sem and longitudinal data latent growth models
DESCRIPTION
SEM and Longitudinal Data Latent Growth Models. UTD 07.04.2006. 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. - PowerPoint PPT PresentationTRANSCRIPT
1
SEM and Longitudinal DataLatent Growth Models
UTD
07.04.2006
2
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).
3
Number of cigarettes smoked after meal as a function of the day of the
course
0
1
2
3
4
5
6
person1
person2
person3
person1 1 2 3
person2 2 3 4
person3 3 4 5
0 1 2
4
• 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.
5
• 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
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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
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• 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.
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A two-factor LGM for anomia for 2 time points
VarIntercept
Intercept
VarSlope
Slope
Anomia1 Anomia2
1
1 01
e1 e2
1 1
9
• 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.
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• 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.
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• 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
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• 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.
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A two-factor LGM for anomia for 3 time points
VarIntercept
Intercept
VarSlope
Slope
Anomia1 Anomia2
1
1 01
e1 e2
1 1
Anomia3
e2
1
2
1
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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
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A two-factor LGM for anomia. 3rd time point free
VarIntercept
Intercept
VarSlope
Slope
Anomia1 Anomia2
1
1 01
e1 e2
1 1
Anomia3
e2
1
m
1
16
level
level
level
level
timetime
time
Parallel stability Linear stability
Strict stabilityMonotone stability
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• 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.
18
3-factor polynomial LGM
VarIntercept
Intercept
VarSlope
Slope- linearcomponent
Anomia1 Anomia2
1
1 01
e1 e2
1 1
Anomia3
e2
1
1
VarSlope
Slope- quadr.component
2 0
14
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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.
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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.
21
Single-indicator model vs. multiple-indicator 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.
22
aut1
aut2
aut3
0,
e2
0,
e3
1
1
0,
Slope
0,
Intercept
0
1
m
1
1
1
0,
e11
Single-indicator model without auto-correlation
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0,
Slope
0,
Intercept
0
Authori1
0
Authori3
0
Authori2
au1_1
0,
e1
1
1
au2_1
0,
e21
au3_1
0,
e31
au1_2
0,
e4
1
1
au2_2
0,
e51
au3_2
0,
e61
au3_3
0,
e9
1
1au2_3
0,
e8
1au1_3
0,
e7
1
0,
res1
0,
res2
0,
res3
1
1
1
1
m
0
1
1
1
multiple-indicator model without auto-correlation
24
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).
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t2
=1
=1
t1
Item a
Item b
Item c
Item d
Item e
Item f
Measurement Invariance:
Equal factor loadings across groups
t2
=1
=1
t1
Item a
Item b
Item c
Item d
Item e
Item f
Group A Group B
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• Configural Invariance
• Metric Invariance
• Scalar Invariance
• Invariance of Factor Variances
• Invariance of Factor Covariances
• Invariance of latent Means
• Invariance of Unique Variances
Steps in testing for Measurement Invariance between groups and/or over time
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• 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
Steps in testing for Measurement Invariance
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• 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
Full vs. Partial Invariance
29
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).
30
0
anom1
0AN01W1R
0,
e1_t1 0AN02W1R
0,
e2_t11
0
anom2
0AN01W2R
0,
e1_t210
AN02W2R
0,
e2_t21
0,
res_an2
0
anom3
0AN01W3R
0,
e1_t30
AN02W3R
0,
e2_t3
0,
res_an3
1 1
1 1
1
1 11
intercept_anom slope_anom
0,
res_an1
1
1 11
1m
0
Latent Curve Model with Autocorrelations
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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.
32
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.
33
Growth MIMIC model – anomia
0
anom1
0AN01W1R
0,
e1_t1 0AN02W1R
0,
e2_t11
0
anom2
0AN01W2R
0,
e1_t210
AN02W2R
0,
e2_t21
0,
res_an2
0
anom3
0AN01W3R
0,
e1_t30
AN02W3R
0,
e2_t3
0,
res_an3
1 1
1 1
1
1 11
intercept_anom slope_anom
0,
res_an1
1
1 11
1m
0,D21
0
age education east/west right/left
0,
D11
34
• 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.
35
an1
AN01W1R
e1
1
1
AN02W1R
e2
1
1
an2
AN01W2R
e3
1
1
AN02W2R
e4
1
1
an3
AN01W3R
e5
1
1
AN02W3R
e6
1
1
Int_an Slope_an
11 1 1 2
res1 res2res3
1 11
ALTER
resint resslo1 1
Satisfactione7
36
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.
37
The coding of time (Biesanz, Deeb-Sossa, 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.
38
• 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.
39
• 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?
40
• 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.
41
time
Level of anomia
1.0
1.5
2.0
42
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.
43
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.
44
0
anom1
0AN01W1R
0,
e1_t1 0AN02W1R
0,
e2_t11
0
anom2
0AN01W2R
0,
e1_t210
AN02W2R
0,
e2_t21
0,
res_an2
0
anom3
0AN01W3R
0,
e1_t30
AN02W3R
0,
e2_t3
0,
res_an3
1 1
1 1
0,
a_res1
0,
a_res2
0
auto1
0AU03W1R
0,
a1
1
1 0AU04W1R
0,
b11
0
auto2
0AU03W2R
0,
a20
AU04W2R
0,
b2
1
1 1
0
auto3
0AU03W3R
0,
a30
AU04W3R
0,
b3
1
1 1
1 1
1
1 11
intercept_anom slope_anom
intercept_aut
0,
a_res0
1
0,
res_an1
1
1 1 1
1 11
1m
0
0,
slope_aut
0 1 n
45
• 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.
46
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.
47
Factor-of-curves LGM
an1
AN01W1R
e1
1
1
AN02W1R
e2
1
1
an2
AN01W2R
e3
1
1
AN02W2R
e4
1
1
an3
AN01W3R
e5
1
1
AN02W3R
e6
1
1
Int_an Slope_an
11 1 1 2
res1 res2res3
1 11
au1
AU04W1R
e8
1
1AU03W1R
e7
1
1
au2
AU04W2R
e10
1
1AU03W2R
e9
1
1
au3
AU04W3R
e12
1
1AU03W3R
e11
1
1
Int_au
1 11
res4 res5res61
1 1
Slope_au
1 2
commonintercept
commonslope
1
L
1
L
d1
d3d4
d2
48
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.
49
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.
50
Ex10: Means and intercepts comparison of two groups-west
0
an1
In_an1_west
AN01W1R
0;
e1
1
1 0
AN02W1R
0;
e2
1
1
0
an2
In_an1_west
AN01W2R
0;
e3
1
1 0
AN02W2R
0;
e4
1
1
0
an3
In_an1_west
AN01W3R
0;
e5
1
1 0
AN02W3R
0;
e6
1
1
Int_an Slope_an
11 1 1 2
0;
res1
0;res2 0;
res3
1 11
51
Ex10: Means and intercepts comparison of two groups- east
0
an1
In_an1_east
AN01W1R
0;
e1
1
1 0
AN02W1R
0;
e2
1
1
0
an2
In_an1_east
AN01W2R
0;
e3
1
1 0
AN02W2R
0;
e4
1
1
0
an3
In_an1_east
AN01W3R
0;
e5
1
1 0
AN02W3R
0;
e6
1
1
Int_an Slope_an
11 1 1 2
0;
res1
0;res2 0;
res3
1 11
52
Ex10: Means and intercepts comparison of two groups-west
0
an1
,03
AN01W1R
0; ,24
e1
1,00
1 0
AN02W1R
0; ,23
e2
1,00
1
0
an2
,03
AN01W2R
0; ,19
e3
1,00
1 0
AN02W2R
0; ,14
e4
1,00
1
0
an3
,03
AN01W3R
0; ,18
e5
1,00
1 0
AN02W3R
0; ,13
e6
1,00
1
2,44; ,40
Int_an,18; ,04
Slope_an
1,001,00 1,00 1,00 2,00
0; ,15
res1
0; ,20res2 0; ,08
res3
1 11
53
Ex10: Means and intercepts comparison of two groups- east
0
an1
,05
AN01W1R
0; ,31
e1
1,00
1 0
AN02W1R
0; ,26
e2
1,00
1
0
an2
,05
AN01W2R
0; ,21
e3
1,00
1 0
AN02W2R
0; ,20
e4
1,00
1
0
an3
,05
AN01W3R
0; ,15
e5
1,00
1 0
AN02W3R
0; ,20
e6
1,00
1
2,73; ,37
Int_an,14; ,00
Slope_an
1,001,00 1,00 1,00 2,00
0; ,14
res1
0; ,21res2 0; ,18
res3
1 11
54
Ex10: Means and interceptsone group
0
an1
in_an1
AN01W1R
0;
e1
1
1 0
AN02W1R
0;
e2
1
1
0
an2
in_an1
AN01W2R
0;
e3
1
1 0
AN02W2R
0;
e4
1
1
0
an3
in_an1
AN01W3R
0;
e5
1
1 0
AN02W3R
0;
e6
1
1
Int_an Slope_an
11 1 1 2
0;
res1
0;res2 0;
res3
1 11
55
Ex10: Means and interceptsone group
0
an1
,04
AN01W1R
0; ,26
e1
1,00
1 0
AN02W1R
0; ,24
e2
1,00
1
0
an2
,04
AN01W2R
0; ,20
e3
1,00
1 0
AN02W2R
0; ,16
e4
1,00
1
0
an3
,04
AN01W3R
0; ,17
e5
1,00
1 0
AN02W3R
0; ,15
e6
1,00
1
2,54; ,40
Int_an,16; ,02
Slope_an
1,001,00 1,00 1,00 2,00
0; ,16
res1
0; ,20res2 0; ,12
res3
1 11
56
Additional uses of LGM models-Intervention 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.
57
First questionnaire was sent
Second questionnaire was sent
Third questionnaire was sent
The move
2-3 weeks
6-7 weeks4 weeks
The intervention
Figure 3. The development of the experiment before and after the move to Stuttgart
58
intervention
bb1
.03
bb2
.11
bb3
.01e2
e3
.33
.17
Standardized estimateschi-square=2.887 df=3 p-value=.409rmsea=.000 pclose=.638 aic=24.887
59
intervention
.35
bb1
.42
bb2
.32
bb3
e2
e3
Slope
Intercept
.00
.68
.46
.59
.56
.61
-.47
e1
.00
Standardized estimateschi-square=2.195 df=3 p-value=.533 rmsea=.000 pclose=.736 aic=24.195
60
Figure 7a. The latent curve model with a multi group analysis for the low intention group (standardized coefficients).
intervention
.36
bb1
.45
bb2
.33
bb3
e2
e3
Slope
Intercept
.00
.72
.49
.60
.56
.62
-.48
e1
.09
Standardized estimateschi-square=13.199 df=8 p-value=.105 rmsea=.054 pclose=.399 aic=53.199
61
Figure 7b. The latent curve model with a multi group analysis for the high intention group (standardized coefficients).
intervention
.25
bb1
.43
bb2
.25
bb3
e2
e3
Slope
Intercept
.00
.74
.52
.50
.44
.51
-.52
e1
-.18
Standardized estimateschi-square=13.199 df=8 p-value=.105 rmsea=.054 pclose=.399 aic=53.199
62
ICEPT SLOPE
AN01W1R
1
AN01W2R
11
AN01W3R
1
2
E1
1
E2
1
E3
1
Control and intervention/treatment groupsCurran and Muthen 1999
E1 E2E3
AN01W1R AN01W2R AN01W3R
ICEPT SLOPE TreatmentSLOPE
1
1
1 1 1
2
1 21
1
63
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.
64
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 trajectory factors in the synthesized model.
65
• 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.
66
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 wave.
67
0,
Slope
0,
Intercept
0,
Authori1
0
Authori3
0
Authori2
au1_1
0,
e1
1
1
au2_1
0,
e21
au3_1
0,
e31
au1_2
0,
e4
1
1
au2_2
0,
e51
au3_2
0,
e61
au3_3
0,
e9
1
1au2_3
0,
e8
1au1_3
0,
e7
1
0,
res2
0,
res3
1
1
1
m
1
1
Unconditional ALT model- exogenous time 1 construct
68
Conditional ALT model- endogenous time 1 construct
0
Slope
0
Intercept
0
Authori1
0
Authori3
0
Authori2
au1_1
0,
e1
1
1
au2_1
0,
e21
au3_1
0,
e31
au1_2
0,
e4
1
1
au2_2
0,
e51
au3_2
0,
e61
au3_3
0,
e9
1
1au2_3
0,
e8
1au1_3
0,
e7
1
0,
res2
0,
res3
1
1
1
m
1
1
education age east/west
0,
eb
0,
ea
1
1
0
1
0,
res1
69
0
Slope
0
Intercept
0,
Authori1
0
Authori3
0
Authori2
au1_1
0,
e1
1
1
au2_1
0,
e21
au3_1
0,
e31
au1_2
0,
e4
1
1
au2_2
0,
e51
au3_2
0,
e61
au3_3
0,
e9
1
1au2_3
0,
e8
1au1_3
0,
e7
1
0,
res2
0,
res3
1
1
1
m
1
1
education age east/west
0,
eb
0,
ea
1
1
Conditional ALT model- exogenous time 1 construct
70
Bivariate unconditional ALT model
0,
anom1
0AN01W1R
0,
e1_t1 0AN02W1R
0,
e2_t11
0
anom2
0AN01W2R
0,
e1_t210
AN02W2R
0,
e2_t21
0,
res_an2
0
anom3
0AN01W3R
0,
e1_t30
AN02W3R
0,
e2_t3
0,
res_an3
1 1
1 1
0,
a_res1
0,
a_res2
0,
auto1
0AU03W1R
0,
a1
1
1 0AU04W1R
0,
b11
0
auto2
0AU03W2R
0,
a20
AU04W2R
0,
b2
1
1 1
0
auto3
0AU03W3R
0,
a30
AU04W3R
0,
b3
1
1 1
1 1
1
1 11
0,
intercep_aut0,
slope_aut
0,
intercep_ano0,
slope_ano
1 1 1 m
1 1 1 n
71
0
anom1
0AN01W1R
0,
e1_t1 0AN02W1R
0,
e2_t11
0
anom2
0AN01W2R
0,
e1_t210
AN02W2R
0,
e2_t21
0,
res_an2
0
anom3
0AN01W3R
0,
e1_t30
AN02W3R
0,
e2_t3
0,
res_an3
1 1
1 1
0,
a_res1
0,
a_res2
0
auto1
0AU03W1R
0,
a1
1
1 0AU04W1R
0,
b11
0
auto2
0AU03W2R
0,
a20
AU04W2R
0,
b2
1
1 1
0
auto3
0AU03W3R
0,
a30
AU04W3R
0,
b3
1
1 1
1 1
1
1 11
age
educ
0
intercep_aut0
slope_aut
0
intercep_ano0
slope_ano
1 1 1 m
1 1 1 n
0,
e2
0,
e3
0,
e1
0,
e40,
e50,
e6
1
1 1
1
1 1
Bivariate Conditional
72
Third order LGM
• An example of a third order LGM.
73
Interceptanomia
Slopeanomia
anomia t1
anomia t2
anomia t3
InterceptGFE
SlopeGFE
GFE t1
GFE t2
GFE t3
1
1
1
1
2
1
1
1
1
2
e41
e51
e61
e31
e21
e11
74
GFE
neg.attitudetoward
foreigners
racismantisemitism
homophobia
exclusion ofhomeless
people
islamophobia
rights of theestablished
75
0
Ras
0
r3
0;
d2
1
1 In_r
r1
0;
d1
r
10
frem
In_f
f4
0;
d5
1
1 0
f8
0;
d4
f
1
0
anti
0
a2
0;
d7
1
1 In_a
a1
0;
d6
a
1
0
hom0
h20;
d9
11
In_h
h10;
d8h1
0
obd0
o20;
d13
1
1
In_o
o10;
d12
o
1
0
Isla0
i20;
d15
1
1
In_i
i10;
d14
i
1
0
eta
0
e40;
d171 1
In_e
e30;
d16
e 1
0
GRE1
ho
iset
0;
z1
0;
z2
0;
z3
0;
z4
0;
z6
0;
z7
0;
z8
1
11
1
11
ob
an fr
1
0
Ras2
0
2r3
0;
2d2In_r
2r1
0;
2d1
0
frem2
In_f
2f4
0;
2d5 0
2f8
0;
2d4
0
anti2
0
2a2
0;
2d7 In_a
2a1
0;
2d6
0
hom20
2h20;
2d9
In_h
2h10;
2d8
0
obd20
2o20;
2d13
In_o
2o10;
2d12
0
Isla20
2i50;
2d15
In_i
2i10;
2d14
0
eta2
0
2e40;
2d17
In_e
2e30;
2d16
0
GRE2
0;
2z1
0;
2z2
0;
2z3
0; 0
2z4
0;
2z6 0;
2z7
0;
2z8
1
1
ra
11
1
fa
1
1
1
aa
1
11
ha1
1
1
oa 1
1
ia
1
11
ea1
isa
1
11
1
11
oba
fra
1
eta1
hoa
1
0
Ras3
0
3r3
0;
3d2In_r
3r1
0;
3d10
frem3
In_f
3f4
0;
3d50
3f8
0;
3d4
0
anti3
0
3a2
0;
3d7In_a
3a1
0;
3d6
0
hom30
3h20;
3d9
In_h
3h10;
3d8
0
obd3
0
3o20;
3d13
In_o
3o10;
3d12
0
Isla3
0
eta3
0
3e40;
3d17In_e
3e30;
3d16
0
GRE3
0;
3z1
0;
3z2
0;
3z30;
3z4
0;
3z6
0;
3z7
0;
3z8
1
1
rb
11
1
fb
1
1
1
ab
1
11
hb1
1
1
1 1
eb 1
hob
isbetb
1
11
1
1
obb
anbfrb
1
1
1
ob
1 1
0
3i50;
3d15
In_i
3i10;
3d14
11
ib 1
ana
0;
2z
0;
3z
1
1
Intercept
Slope
1
1
1
1
2
0;
1z
1
76
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 non-established (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
77
Racism:
ra01r
Aussiedler (Russian immigrants with German ancestors) should be better employed than foreigners, since they have a German origin.
ra03r
The white people are justifiably leading in the world.
Foreigners Enmity
ff04d1r
Too many foreigners live in Germany.
ff08d1r
If working places become scarce, one should send foreigners living in Germany back to their home country.
Antisemitism
as01r
Jews have too much influence in Germany.
as02r
Jews are to be blamed due to their behavior for their persecution.
78
Heterophobia1. 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
He01brOne feels sometimes not comfortable in the presence of disabled 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.
79
Islamphobia
he01m
The Muslims in Germany should have the right to live according to their belief.
he02m
It is only a matter of Muslims, if they call to pray over loudspeakers.
Rights of the established
ev03r
One who is new somewhere should be at first satisfied with less.
ev04r
Those who have always lived here should have more rights than those who came later.
Classical sexism
sx03r Women should take again the role of wives and mothers.
80
Growth of group related enmity (GFE)- slope and intercept
3rd level
1st level
2nd level GFE 1st time point
GFE 2nd time point
GFE 3rd time point
r, f, a, h, ob, i, eta- 1st time point
r, f, a, h, ob, i, eta- 2nd time point
r, f, a, h, ob, i, eta- 3rd time point
81
SUMMARY4.0) 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.
82
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 assumptions into one framework.
83
• 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.
84
• Thank you very much for your attention!!!!