SEM and Longitudinal Data Latent Growth Models

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SEM and Longitudinal DataLatent Growth Models

• UTD
• 07.04.2006

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Why Growth models?

• Arent 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).

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Number of cigarettes smoked after meal as a
function of the day of the course
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• 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.

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• 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
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• 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 individuals 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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• Exerciseis 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
• Anomia11Intercept 0Slope e1
• Anomia21Intercept 1Slope 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
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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
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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
• Anomiaintercept slope1tslope2t2
• However, this is more rare in sociology and
  political sciences. It might be reasonable in
  contexts such as learning, tobacco reduction etc.

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3-factor polynomial LGM
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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.

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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.

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Single-indicator model without auto-correlation
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multiple-indicator model without auto-correlation
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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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Measurement Invariance Equal factor loadings
across groups
Group A
Group B
dB11
Item a
dA11
Item a
lB111
lA111
fB11 k B1
fA11 k A1
?Bt1
?At1
lB21
lA21
dB22
dA22
Item b
Item b
lB31
lA31
Item c
Item c
dB33
dA33
fB21
fA21
dB44
dA44
Item d
Item d
lB421
lA421
?Bt2
?At2
lB52
lA52
Item e
Item e
dB55
dA55
lB62
lA62
fB22 k B2
fA22 k A2
dB66
dA66
Item f
Item f
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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

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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

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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

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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).

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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.

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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.

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Growth MIMIC model anomia
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• 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.

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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.

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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.

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• 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.

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• 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?

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• 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.

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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.

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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.

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• 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.

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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.

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Factor-of-curves LGM
d2
d1
d3
d4
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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.

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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.

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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.

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Figure 3. The development of the experiment
before and after the move to Stuttgart
The intervention
2-3 weeks
6-7 weeks
4 weeks
The move
First questionnaire was sent
Second questionnaire was sent
Third questionnaire was sent
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Figure 7a. The latent curve model with a multi
group analysis for the low intention group
(standardized coefficients).
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Figure 7b. The latent curve model with a multi
group analysis for the high intention group
(standardized coefficients).

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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.

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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.

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• 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.

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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.

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Unconditional ALT model- exogenous time 1
construct
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Conditional ALT model- endogenous time 1 construct
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Conditional ALT model- exogenous time 1 construct
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Bivariate unconditional ALT model
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Bivariate Conditional
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Third order LGM

• An example of a third order LGM.

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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
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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.
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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
He01br One 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.
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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.
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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.

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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.

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• 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.

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• Thank you very much for your attention!!!!