Latent Growth Models Brad Verhulst amp Lindon Eaves Two Broad Categories of Developmental Models Autoregressive Models The things that happened yesterday affect what happens today which affect what happens tomorrow ID: 565176
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Slide1
Developmental Models: Latent Growth Models
Brad Verhulst &
Lindon
EavesSlide2
Two Broad Categories of Developmental Models
Autoregressive Models:
The things that happened yesterday affect what happens today, which affect what happens tomorrow
Simplex Models
Growth Models
Latent parameters are estimated for the level (stable) and the change over time (dynamic) components of the traitsSlide3
The Univariate Simplex Model (in singletons)
Y
1
ε
1
ε
1
Y
2
ε
1
ε
2
Y
3
ε
1
ε
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Y
4
ε
1
ε
4
With a
Univariate
Simplex Model, the
Y
t
causes Y
t+1
and is caused by Yt-1
Note that the disturbance terms are uncorrelated
This is also called an AR1 model as the “lag” is only 1 time pointSlide4
Latent Growth Model
C
1
L
1
Q
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X
1X2X4X3
Δ
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0
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ψ
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ψ
33
ψ
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ψ
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ψ
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ψ
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μ
C
μ
L
μ
Q
Note that the means for the latent variables are being estimated within the modelSlide5
Identification of Mean Structures
SEMs with Mean Structures must be identified both at the level of the Mean and at the level of the Covariance.
You can only estimate each mean once
If your model is unidentified at either the mean or the covariance level, your model is unidentified
An
overidentified covariance structure will not help identify the mean structure and vice versa.Slide6
Mean Structures in Factor Models
ξ
1
Y
1
Y
2
ψ
1
Y
2
μ
1
μ
2
μ3
ψ
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ψ
3
1
V
F
λ
1
λ
2
λ
3
μ
F
1
You must choose one of the other, as both ΛE(
ξ
) and Μ are not simultaneously identifiedSlide7
Latent Growth Models (LGM)
Latent Growth Models are (probably) the most common SEM with mean structures in a single sample.
Data requirements for LGM:
Dependent Variables measured over time
Scores have the same units and measure the same thing across time
Measurement Invariance can be assumedData are time structured (tested at the same intervals)The intervals do not have to be equal6 months, 9 months, 12 months, 18 monthsSlide8
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X13X15X14
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a
1
Growth Model for Two Twins
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a
lSlide9
Intuitive Understanding of the LGM
Each time point is represented as an indicator of all of the latent growth parameters
The constant is analogous to the constant in a linear regression model.
All of the (unstandardized) loadings are fixed to 1.
The linear effect is analogous to the regression of the observations on time (with loadings of 0, 1, …, t)
If latent slope loadings are set to -2,1,0,1,2,…,t , then the intercept will be at the third measurement occasionThe quadratic increase is analogous to a non-linear effect of time on the observed variables and is interpreted in a very similar way to the linear effect.Cubic, Quartic and higher order time effectsBe cautious in your interpretation as they may only be relevant in your sample.
Interpretations of high order non-linear effects are difficult.Slide10
Interpretation of the Latent Growth Factors
Residuals:
The variance in the phenotype that is not explained by the latent growth structure
Factor Loadings:
The same as you would interpret loadings in any factor model (but they are typically not interpreted)
Factor Means: The average effect of the intercept/linear/quadratic in the population (more on this next)Factor Covariance: Random Effects of the Latent Growth Parameters (more on this too)Slide11
Means of the Latent Parametersμ
C
: Mean of the latent Constant
The average level of the latent phenotype when the linear effect is zero
μ
L: Mean of the latent Linear slopeThe average increase (decrease) over timeμQ: Mean of the latent Quadratic slope
The average quadratic effect over timeSlide12
Variances of the Latent Growth Parameters
ψ
ii
: Variance of the latent growth parameters
Dispersion of the values around the latent parameter
Large variances indicate more dispersion
Large variance on a Latent slope may indicate that the average parameter increase but some of the latent trajectories may be negativeTypically the variance of higher order parameters are smaller than the variances of lower order parametersψ11
> ψ22 > ψ
33Slide13
Covariances between the Latent Growth Parameters
ψ
ij
: Covariance
of the latent growth parameters
Generally expressed in terms of correlationsImportant to keep in mind what the absolute variance in the constituent growth parameters areE.G. if the variance of the linear increase is really small, the correlation may be very large as an artifact of the variance
ψ12: Covariance
between the intercept and the linear increaseψ12
> 0: the higher an individual starts, the faster they increaseψ12 < 0: the higher an individual starts, the slower they increaseSlide14
Presenting LGC Results
The basic formula for the average effect of the growth parameters is very similar to the simple regression equation:
These expected values can easily be plotted against time.
It is also possible to include either the standard errors of the parameters or the variances of the growth parameters in the graphs.
Some people like to include the raw observations also.Slide15
Example Presentation of the LGC ResultsSlide16
Alternative SpecificationsInstead of fixing the loadings to 0, 1, 2, 3, if we fix the loadings to -3, -1, 1, 3, it will reduced the (non-essential)
multicolinearity
Importantly, the model fit will not change!
So what correlations are the right ones?Slide17
Caveat
If the starting point is zero, then it might be best to pick a value in the middle of the range for the intercept, or generate an orthogonal set of contrasts
Just because you started your study when people were 8, 14, 22, doesn’t mean that 8, 14 or 22 are meaningful starting points.
If zero is a meaningful then it might be a good idea to keep that value as 0.
Critical Event
Treatment (with pretests and follow-up tests)Slide18
LGM on Latent Factors
η
1
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1
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2
λ
11δδ
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λ31
1
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1
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η
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δ
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1
1
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μ
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QSlide19
Autoregressive LGM
η
1
Y
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λ
11δδ
1
λ
21
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2
δ
λ31
1
1
δ
1
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1
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1
η
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δ
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δ
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1
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η
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1
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η
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δ
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1
λ
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11,5
1
1
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1
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λ
12,5
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1
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Q