Developmental Models:
Latent Growth Models
Brad Verhulst & Lindon Eaves
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 traits
The Univariate Simplex Model
(in singletons)
With a Univariate Simplex Model, the Yt causes Yt+1 and is
caused by Yt-1
This is also called an AR1
model as the “lag” is only
Y1
1 time point
Y2
Y3
Y4
ε1
ε
1
ε2
ε
1
ε3
ε
1
Note that the disturbance terms
are uncorrelated
ε4
ε
1
Latent Growth Model
ψ13
ψ12
ψ11
μC
1
μL
μQ
ψ22
C1
1
ψ23
1
1
ψ33
L1
0 1 2 3
1
0
1
Q1
4
9
X1
X2
X3
X4
δ1
δ2
δ3
δ4
Δ
Δ
Δ
Δ
1
1
1
1
Note that the means for
the latent variables are
being estimated within the
model
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.
Mean Structures in Factor Models
VF
1
μF
ξ1
λ1
ψ1 Y1
μ1
1
λ2
ψ2 Y2
μ2
E(Y) = LE(x ) or E(Y) = M
μ3
E(y1 ) = l1mF or m1
λ3
ψ3 Y2
You must choose one of
the other, as both ΛE(ξ)
and Μ are not
simultaneously
identified
Latent Growth Models (LGM)
• Latent Growth Models are (probably) the most
common SEM with mean structures in a single
sample.
• Data requirements for LGM:
1. Dependent Variables measured over time
2. Scores have the same units and measure the same
thing across time
•
Measurement Invariance can be assumed
3. Data are time structured (tested at the same
intervals)
• The intervals do not have to be equal
– 6 months, 9 months, 12 months, 18 months
Growth Model for Two Twins
1
1
1
1
1
1
1
1
1
1
1
1
A
E
A
E
A
E
A
E
A
E
A
E
ac
al
ec
C1
el
1
0
1
1
2
3
4
X14
X13
E
1
1
A
1
e2
E
1
A
1
e3
A
1
1
E
E
1
1
1
0
1
2
Q2
3
0 1
4
9
A
1
X14
X13
a1
e4
aq eq
1
X12
a4
el
L2
1
X15
a3
a2
e1
C2
9
X12
1
Q1
al
ec
0 1
1
A
ac
L1
1
a1
aq eq
a3
a2
e1
A
1
X15
e2
E
E
1
1
A
1
a4
e3
A
1
e4
E
E
1
1
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 occasion
• The 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 effects
– Be cautious in your interpretation as they may only be relevant in your
sample.
– Interpretations of high order non-linear effects are difficult.
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)
Means of the Latent Parameters
• μC: Mean of the latent Constant
– The average level of the latent phenotype when
the linear effect is zero
• μL: Mean of the latent Linear slope
– The average increase (decrease) over time
• μQ: Mean of the latent Quadratic slope
– The average quadratic effect over time
Variances of the Latent Growth
Parameters
• ψii: 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 negative
• Typically the variance of higher order parameters
are smaller than the variances of lower order
parameters
– ψ11 > ψ22 > ψ33
Covariances between the Latent
Growth Parameters
• ψij: Covariance of the latent growth parameters
– Generally expressed in terms of correlations
– Important to keep in mind what the absolute variance in
the constituent growth parameters are
• E.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
increase
– ψ12 > 0: the higher an individual starts, the faster they
increase
– ψ12 < 0: the higher an individual starts, the slower they
increase
Presenting LGC Results
• The basic formula for the average effect of the growth
parameters is very similar to the simple regression
equation:
Yt = mc + mL lL + mQ lQ + ...+ mk lk
• 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.
3
2
1
0
-1
-2
Substance Use Propensity
4
5
Example Presentation of the LGC
Results
12
13
14
15
Age
16
17
Alternative Specifications
• Instead 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?
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)
LGM on Latent Factors
φ13
φ23
φ12
ξC
φ11
μC
1
ξC
φ22
φ33
ξC
μL
μQ
1
1
1
1
ζ
ζ
ζ
ζ
ζ1
ζ2
η1
ζ3
η2
λ11 λ21 λ31
λ42 λ52 λ62
η3
η4
λ73 λ83 λ94
λ10,5 λ11,5 λ12,5
Y
Y
Y
Y
X
X
X
X
X
1
2
2
4
5
6
7
8
9
δ2
δ
δ3
δ
δ5
δ
δ6
δ
δ
δ8
δ
δ9
δ
1
1
1
δ1
δ
1
1
1
δ4
δ
1
1
1
ζ4
δ7
X10
δ10
X11
X12
δ
δ11
δ
δ12
δ
1
1
1
Autoregressive LGM
φ13
φ23
φ12
ξC
φ11
μC
1
ξC
φ22
φ33
ξC
μL
μQ
1
1
1
1
ζ
ζ
ζ
ζ
ζ1
η1
β1
ζ2
η2
λ11 λ21 λ31
β2
ζ3
η3
λ10,5 λ11,5 λ12,5
Y
Y
Y
Y
X
X
X
X
X
1
2
2
4
5
6
7
8
9
δ2
δ
δ3
δ
δ5
δ
δ6
δ
δ
δ8
δ
δ9
δ
1
1
1
δ1
δ
1
1
1
δ4
δ
1
1
1
δ7
η4
β3
λ73 λ83 λ94
λ42 λ52 λ62
ζ4
X10
δ10
X11
X12
δ
δ11
δ
δ12
δ
1
1
1