Γ
describes relationships between level-2 covariates ( X i ) and random growth parameters, and  i is
a vector of errors in equations, which are contained in ΨB. The corresponding between-institution
matrix specification follows.
ALPHA
_______________________________________________________________________________
S
GRADUATE
GROWRATE
PRIVATE
PRESTIGE
_______________________________________________________________________________
2
3
0
0
0
_______________________________________________________________________
BETA
________________________________________________________________________________
S
GRADUATE
GROWRATE
PRIVATE
PRESTIGE
________________________________________________________________________________
S
0
0
0
4
5
GRADUATE
0
0
0
6
7
GROWRATE
0
0
0
0
0
PRIVATE
0
0
0
0
0
PRESTIGE
0
0
0
0
0
_______________________________________________________________________
PSI
_________________________________________________________________________________
S
GRADUATE
GROWRATE
PRIVATE
PRESTIGE
_________________________________________________________________________________
S
8
GRADUATE
9
10
GROWRATE
0
0
0
PRIVATE
0
0
0
0
PRESTIGE
0
0
0
0
0
_________________________________________________________________________
Ronald H. Heck
12
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
Ronald H. Heck
13
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
The Mplus input file for the random-coefficients growth model is summarized below.
________________________________________________________________________
TITLE:
DATA:
Two-Level Random-Coefficients Growth Model;
File is C:\mplus\ch7ex1.dat;
Format is 3f8.0, f4.0,2f8.0;
Names are id private prestige index1 graduate growrate;
Usevariables are id graduate growrate private prestige;
cluster = id;
within= growrate;
between = private prestige;
center private prestige(grand);
Type = twolevel random;
Estimator is MLR;
VARIABLE:
define:
ANALYSIS:
Model:
%between%
graduate on private prestige;
S on private prestige;
graduate with S;
%within%
S | graduate on growrate;
OUTPUT:
sampstat tech1;
_________________________________________________________________________________
The model results are presented next in Table 1. The results suggest the adjusted initial status
graduation level (grand-mean centered) was 46.048 percent (in six years). The average linear
growth rate was 4.156. Moreover, private schools had higher initial status graduation rates
(6.529) than their public counterparts. Prestige did not affect initial status, however. Regarding
growth , private schools made more growth over time (6.059) than public institutions.
Table 1. Variables explaining institutional growth over time.
_______________________________________________________________________
Estimate
S.E.
Est./S.E.
P-Value
_______________________________________________________________________
Within Level
Residual Variances
GRADUATE
59.855
13.388
4.471
0.000
PRIVATE
PRESTIGE
6.059
-0.003
1.954
1.985
3.100
-0.001
0.002
0.999
GRADUATE
ON
PRIVATE
PRESTIGE
6.529
1.313
8.052
8.177
0.811
0.161
0.417
0.872
GRADUATE WITH
S
-25.698
19.146
-1.342
0.180
46.028
4.156
3.837
0.931
11.995
4.463
0.000
0.000
252.587
93.582
2.699
0.007
Between Level
S
ON
Intercepts
GRADUATE
S
Residual Variances
GRADUATE
Ronald H. Heck
14
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
S
5.374
6.101
0.881
0.378
____________________________________________________________
References
Mehta, P.D. & Neale, M. C. (2005). People are variables too. Multilevel structural equations
modeling. Psychological Methods, 10(3), 259-284.
Muthén, B. O. & Asparouhov, T. (2003). Advances in latent variable modeling. Part I:
Integrating multilevel and structural equation modeling using Mplus. Unpublished paper.
Muthen & Asparouhov, T. (2011). Beyond multilevel regression modeling: Multilevel analysis
in a general latent variable framework. In J. Hox and J. K. Roberts (Eds.), Handbook of
advanced multilevel analysis (pp. 15-40). New York: Taylor and Francis.
Preacher, K. J., Zyphur, M. J., & Zhang, Z. (2010). A general multilevel SEM framework for
assessing multilevel mediation. Psychological Methods, 15, 209–233.
Raudenbush, S. W. (2001). Comparing personal tranectories and drawing causal inferences from
longitudinal data. Annual Review of Psychology, 52, 501-525.
Raykov, T. & Marcoulides, G. A. (2006). A first course in structural equation modeling (2nd
edition). Mahwah, NJ: Lawrence Erlbaum.
Willett, J. & Sayer, A. (1996). Cross-domain analysis of change over time: Combining growth
modeling and covariance structure analysis. In G. Marcoulides and R. Schumacker
(Eds.), Advanced structural equation modeling: Issues and techniques (pp. 125-158).
Mahwah, NJ: Lawrence Erlbaum.
Endnotes
i
The measurement part of the model can therefore be specified as follows:
 y1 
 y2
 
 y3
 
 y4
ii
1
1

1

1
0
1 
2

3
 11



 0  0  22

+
 1  0 0  33 
 


0 0 0  44 
(A1)
The psi matrix corresponds to:
11
Ψ = 
 21

,
 22
(A2)
Ronald H. Heck
15
EDEP 768 (S2011): Seminar in Structural Equation Modeling
April 4, 2011
University of Hawai‘i at Mānoa Specifying Latent Curve and Other Growth (Rev. 12-1-2014)
iii
The structural model can be referred to as follows:
0

0   0  0


    0
 1  1

0
1,3 1,4 
0  2,3  2,4  0   0 

0 0
0  1   1 
0
0
0

0 
(A3)