1 latent growth curve models patrick sturgis, department of sociology, university of surrey
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Latent Growth Curve Models
Patrick Sturgis,
Department of Sociology, University of Surrey
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Overview
• Random effects as latent variables• Growth parameters• Specifying time in LGC models• Linear Growth• Non-linear growth• Explaining Growth• Fixed and time-varying predictors• Benefits of SEM framework
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SEM for Repeated Measures
• The SEM framework can be used on repeated measured data to model individual growth trajectories.
• 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.
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LV
X11 X12 X13 X14
E1
1
E2
1
E3
1
E4
1
A Single Latent Variable Model
4 different items
same item at 4 time points
Estimate mean and variance of underlying factor
Estimate mean and variance of trajectory of change over time
Estimate factor loadingsConstrain factor loadings
LV
X1 X2 X3 X4
E1
1
E2
1
E3
1
E4
1
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Repeated Measures & Random Effects
• We have average (or ‘fixed’) effects for the population as a whole
• And individual variability (or ‘random’) effects around these average coefficients
ittiiit xy
ii ii
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Random Effects as Latent Variables
• In LGC:• 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.
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Growth Parameters
• The earlier path diagram was an over-simplification.
• In practice we require at least two latent variables to describe growth.
• One to estimate the mean and variance of the intercept.
• And one to estimate the mean and variance of the slope.
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Specifying Time in LGC Models
• In random effects models, time is included as an independent variable:
• 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.
ijijiiij xy 10
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ICEPT SLOPE
X1t1 X1t2 X1t3 X1t4
E1
1
E2
1
E3
1
E4
1
A Linear Growth Curve Model
11 1
1 1
0
2 3
Constraining values of the intercept to 1 makes this parameter indicate initial status
Constraining values of the slope to 0,1,2,3 makes this parameter indicate linear change
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Quadratic Growth
ICEPT SLOPE
X1 X2 X3 X4
E1 E2 E3 E4
QUAD
1
0
2 311 1
1
01 4 9
Add additional latentvariables with factor loadings constrained to powers of the linearslope
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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.
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A (made up) Example
• We are interested in the development of knowledge of longitudinal data analysis.
• 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.
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Linear Growth Example
mean=11.2 (1.4) p<0.001
variance =4.1 (0.8) p<0.001
mean=1.3 (0.25) p<0.001
variance =0.6 (0.1) p<0.001
11 1
1 1
0
2 3
ICEPT SLOPE
X1t1 X1t2 X1t3 X1t4
E1
1
E2
1
E3
1
E4
1
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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.
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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.• This is equivalent to fitting an interaction
between time and predictor variables in random effects models.
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Time-Invariant Predictors
11 1
1 10
2 3
Do men have a different initial status than women?
Do men grow at a different rate than women?
Gender
(women = 0; men=1)
Does initial status influence rate of growth?
ICEPT SLOPE
X1t1 X1t2 X1t3 X1t4
E1
1
E2
1
E3
1
E4
1
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Time-varying predictors of growth
ICEPT
SLOPE
X1 X2 X3 X4
E1 E2 E3 E4
2
1
1 11
01
3
w1 w2 w3 w4
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Why SEM?
• Most of this kind of stuff could be done using random/fixed effects.
• SEM has some specific advantages which might lead us to prefer it over potential alternatives:– SPSS linear mixed model– HLM– MlWin– Stata (RE, FE)
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Fixed Effects/Unit Heterogeneity
• A fixed effects specification removes ‘unit effects’• This controls for all observed and unobserved invariant
unit characteristics• Highly desirable when one’s interest is in the effect of
time varying variables on the outcome• This is done by allowing the random effect to be
correlated with all observed covariates• Downside=no information about effect of time invariant
variables, possible efficiency loss• SEM allows various hybrid models which fall between
the classic random and fixed effect specifications
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Random effect model
ICEPT
SLOPE
X1 X2 X3 X4
E1 E2 E3 E4
2
1
1 11
01
3
w1 w2 w3 w4
b b b b
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Fixed effect model
ICEPT
SLOPE
X1 X2 X3 X4
E1 E2 E3 E4
2
1
1 11
01
3
w1 w2 w3 w4
b b b b
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Hybrid model
ICEPT
SLOPE
X1 X2 X3 X4
E1 E2 E3 E4
2
1
1 11
01
3
w1 w2 w3 w4
b b b b
Z
Introduce Time-Invariant Covariate that has indirect Effect on X
Remove equality constrainton beta weights
Allow correlatederrors
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Multiple Indicator LGC Models
• Single indicators assume concepts measured without error
• Multiple indicators allow correction for systematic and random error
• Reduced likelihood of Type II errors (failing to reject false null)
• Tests for longitudinal meaning invariance• Allows modeling of measurement error
covariance structure
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Multiple Indicator LGC Models
MI1
X11
e1
X21
e2
X31
e3
MI2
X12
e4
X22
e5
X32
e6
MI3
X13
e7
X23
e8
X33
e9
MI4
X14
e10
X24
e11
X34
e12
INT SLOPE
1 11
2
3011
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Other Benefits of SEM
• Global tests and assessments of model fit
• Full Information Maximum Likelihood for missing data
• Decomposition of effects – total, direct and indirect
• Probability weights
• Complex sample data
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SLO
PE
1IC
EP
T1
X1t
1X
1t2
X1t
3X
1t4
E1 1
E2 1
E3 1
E4 1
01
23
11
11
ICE
PT
2S
LOP
E2
Y1t1
Y1t2
Y1t3
X1t4
E51
E61
E71
E81
11
11
10
23
Does initial status on one variable influence development on the other?
Does rate of growth on one variable influence rate of growth on the other?
Multiple Process Models