Transcript Random effects as latent variables: SEM for repeated

Random effects as latent variables:
SEM for repeated measures data
Dr Patrick Sturgis
University of Surrey
Overview
• Random effects (multi-level) models for
repeated measures data.
• Random effects as latent variables.
• Specifying time in LGC models.
• Growth parameters.
• Linear Growth.
• Plotting observed and fitted growth.
• Predicting Growth.
• An example: Issue Voting
Repeated Measures & Random Effects
• A problem when analysing panel data is how to account for
the correlation between observations on the same subject.
• Different approaches handle this problem in different ways.
• E.g. impose different structures on the residual correlations
(exchangeable, unstructured, independent).
• Assume correlations between repeated observations arise
because the regression coefficients vary across subjects.
• So, we have average (or ‘fixed’) effects for the population as
a whole.
• And individual variability (or ‘random’) effects around these
average coefficients.
• This is sometimes referred to as a ‘random effects’ or ‘multilevel’ model.
SEM for Repeated Measures
• The primary focus of this course has been on how latent
variables can be used on cross-sectional data.
• The same framework can be used on repeated
measured data to overcome the correlated residuals
problem.
• The mean of a latent variable is used to estimate to the
‘average’ or fixed effect.
• The variance of a latent variable represents individual
heterogeneity around the fixed coefficient – the ‘random’
effect.
• For cross-sectional data latent variables are specified as
a function of different items at the same time point.
• For repeated measures data, latent variables are
specified as a function of the same item at different
time points.
A Single Latent Variable Model
E1
E1
1
X1
X11
E2
E2
E3
E3
1
1
X2
X12
X3
X13
E4
E4
1
X4
X14
same item at 4
4 different items
time points
Constrain
Estimate factor
factor loadings
loadings
LV
LV
Estimate mean and
variance of trajectory of
underlying
change
overfactor
time
Random Effects as Latent Variables
• So, it turns out that another way of estimating
growth trajectories on repeated measures data
is as latent variables in SEM models.
• The mean of the latent variable is the fixed
part of the model.
– It indicates the average for the parameter in the
population.
• The variance of the latent variable is the
random part of the model.
– It indicates individual heterogeneity around the
average.
– Or inter-individual difference in intra-individual
change.
Growth Parameters
• The earlier path diagram was an oversimplification.
• In practice we require at least two latent
variables to describe growth.
• One to estimate the mean and variance of
the intercept (usually denoting initial
status).
• And one to estimate the mean and variance
of the slope (denoting change over time).
Specifying Time in LGC Models
• In random effect models, time is included as an
independent variable:
yit  oi  1i xt  it
• In LGC models, time is included via the factor loadings
of the latent variables.
• We constrain the factor loadings to take on particular
values.
• The number of latent variables and the values of the
constrained loadings specify the shape of the trajectory.
A Linear Growth Curve Model
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E2
1
1
X1t1
Constraining values of
the intercept to 1 makes
this parameter indicate
initial status
E3
1
X1t2
1
1
ICEPT
E4
1
1
X1t3
1
1
X1t4
2
3
0
SLOPE
Constraining values of
the slope to 0,1,2,3
makes this parameter
indicate linear change
File structure for LGC
• For random effect models, we use ‘long’
data file format.
• There are as many rows as there are
observations.
• For LGC, we use ‘wide’ file formats.
• Each case (e.g. respondent) has only one
row in the data file.
An Example
• We are interested in the development of
knowledge of SEM during a course.
• We have measures of knowledge on
individual students taken at 4 time points.
• Test scores have a minimum value of zero
and a maximum value of 25.
• We specify linear growth.
Linear Growth Example
E1
E2
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1
X1t1
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1
1
E4
1
X1t2
mean=11.2 (1.4) p<0.001
variance =4.1 (0.8) p<0.001
E3
1
X1t3
1
1
2
X1t4
3
0
mean=1.3 (0.25) p<0.001
ICEPT
SLOPE variance =0.6 (0.1) p<0.001
Interpretation
• The average level of knowledge at time point
one was 11.2
• There was significant variation across
respondents in this initial status.
• On average, students increased their knowledge
score by 1.2 units at each time point.
• There was significant variation across
respondents in this rate of growth.
• Having established this descriptive picture, we
will want to explain this variation.
Graphical Displays
• It is useful to graph observed and fitted growth
trajectories.
• This gives us a clear picture of heterogeneity in
individual development.
• This is useful for determining which time
function(s) to specify.
• And can highlight model mis-specifications in a
way that is difficult to spot with just the numerical
estimates.
Observed Individual Trajectories
Fitted Trajectories
Explaining Growth
• Up to this point the models have been
concerned only with describing growth.
• These are unconditional LGC models.
• We can add predictors of growth to explain
why some people grow more quickly than
others.
• These are conditional LGC models.
Predicting Growth
• Some predictors of growth do not change during
the period of observation.
• E.g. sex, parental social class, date of birth.
• These are referred to as ‘fixed’ or time-constant.
• Other predictors change over time and may
influence the outcome variable.
• E.g. parental status, health status.
• These are referred to as time-varying covariates.
Fixed Predictors of Growth
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E2
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1
X1t1
1
Do men have a
different initial
status than
women?
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1
X1t2
1
1
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E4
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X1t3
1
2
X1t4
3
0
ICEPT
SLOPE
Gender
Do
men
grow
at
Does
initial
status
a differentrate
rateof
influence
than women?
growth?
(women = 0; men=1)
Example: Issue Voting
• Proximity to parties on issue dimensions strongly
related to political preferences...
• All previous investigations use between-person
analysis of cross-sectional data.
• Is individual change in issue proximity
correlated with individual change in party
evaluation over the 5 years of the panel?
• Is this relationship moderated by level of political
knowledge?
• British Election Panel Study 1997-2001.
Direction and Proximity
Direction –
voter prefers
party strongest
on same side of
issue as them
Dij 
Proximity –
voter prefers
party closest to
them
nki
 (x
* y jk ) / nki
V
P2
ik
P3
P1
Pij   xik  y jk / nki
k 1
k 1
Left
nki
Penalty applied to parties outside ‘region of
acceptability’
Right
Issue Dimensions
• European Integration
– Some people feel that Britain should do all it can to
unite fully with the European Union. Other people feel
that Britain should do all it can to protect its
independence from the European Union.
• Taxation and spending
– Some people feel that the government should put up
taxes a lot and spend much more on health and
social services. Other people feel that the government
should cut taxes a lot and spend much less on health
and social services.
Issue Dimensions
• Income redistribution
– Some people feel that government should make
much greater efforts to make people’s incomes more
equal. Other people feel that government should be
much less concerned about how equal people’s
incomes are.
• Unemployment and Inflation
– Some people feel that getting people back to work
should be the government's top priority. Other people
feel that keeping prices down should be the
government's top priority.
Spatial & Directional Scores 97-01
Figure 1b spatial respondent party placement
Figure 1a spatial mean party placement
6.00
6.00
5.00
5.00
4.00
4.00
labour
labour
3.00
tory
3.00
tory
Lib Dem
Lib Dem
2.00
2.00
1.00
1.00
0.00
0.00
97
98
99
00
1
01
2
3
4
5
Figure 1d directional respondent party placement
Figure 1c directional mean party placement
5.00
7.00
6.00
4.00
5.00
3.00
4.00
2.00
labour
1.00
tory
Lib Dem
0.00
97
98
99
00
01
3.00
Lib Dem
0.00
-1.00
-3.00
tory
1.00
-1.00
-2.00
labour
2.00
-2.00
-3.00
97
98
99
00
01
Party Evaluations 1997-2001
Choose a phrase from this scale
to say how you feel about the
Labour/Conservative/Liberal
Democrat party
5. Strongly Against
4. Against
3. Neither/Nor
2. In Favour
1. Strongly in Favour
Path Diagram for LGC Models
Cross-Sectional Betas of party
evaluation on proximity by
Knowledge of Party Positions
Sample
full sample
n=2034
low knowledge
n=513
high knowledge
n=579
spatial
Party
(mean)
Conservative
0.35*
Labour
0.27*
Conservative
0.17*
Labour
0.11*
Conservative
0.54*
Labour
0.46*
*=significant at 95% confidence level.
spatial
direction direction
(personal ) (mean)
(personal)
0.53*
-0.46*
-0.48*
0.42*
-0.39*
-0.43*
0.36*
-0.25*
-0.32*
0.24*
-0.21*
-0.27*
0.69*
-0.64*
-0.64*
0.58*
-0.58*
-0.57*
Betas of party evaluation slope on
proximity slope from LGC models by
Knowledge of Party Positions
Sample
full sample
n=2034
low knowledge
n=513
high knowledge
n=579
spatial
Party
(mean)
Conservative
-1.72
Labour
0.66*
Conservative
-0.45
Labour
0.56
Conservative
-4.06
Labour
0.86*
*=significant at 95% confidence level;
spatial
direction direction
(personal ) (mean)
(personal)
0.63*
-0.56*
-0.65*
0.43*
-0.88*
-1.03*
0.23
-0.16
-0.29
1.05
-0.52
-1.21
0.83*
-0.81*
-0.77*
0.79*
-1.30*
-1.08*
Conclusions
• For more sophisticated voters, change in
policy proximity correlated with change in
evaluation.
• No relationship between change in policy
proximity and evaluation for least
sophisticated.
• Cross-sectional parameters tell us nothing
about temporal dimension of relationships.