Transcript SEM and Longitudinal Data Latent Growth Models

SEM and Longitudinal Data
Latent Growth Models
UTD
07.04.2006
1
Why Growth models?
• Aren‘t autoregressive and cross-lagged models
enough to test change and relationships over
time?
• 1) In autoregressive models we can see stability
over time but not type of development.
• We might have a stability of 1 – that is the
relative placement of people is unchanged, and
still everyone increases (or decreases).
2
Number of cigarettes smoked after
meal as a function of the day of the
course
6
5
4
person1
3
person2
2
person3
1
0
0
1
2
person1
1
2
3
person2
2
3
4
person3
3
4
5
3
• Stability is 1 in an autoregressive model. Higher
ones remain higher, and lower ones remain
lower.
• However, there is a development. They all
increase the number of cigarettes smoked. We
cannot see it in the autoregressive model.
• We need a developmental model, which takes
into account this development, but- also the
differences in development across individuals.
4
• In the example, each individual had an
intercept and a slope.
• Person1 had a slope 1, and an intercept 1
• Person2 had a slope 1, and an intercept 2
• Person3 had a slope 1, and an intercept 3
• The mean of their slope is 1
• The mean of their intercept is 2
• The developmental model should take this
individual information into account
• Still, the model should allow us to study
development at the group level
5
The Latent Growth Curve Model
• These criteria are met by the growth curve
model. Meredith and Tissak (1990) belonged to
the first to develop the growth model
mathematically.
• The model uses an SEM methodology
• The results are meaningful when there is time
gap between the measurements, and not just
repeated measures
• How long the time gap is between the time
points- is also meaningful
• The number of time points and the spacing
between time points across individuals should
be the same
6
• The latent factors in the growth model are
interpreted as common factors representing
individual differences over time.
• Remark: Latent growth model was developped
from ANOVA, and expanded over time.
• Basically, with two time points we can have only a
linear process of change. However, for deductive
purpose, we will start with modeling a growth
model for two time points, and then expand it to
more points in time.
7
A two-factor LGM for anomia for 2
time points
VarSlope
VarIntercept
Slope
Intercept
1
0
1
1
Anomia1
1
e1
Anomia2
1
e2
8
• Intercept: The intercept represents the common or
mean intercept for all individuals, since it has a
factor loading of 1 to all the time points. In the
previous example it will have a mean 2. It is the
point where the common line for all individuals
crosses the y axis.
• It presents information in the sample about the
mean and variance of the collection of intercepts
that characterize each individual‘s growth curve.
9
• Slope: It represents the slope of the sample. In
this case it is the straight line determined by the
two repeated measures. It also has a mean and
a variance, that can be estimated from the data.
• Slope and intercept are allowed to covary.
• In this example with two time points, in order to
get the model identified, the coefficients from the
slope to the two measures have to be fixed. For
ease of interpretation of the time scale, the first
coefficient is fixed to zero.
• With a careful choice of factor loadings, the
model parameters have familiar straightforward
interpretations.
10
• Exercise:is the model identified? How many df?
How many parameters are to be estimated?
• In this example:
• The intercept factor represents initial status
• The slope factor represents the difference
scores anomia2-anomia1 since:
• Anomia1=1*Intercept + 0*Slope + e1
• Anomia2=1*Intercept + 1*Slope + e2
• If errors are the same then
• Anomia2 – Anomia1 = Slope
11
• This model is just identified (if we set the
measurement errors to zero). By expanding the
model to include error variances, the model
parameters can be corrected for measurement
error, and this can be done when we have three
measurement time points or more.
• Three or more time points provide an
opportunity to check non linear trajectories.
• For those interested, Duncan et al. Shows the
technical details for this model on p. 15-19.
12
A two-factor LGM for anomia for 3
time points
VarSlope
VarIntercept
Intercept
Slope
1
0
1
2
1
1
Anomia1
1
Anomia2
Anomia3
1
1
e1
e2
e2
13
Representing the shape of growth
• With three points in time, the factor
loadings carry information about the shape
of growth over time.
• In this example we specify a linear model.
We have reasons to believe that anomia is
increasing as a linear process, and this
way we can test it.
• If we are not sure, we can test a model
where the third factor loading is free
14
A two-factor LGM for anomia. 3rd
time point free
VarSlope
VarIntercept
Intercept
Slope
1
0
1
m
1
1
Anomia1
1
Anomia2
Anomia3
1
1
e1
e2
e2
15
Parallel stability
Linear stability
level
level
time
time
Strict stability
Monotone stability
level
level
time
16
• Sometimes there are reasons to believe that the
process is not linear. For example, a process
might take a quadratic form.
• In this case, one can model a three-factor
polynomial LGM
• Anomia=intercept +slope1*t+slope2*t2
• However, this is more rare in sociology and
political sciences. It might be reasonable in
contexts such as learning, tobacco reduction etc.
17
3-factor polynomial LGM
VarIntercept
Slope- linear
component
Intercept
1
1
VarSlope
2
0
1
1
Anomia1
1
Anomia2
VarSlope
Slope- quadr.
component
0
1
4
Anomia3
1
1
e1
e2
e2
18
Summary1
• In all the examples shown we use LGM when we
believe that the process at hand is a function of
time.
• What is the meaning of the covariance between
slope and intercept? Intercept represents the
initial stage, and slope the change. A negative
covariance suggests that people with a lower
initial status, change more and people with a
higher initial status change less.
• For positive covariances: people with a higher
initial status change more, and people with a
lower initial status change less.
19
Summary 2
• There is no direct test for cross lagged
effects.
• The means of the latent slope and the
latent intercept represent the
developmental process over time for the
whole group; their variance represents the
individual variability of each subject
around the group parameters.
20
Single-indicator model vs. multipleindicator model
• Instead of using a single-scale score to
measure at each time point
authoritarianism or anomie for example,
we could use latent factors to estimate
these constructs, and could therefore be
purged from measurement error.
21
Single-indicator model
without auto-correlation
1
aut1
0,
e1
0
0,
1
Slope
1
aut2
1
1
0,
e2
0,
m
Intercept
0,
1
aut3
1
e3
22
multiple-indicator model without autocorrelation
e1
e2
e3
0,
0,
0,
1
1
1
au1_1au2_1au3_1
1
0
Authori1
0
0,
0,
1
1
0,
0,
res1
0,
e4
e5
e6
1
1
1
au1_2au2_2au3_2
Slope
1
1
0
0,
1
Authori2
res2
1
0,
m
Intercept
0
1
0,
1
Authori3
res3
1
au1_3au2_3au3_3
1
0,
e7
1
0,
e8
1
0,
e9
23
In a 2nd order LGM
• The same 1st order variable is chosen as
the scale indicator for each first-order
factor. Corresponding variables whose
loadings are free have those loadings
constrained to be equal across time. This
ensures a comparable definition of the
construct over time (referred to as
„stationarity“, Hancock, Kuo & Lawrence
2001, Tisak and Meredith 1990).
24
Measurement Invariance:
Equal factor loadings across groups
Group A
dA11
dA22
Item a
Item b
lA11=1
lA21
lA31
dA33
Group B
dB11
f 11
k A1
A
At1
dB22
Item a
Item b
lB11=1
lB21
lB31
dB33
Item c
Bt1
fB11
k B1
Item c
fB21
fA21
dA44
d
A
55
dA66
Item d
Item e
Item f
dB44
lA42=1
lA52
lA62

dB55
A
t2
fA22
k A2
dB66
Item d
Item e
lB42=1
lB52
lB62
Item f
Bt2
fB22
k B2
25
Steps in testing for Measurement Invariance
between groups and/or over time
• Configural Invariance
• Metric Invariance
• Scalar Invariance
• Invariance of Factor Variances
• Invariance of Factor Covariances
• Invariance of latent Means
• Invariance of Unique Variances
26
Steps in testing for Measurement Invariance
• Configural Invariance
• Metric Invariance
• Equal factor loadings
• Same scale units in both groups/time points
• Presumption for the comparison of latent means
• Scalar Invariance
• Invariance of Factor Variances
• Invariance of Factor Covariances
• Invariance of latent Means
• Invariance of Unique Variances
27
Full vs. Partial Invariance
• Concept of ‘partial invariance’ introduced by
Byrne, Shavelson & Muthén (1989)
• Procedure
• Constrain complete matrix
• Use modification indices to find non-invariant
parameters and then relax the constraint
• Compare with the unrestricted model
• Steenkamp & Baumgartner (1998): Two indicators
with invariant loadings and intercepts are
sufficient for mean comparisons
• One of them can be the marker + one further
invariant item
28
Autocorrelation
• As in the autoregressive model, we
believe that measurement errors of
repeated measures are related to one
another. Therefore, we correlate them
(Hancock, Kuo & Lawrence 2001, Loehlin
1998).
29
Latent Curve Model with Autocorrelations
intercept_anom
slope_anom
0
1
1
1
1
0,
0,
0,
e1_t2
1
e2_t1
e1_t1
1 0
10
AN01W1R
1 AN02W1R
0
anom1
0,
0,
0
AN01W2R
1
e2_t2
1
0
AN02W2R
0
anom2
0,
res_an1
e1_t3
1
0,
e2_t3
1
0
0
AN01W3R1 AN02W3R
0
anom3
1
1
m
0,
res_an2
1
0,
res_an3
30
Intercepts
• In a 2nd order factor LGM, intercepts for
corresponding 1st order variables at
different time points are constrained to be
equal, reflecting the fact that change over
time in a given variable should start at the
same initial point.
31
MIMIC and LGM, time-invariant covariates
in the latent growth modeling
• Sometimes a model in which longitudinal
development is predicted by an intercept
and growth curve is too restrictive. Such a
model is called unconditional. In such a
case we may try to predict the latent slope
and intercept by background variables (for
example demographic variables), which
are time invariant. This would be called a
conditional model.
32
Growth MIMIC model – anomia
age
east/west
education
right/left
0,
D1
0,
1
1
intercept_anom
slope_anom
0
1
1
e1_t2
1
e2_t1
e1_t1
1 0
10
AN01W1R
1 AN02W1R
0
anom1
0,
0,
0
AN01W2R
1
e2_t2
1
0
AN02W2R
0
anom2
0,
res_an1
e1_t3
1
0,
e2_t3
1
0
0
AN01W3R1 AN02W3R
0
anom3
1
1
m
1
1
0,
0,
0,
D2
0,
res_an2
1
0,
res_an3
33
• Another complication: the intercept and
slope may be not only conditioned on
some other variables, they could also
cause them. For example, the intercept of
anomia could be a cause of a variable
named satisfaction in life.
34
e1
e2
1
e3
1
e4
1
e5
1
e6
1
1
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1
1
1
1
1
an1
an2
1
1
resint
an3
res2
res1
1
1
1
1
1
res3
2
1
1
Int_an
1
Slope_an
resslo
ALT ER
e7
Satisfaction
35
Analyzing growth in multiple
populations
• Sometimes our data contains information on
several populations: males and females,
different age cohorts, people from former east
and west Germany, voters of right and left wing
parties, ethnicities, treatment and control groups
etc.
• The SEM methodology to analyze multiple
groups can be applied also here. We can
compare the means of the slope and intercept
latent variables as well as growth parameters,
equality of covariance between slope and
intercept etc.
36
The coding of time (Biesanz, DeebSossa, Papadakis, Bollen and Curran,
2004)
• Misinterpretation regarding the relationships among growth
parameters (intercepts and slopes) appear frequently.
Therefore it is important to pay attention to the coding of time
• Covariance between intercept and slope, and variance of the
intercept and slope are directly determined by the choice of
coding
• When the coefficient between slope and the first time point is
set to zero, the covariance and variances are related to the first
time point. For example, a negative covariance between the
slope and intercept indicates that at the first (0) time point
people with a lower starting point change more quickly. It is not
necessarily true for later time points.
37
• If we are interested at the relation between the
intercept and the slope at a later time point, for
example the second one, we have to fix at this
point the coefficient from the slope to zero. The
first coefficient will change from 0 to -1, and the
third coefficient will change from 2 to 1.
• It is useful to code the coefficients from the slope
according to the time interval on a yearly basis, if
we believe in a linear process.
38
• Example: if we have data collected in January,
then in July, and then again in July in the
following year, a possible coding of the
coefficients from the slopes to the
measurements could be:
• 0, 0.5 (since the measurement took place half a
year later) and 1.5.
• Exercise: If we are interested in the relations
between the slope and the intercept at the
second time point, how could we code the
coefficients?
39
• Answer: -0.5, 0 and 1.
• Using a yearly basis, we keep the interpretation
simple.
• If we have a quadratic model, the interpretation of
the highest order coefficient (for example its
variance) does not change with different codings
and placement of time origins. But the interpretation
of lower order terms (intercept and linear slope)
does.
• The choice of where to place the origin of time has
to be substantially driven. This choice determines
that point in time at which individual differences will
be examined for the lower order coefficients.
40
Level of
anomia
2.0
1.5
1.0
time
41
Coding and Mimic
• The meaning of the variance of the
intercept and the slope changes in Mimic
models. If the intercept is explained
(conditioned) by age for example, the
residual variance of the intercept indicates
the variability across individuals in the
starting point not accounted for by age.
• This should be taken into account when
we interpret our results.
42
The bivariate latent trajectories (growth
curve) analysis
• We can extend the univariate latent
trajectory model to consider change in two
or more variables over time.
• The bivariate trajectory model is simply the
simultaneous estimation of two univariate
latent trajectory models.
• The relation between the random
intercepts and slopes is evaluated for each
series. Then it is possible to determine
whether development in one behavior
covaries with other behaviors.
43
0,
0,
0,
a1
1
a2
1
b1
0,
0,
b2
a3
1
0,
b3
1 0
0
AU03W1RAU04W1R
1
0
1 0
0
AU03W2RAU04W2R
1
0
1 0
0
AU03W3R
AU04W3R
1
0
auto1
auto2
auto3
1
1
0
0,
a_res0
1
1
1
0,
1
a_res1
slope_anom
0
1
e1_t1
10
e1_t2
1
e2_t1
1 0
AN01W1R
1 AN02W1R
0
anom1
0,
0,
0
AN01W2R
1
e2_t2
1
0
AN02W2R
0
anom2
0,
res_an1
e1_t3
1
0,
e2_t3
1
0
0
AN01W3R
1 AN02W3R
0
anom3
1
1
m
1
1
0,
0,
a_res2
slope_aut
intercept_anom
0,
0,
0,
intercept_aut
1
1
n
0,
res_an2
1
0,
res_an3
44
• So far we could demonstrate LGM which allow
multiple measures, multiple occasions and
multiple behaviors simultanuously over time.
• We could estimate the extent of covariation in
the development of pairs of behaviors.
• We can go one level higher, and extend the test
of dynamic associations of behaviors by
describing growth factors in terms of common
higher order constructs.
45
Factor-of-curves LGM
• To test whether a higher order factor could
describe the relations among the growth factors
of different processes, the models can be
parameterized as a factor of curves LGM.
• The covariances among the factors are
hypothesized to be explained by the higher
order factors (McArdle 1988).
• The method is useful in determining the extent to
which pairs of behaviors covary over time.
• Rarely used. The test if the approach is better
can be done by comparing the fit measures of
alternative models.
46
Factor-of-curves LGM
e1
e2
1
e3
1
e4
1
e5
1
e6
1
1
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1
1
1
1
1
an1
an2
1
1
1
1
1
common
intercept
d1
1
1
res3
2
d2
Slope_an
Int_an
1
common
slope
L
L
d4
Slope_au
Int_au
1
1
1
1
res4
1
1
1
res6
au3
au2
1
d3
2
1 res5
au1
1
an3
res2
res1
1
1
1
1
1
AU04W3R
AU03W1R
AU04W1R AU03W2R
AU04W2RAU03W3R
1
1
1
e7
e8
e9
1
e10
1
e11
1
e12
47
Missing values and LGM
• As in AR models, missing data constitute a
problem in LGM. Also here we distinguish
between 3 kinds of MD: MCAR, MAR and
MNAR.
• The diagnosis and solutions discussed in
the AR apply also for LGM models.
48
Estimating Means and Getting the
model identified
• As Sörbom (1974) has shown, in order to estimate the means,
we must introduce some further restrictions:
• 1) setting the mean of the latent variable in one group-the
reference group- to zero. The estimation of the mean of the
latent variable in the other group is then the mean difference
with respect to the reference group. In the growth model, one
could alternatively set all intercepts of the constructs in both
groups to zero and intercept of one indicator per construct to
zero (constraining the second to be equal across time points),
and then compare the means of the latents mean and
intercepts in both groups.
• 2) in case of a one group analysis: setting the measurement
models invariant across time, since it makes no sense to
compare the means of constructs having a different
measurement model over time. At least one intercept (of an
indicator) per construct has to be set equal across time.
49
Ex10: Means and intercepts
comparison of two groups-west
0;
0;
e1
0;
e2
In_an1_west 1
0;
e3
1 0
0;
e4
0;
e5
1
1 0
In_an1_west
e6
1
1 0
In_an1_west
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1
1
1
1
1
an1
1
0;
an2
res1
0
1
1
Int_an
0
1
1 0;
an3
res2
1
1
0
1 0;
res3
2
Slope_an
50
Ex10: Means and intercepts
comparison of two groups- east
0;
0;
e1
0;
e2
In_an1_east 1
0;
e3
1 0
0;
e4
0;
e5
1
1 0
In_an1_east
e6
1
1 0
In_an1_east
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1
1
1
1
1
an1
1
0;
an2
res1
0
1
1
Int_an
0
1
1 0;
an3
res2
1
1
0
1 0;
res3
2
Slope_an
51
Ex10: Means and intercepts
comparison of two groups-west
0; ,24
e1
,03
0; ,23
0; ,19
e2
1
e3
1 0
1,03
0; ,14
0; ,18
e4
e5
1 0
1 ,03
0; ,13
e6
1 0
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1,001,00
1,001,00
1,001,00
an1
1
0; ,15
res1
0
1,00
1,00
Int_an
2,44; ,40
an2
0
1,00
1 0; ,20
an3
res2
1,00
0
1 0; ,08
res3
2,00
Slope_an
,18; ,04
52
Ex10: Means and intercepts
comparison of two groups- east
0; ,31
e1
,05
0; ,26
0; ,21
e2
1
e3
1 0
1,05
0; ,20
0; ,15
e4
e5
1 0
1 ,05
0; ,20
e6
1 0
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1,001,00
1,001,00
1,001,00
an1
1
0; ,14
res1
0
1,00
1,00
Int_an
2,73; ,37
an2
0
1,00
1 0; ,21
an3
res2
1,00
0
1 0; ,18
res3
2,00
Slope_an
,14; ,00
53
Ex10: Means and intercepts
one group
0;
0;
e1
0;
e2
in_an1 1
0;
e3
1 0
1
in_an1
0;
e4
0;
e5
1 0
1in_an1
e6
1 0
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1
1
1
1
1
an1
1
0;
an2
res1
0
1
1
Int_an
0
1
1 0;
an3
res2
1
1
0
1 0;
res3
2
Slope_an
54
Ex10: Means and intercepts
one group
0; ,26
e1
,04
0; ,24
0; ,20
e2
1
e3
1 0
1,04
0; ,16
0; ,17
e4
e5
1 0
1 ,04
0; ,15
e6
1 0
AN01W1R
AN02W1R AN01W2R
AN02W2R AN01W3R
AN02W3R
1,001,00
1,001,00
1,001,00
an1
1
0; ,16
res1
0
1,00
1,00
Int_an
2,54; ,40
an2
0
1,00
1 0; ,20
an3
res2
1,00
0
1 0; ,12
res3
2,00
Slope_an
,16; ,02
55
Additional uses of LGM modelsIntervention studies
• 1) Using the multiple group option to test
effects of intervention programs
• 2) The effect of interventions in
experimental settings can be done also as
a mimic model
• See Curran and Muthen, 1999.
56
Figure 3. The development of the experiment before and after the move to Stuttgart
The
intervention
6-7 weeks
2-3
weeks
First
questionnair
e was sent
The
move
4 weeks
Second
questionnair
e was sent
Third
questionnair
e was sent
57
bb1
.17
.03
bb2
e2
.01
intervention
.33
.11
bb3
e3
Standardized estimates
chi-square=2.887 df=3 p-value=.409
rmsea=.000 pclose=.638 aic=24.887
58
.35
e1
bb1
.00
intervention
.59
Slope
.42
.00
.68
bb2
e2
.56
-.47
.46
.32
Intercept
.61
bb3
e3
Standardized estimates
chi-square=2.195 df=3 p-value=.533
rmsea=.000 pclose=.736 aic=24.195
59
Figure 7a. The latent curve model with a multi group analysis for the low intention group (standardized coefficients).
.36
e1
bb1
.00
intervention
.60
Slope
.45
.09
.72
bb2
e2
.56
-.48
.49
Intercept
.33
.62
bb3
e3
Standardized estimates
chi-square=13.199 df=8 p-value=.105
rmsea=.054 pclose=.399 aic=53.199
60
Figure 7b. The latent curve model with a multi group analysis for the high intention group (standardized coefficients).
.25
e1
bb1
.00
intervention
.50
Slope
.43
-.18
.74
bb2
e2
.44
-.52
.52
Intercept
.25
.51
bb3
e3
Standardized estimates
chi-square=13.199 df=8 p-value=.105
rmsea=.054 pclose=.399 aic=53.199
61
Control and intervention/treatment groups
Curran and Muthen 1999
E3
E2
E1
1
1
1
1
1
AN01W3R
AN01W2R
AN01W1R
2
1
1
SLOPE
ICEPT
1
1
1
1
1
1
1
ICEPT
AN01W3R
AN01W2R
AN01W1R
1
E3
E2
E1
2
2
SLOPE
Treatment
SLOPE
62
ALT/Hybrid Modeling
Goals
• Combining features of both autoregressive
and latent growth curve models to result in
a more comprehensive model for
longitudinal data than either the
autoregressive or latent trajectory model
provide alone.
63
Model specification:
unconditional model
• We incorporate key elements from the latent
trajectory and autoregressive models in the
development of the univariate ALT model: from the
latent trajectory model we include the random
intercept and random slope factors to capture the
fixed and random effects of the underlying
trajectories over time. From the autoregressive
model we include the standard fixed autoregressive
parameters to capture the time specific influences
between the repeated measures themselves.
• The mean structure enters solely through the latent
64
trajectory factors in the synthesized model.
• Usually we will treat the first time point
measurement as predetermined in the ALT
model and it can be expressed simply by an
unconditional mean and an individual deviation
from the mean. It will correlate with the intercept
and the slope.
• There are some instances where treating the
initial measure as endogenous will be required
in order to achieve identification (For equations,
see Bollen/Curran 2004 page 349-352).
• we assume the residuals have zero means and
are uncorrelated with the exogenous variables.
65
Identifying the ALT model
• 1) With five or more waves of data, the model is
identified while treating the wave one y variable as
predetermined without making any further
assumptions.
• 2) With four waves we need a constant
autoregressive parameter.
• 3) If we have only three waves of data, we can
have an identified model when we assume an
equal autoregressive parameter throughout the
past, make the wave one endogenous, and
introduce further (nonlinear) constraints for the first
66
wave.
Unconditional ALT model- exogenous time 1 construct
0,
0,
e1
0,
e2
1
e3
1
1
au1_1au2_1au3_1
1
0,
Authori1
0,
0,
0,
0,
e4
e5
e6
1
1
1
au1_2au2_2au3_2
Slope
1
1
0
0,
1
Authori2
res2
1
0,
m
Intercept
0
1
0,
1
Authori3
res3
1
au1_3au2_3au3_3
1
0,
e7
1
0,
e8
1
0,
e9
67
Conditional ALT model- endogenous time 1 construct
0,
0,
e1
e2
1
education age
0,
e3
1
1
au1_1au2_1au3_1
east/
west
1
0,
0
res1
Authori1
0,
0
1
0,
0
eb
0,
0,
e4
e5
e6
1
1
1
au1_2au2_2au3_2
Slope
1
1
1
0
0,
1
Authori2
1
0,
0 1
res2
ea
m
Intercept
0
1
0,
1
Authori3
res3
1
au1_3au2_3au3_3
1
0,
e7
1
0,
e8
1
0,
e9
68
Conditional ALT model- exogenous time 1 construct
0,
0,
e1
e2
1
education age
0,
e3
1
1
au1_1au2_1au3_1
east/
west
1
0,
Authori1
0,
0,
0
1
eb
0,
0,
e4
e5
e6
1
1
1
au1_2au2_2au3_2
Slope
1
1
0
0,
1
Authori2
1
0,
0 1
res2
ea
m
Intercept
0
1
0,
1
Authori3
res3
1
au1_3au2_3au3_3
1
0,
e7
1
0,
e8
1
0,
e9
69
Bivariate unconditional ALT model
0,
0,
intercep_aut
0,
slope_aut
0,
0,
a1
1
1
b1
a2
1
0,
1 1
b2
1 0
0,
m
a3
1 0
0,
b3
1
1 0
0
AU03W1R
AU04W1R
1
0,
0
AU03W2R
AU04W2R
1
0
0
AU03W3R
AU04W3R
1
0
auto1
auto2
auto3
1
1
0,
0,
a_res2
a_res1
0,
0,
e1_t1
10
e2_t1
1 0
AN01W1R
1 AN02W1R
0,
0,
e1_t2
1
0
AN01W2R
1
anom1
e2_t2
1
0
AN02W2R
0
0,
intercep_ano
e2_t3
e1_t3
1
1
0
0
AN01W3R
1 AN02W3R
0
anom2
1
0,
0,
0,
anom3
1
1
0,
res_an2
1
n
0,
slope_ano
1
0,
res_an3
70
Bivariate Conditional
0,
0,
e3
e2
1
1
0
0
intercep_aut
0,
slope_aut
e1
0,
0,
0,
1
0,
0,
1
b2
1
0,
m
a3
1
1 0
0
AU03W1R
AU04W1R
1
0
1
0
0
AU03W2R
AU04W2R
1
0
0
0
AU03W3R
AU04W3R
1
0
auto1
auto2
auto3
a1
11
b1
a2
1
b3
1
1
1
0,
0,
a_res2
a_res1
age
educ
0,
0,
e1_t1
10
e2_t1
1 0
AN01W1R
1 AN02W1R
0
0,
e1_t2
1
0
AN01W2R
1
anom1
1
0
AN02W2R
0
1
0
0,
1 intercep_ano
e5
e2_t3
1
e1_t3
1
0
0
AN01W3R
1 AN02W3R
0
anom2
0,
e4
e2_t2
1
0,
0,
0,
anom3
1
1
0,
res_an2
0, 1
e6
1
n
0
slope_ano
1
0,
res_an3
71
Third order LGM
• An example of a third order LGM.
72
e1
1
e4
1
GFE t1
anomia t1
1
1
e5
1
1
Intercept
anomia
Intercept
GFE
1
e2
1
1
anomia t2
GFE t2
1
e6
11
anomia t3
1
Slope
anomia
2
Slope
GFE
e3
1
2
GFE t3
73
neg.attitude
toward
foreigners
antisemitism
racism
GFE
homophobia
rights of the
established
exclusion of
homeless
people
islamophobia
74
0;
d7
0;
0;
d6
10
d8
0;
1
d9
f4
f8f
0
0;
In_h 1 z4
1
0
h
anti z3
h10
an fr
1 hom
0
0;
ho
h2
z6
1
0
ob
GRE1
obd
o
1
1
In_o 0
o11 o21 z7
0;
d12
i
In_i
d13
1
10
0;
0
10
Intercept
d14
r1
0
0;
1
Ras
0
1
0
1
1In_r
1
r3
1 0;
1 r
z2
1
1z
d1
z1
0;
e4
d17
e3
d16
In_e 0;
e 1
eta
0;
2d7
0
1
i11 i21
0;
0;
d2
frem
z8
Isla
0;
1
et
10;
is0
0;
1
0;
d4
1
1
In_a In_f
a2
0; 1a a1
0;
1
0;
d5
0;
01
2z3
0; ana
0
frem2
Slope
obd2
oa
1
0;
3d7
10
0;
3d5
2o1
2o2
11
0;
0;
3d4
1
In_a 1
In_f
10
0;
3a23a1 3f4 3f8
ab
1
1
2
0
fb
0
0;
In_h
1
0 0;
hb
1 anti3 3z3
3d8 3h1
0;
3z4
f rb
10 hom3
anb
1
0
hob
3d9 3h2
0;
0
1
3z6
obb
GRE3
0;
1
1
In_o 0
3o1
1 3o2
1
0;
3d12
0;
3z7
0; ib
3d13
Isla3
In_i
10
2d12
2d13
2r1
1 ra
2z2
ia
In_i
0
Ras2
0;
2z8
eta
1 01
eta2
Isla2
2z7
1
0
In_r
1
0;
1
2z1
0
0;
1
2e42d17
ea In_e0;
1
2e32d16
2i110; 2i510;
2d14
0;
2d15
2z
3d1
1In_r
0
1
etb
isb
0
1
3d2
0;
2d1
0 1
3r3 3r1
frem31 0;
1rb
obd3
ob
0;
1
In_o 0
0;
2r3
0;
In_h 1 2z4
0
ha
2d8 2h1
hoa
0
0;
1 1 hom2
GRE2
0;
2d9 2h2
oba
isa0
2z6 1
0
0;
1
1
3d6
1
fra
0
1
0;
1
0;
2d2
1
0
0;
1
anti2
1
0;
2f4 12f8fa
0
d15
0;
2d4
In_a2d51
In_f 10
2a2
2a1
aa
1
0;
0;
2d6
3z2
0
1
eta3
1
0
13i1
0;
3i51
3d14
3d15
0;
1
Ras3
0
3e4
1
0;
0;
3z1
3d17
0;
eb In_e
1
1 3e3
1
0;
3d16
3z8
0;
3z
75
Level of latent variables
Content
First order
Latent variables of different aspects of
group related enmity, each measured
by two indicators: racism (r), enmity
towards foreigners (f), anti-Semitism
(a), enmity towards homosexuals (h),
enmity towards homeless people (ob),
Islam-phobia (i)and enmity of the nonestablished (eta)
Measured in 2002, 2003 and 2004 in
Germany on a representative sample of
the German population
Second order
GRE- Second order variable of group
related enmity
Third order
Growth variables- slope and intercept
76
Racism:
ra
01
r
Aussiedler (Russian immigrants with German ancestors) should be better
employed than foreigners, since they
have a German origin.
ra
03
r
The white people are justifiably leading in the world.
Foreigners Enmity
ff04 Too many foreigners live in Germany.
d1r
ff08 If working places become scarce, one should send foreigners
d1r living in Germany back to their home country.
Antisemitism
as Jews have too much influence in
01r Germany.
as Jews are to be blamed due to their
02r behavior for their persecution.
77
Heterophobia
1. Rejection of homosexuals
he01h
Marriage between two women or two men should be permitted.
he02hr
It is disgusting, when homosexuals kiss in public.
2. Rejection of disabled
One feels sometimes not comfortable in the presence of disabled
He01br people.
He02br Sometimes on is not sure how to behave with disabled people.
3. Rejection of homeless people
he01o
Homeless beggars should be removed from pedestrian
zones.
he02or The homeless people in towns are unpleasant.
78
Islamphobia
he01
m
he02
m
The Muslims in Germany should have the right to live according to
their belief.
It is only a matter of Muslims, if they call to pray over loudspeakers.
Rights of the established
ev03 One who is new somewhere should be at first satisfied with less.
r
ev04 Those who have always lived here should have more rights than those
r
who came later.
Classical sexism
sx03r
Women should take again the role of
wives and mothers.
79
Growth of group related enmity
(GFE)- slope and intercept
3rd level
2nd level
1st level
GFE 1st
time point
GFE 2nd time
point
r, f, a, h, ob, i,
eta- 1st time
point
r, f, a, h, ob, i,
eta- 2nd time
point
GFE 3rd time
point
r, f, a, h, ob, i,
eta- 3rd time
point
80
SUMMARY
4.0) Evaluation of the different strategies for
analysis of panel data in SEM
• Each of the two models (AR and LGC) has a
distinct approach to modeling longitudinal data.
Each has been widely used in many empirical
applications.
• Two key components of the autoregressive and
cross lagged models are the assumptions of
lagged influences of a variable on itself and that
the coefficients of effects are the same for all
cases, when we do not conduct a multiple-group
analysis.
81
Summary (continuation)
• In contrast, the latent trajectory model has no influences
of the lagged values of a variable on itself. The intercept
and the slope parameters governing the trajectories
differ over subjects in the analysis. Measurements are
modeled alternatively as a function of time.
• The LGM gives us a description of a process. We do not
get it from the AR.
• However, in bivariate Lgm we have the same problem as
in cross section: we have one slope trajectory and one
intercept trajectory variables for each process. It is again
not clear what is the cause of what…
• Each of these assumptions about the nature of changes
is empirically or theoretically plausible.
• The hybrid model combines for these reasons both
82
assumptions into one framework.
• Further SEM applications such as a multiple group
comparison, can also be done with the ALT model.
• In a discussion with Muthen, he criticizes the ALT
model. His critique concentrates in the difficulty to
interpret the parameters in this model.
• An alternative is to use continuous time modelling
with differential equations(Oud, Singer), but it is not
as straight forward to be applied as the AR and Lgm
modeling
• An alternative is to run AR and LGM models
separately. Depending on the research question,
each model would provide complementary answers.
83
• Thank you very much for your attention!!!!
84