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)

Y1

ε1

ε1

Y2

ε1

ε2

Y3

ε1

ε3

Y4

ε1

ε4

With a Univariate Simplex Model, the Yt causes Yt+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 point

Latent Growth Model

C1 L1 Q1

X1 X2 X4X3

Δ1

δ1

Δ1

δ2

Δ1

δ3

Δ1

δ4

10

1 49

1111

1 20 3

ψ11ψ33ψ22

ψ12

ψ13

ψ23

μC μL

μQ

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

ξ1

Y1 Y2ψ1Y2

μ1 μ2 μ3

ψ2ψ3

1

VF

λ1 λ2 λ3

μF1

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 time2. 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

C1 L1 Q1

X12 X13 X15X14

A1

E1

e1A1

a2

E1

e2A1

a3

E1

e3A1

a4

E1

e4

10 1

4

9

11

1

1

1 20 3

a1

Growth Model for Two Twins

A1

E1

ecac

A1

E1

eqaq

A1

E1

elal

C2 L2 Q2

X12 X13 X15X14

A1

E1

e1A1

a2

E1

e2A1

a3

E1

e3A1

a4

E1

e4

0 14

9

111

1

1 20 3

a1

A1

E1

ecac

A1

E1

eqaq

A1

E1

elal

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:

• 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.

Example Presentation of the LGC Results

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

η1

Y1 Y2

λ11

δ δ1

λ21

Y2

δ

λ31

1 1

δ1 δ2 δ3

ζ1

ζ1 η2

Y4 X5

λ42

δ δ1

λ52

X6

δ

λ62

1 1

δ4 δ5 δ6

ζ1

ζ2 η3

X7 X8

λ73

δ δ1

λ83

X9

δ

λ94

1 1

δ7 δ8 δ9

ζ1

ζ3 η4

X10 X11

δ δ1

λ10,5

X12

δ

λ11,5

1 1

δ10 δ11 δ12

ζ1

ζ4

λ12,5

ξC ξC ξCφ11 φ33φ22

φ12

φ13

φ23

1

μC μL

μQ

Autoregressive LGM

η1

Y1 Y2

λ11

δ δ1

λ21

Y2

δ

λ31

1 1

δ1 δ2 δ3

ζ1

ζ1 η2

Y4 X5

λ42

δ δ1

λ52

X6

δ

λ62

1 1

δ4 δ5 δ6

ζ1

ζ2 η3

X7 X8

λ73

δ δ1

λ83

X9

δ

λ94

1 1

δ7 δ8 δ9

ζ1

ζ3 η4

X10 X11

δ δ1

λ10,5

X12

δ

λ11,5

1 1

δ10 δ11 δ12

ζ1

ζ4

λ12,5

β1 β2 β3

ξC ξC ξCφ11 φ33φ22

φ12

φ13

φ23

1

μC μL

μQ