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Modeling longitudinal data Sanja Franić Vrije Universiteit Amsterdam. Introduction Frequently, researchers are faced with the question of how to optimally utilize longitudinal data E.g., one may have collected data on children’s cognitive abilities, at ages 10, 12, 14, 16, and 18 - PowerPoint PPT Presentation
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Modeling longitudinal data
Sanja FranićVrije Universiteit Amsterdam
Introduction
Frequently, researchers are faced with the question of how to optimally utilize longitudinal data
E.g., one may have collected data on children’s cognitive abilities, at ages 10, 12, 14, 16, and 18
Some of the possible questions:
Introduction
How large is the role of factors that act in concert across different time points to cause the observed stability of the variable of interest over time?
How large is the role of those factors that cause individual differences specific to a certain time point?
Is there a stabile driving force behind the growth or decline of a trait over time, or do novel factors relevant to the trait emerge at different time points? If so, how to detect and quantify them?
Introduction
How well can a trait at a certain time point be predicted from a measurement at the preceding time point?
How much do individuals differ in the starting level of the variable of interest (e.g., in mathematical skills prior to formal education)?
How much do individuals differ in their speed of growth or decline over time?
Does the development of a skill follow a linear curve, or is there non-linear change?
Introduction
In today’s workshop, we will cover several types of models aimed at addressing the above questions
Overview
Cholesky decomposition Simplex model Latent growth curve model
Example
You’ve collected longitudinal data on IQ. You applied the Wechsler Intelligence Scale for Children (WISC) at ages 10, 12, 14, and 16.
Example
Structure of the data at each time point:
Item 1
Item 2
Item 3
Item 4
… … … … Item m
Person 1Person 2Person 3Person 4
.
.Person n
Example
Subscale scores:
Verbal Comprehension Index (VCI)
Perceptual Reasoning Index (PRI)
Working Memory Index
(WMI)
Processing Speed Index
(PSI)Person 1Person 2Person 3Person 4
.
.Person n
Example
Subscale scores:
VCI
PRI
WMI
PSI
Age 10
Example
Subscale scores:
VCI
PRI
WMI
PSI
VCI
PRI
WMI
PSI
Age 10
Age 12
Example
Subscale scores:
VCI
PRI
WMI
PSI
VCI
PRI
WMI
PSI
VCI
PRI
WMI
PSI
Age 10
Age 12
Age 14
Example
Subscale scores:
VCI
PRI
WMI
PSI
VCI
PRI
WMI
PSI
VCI
PRI
WMI
PSI
VCI
PRI
WMI
PSI
Age 10
Age 12
Age 14
Age 16
Example
Let us assume all subscale scores at all time points are continuous normally distributed variables (for IQ scores this is a reasonable assumption; with real data, one can test it)
We will demonstrate each of the three methods as applied to this example dataset; we will use path-diagrammatic representations of the data (as presented in the SEM workshops by Dylan Molenaar)
A brief explanation of path diagrams:
Path diagrams
Squares = observed (measured) variables
VCI
PRI
WMI
PSI
Path diagrams
Circles = latent variables
VCI
PRI
WMI
PSI
g
Path diagrams
Single-headed arrows = causal relations in the model
VCI
PRI
WMI
PSI
g
Path diagrams
Path coefficients = strength of the relationship between variables
VCI
PRI
WMI
PSI
g
.5
.6
.7
.8
Path diagrams
Double-headed arrows: variances and covariances
VCI
PRI
WMI
PSI
g
.5
.6
.7
.8
1
Path diagrams
Double-headed arrows: variances and covariances
VCI
PRI
WMI
PSI
g
.5
.6
.7
.8
1.1
Path diagrams
Residuals
VCI
PRI
WMI
PSI
g
.5
.6
.7
.8
1.1
Overview
Cholesky decomposition Simplex model Latent growth curve model
Cholesky decomposition
VCI10
PRI10
WMI10
VCI12
PRI12
WMI12
VCI14
PRI14
WMI14
VCI16
PRI16
WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Cholesky decomposition
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Cholesky decomposition
Age 10
Age 12
Age 14
Age 16
PS10
1
PSI10 PSI12 PSI14 PSI16
Cholesky decomposition
Age 10
Age 12
Age 14
Age 16
PS10
1PS12
1
PSI10 PSI12 PSI14 PSI16
Cholesky decomposition
PSI10 PSI12 PSI14 PSI16
Age 10
Age 12
Age 14
Age 16
PS10
1PS12
1
PS14
1
Cholesky decomposition
PSI10 PSI12 PSI14 PSI16
Age 10
Age 12
Age 14
Age 16
PS10
1PS12
1
PS14
1
PS16
1
Cholesky decomposition
- the first factor (PS10) captures all of the variation in perceptual speed at age 10 (PSI10) and the variation in the other three observed variables (PSI12, PSI14, and PSI16) which they share with PSI10
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
- this factor (PS10) represents what is common to all four observed variables
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
→ factor that causes stability of the observed measure (perceptual speed) across all four ages
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
- this factor (PS12) represents what is common only to the last three observed variables
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
→ factor that causes stability of the observed measure (over and above the stability caused by the first factor, PS10) across the last three ages
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
- the third factor (PS14) represents what is common only to the last two observed variables
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
- the last factor (PS16) represents the variation that is unique to the last variable (PSI16)
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
If we specify a model like this (e.g., in Mplus) and fit it to observed data, we will get estimates of parameters in the model – in this case, the loadings of the observed variables on the latent factors.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
If we specify a model like this (e.g., in Mplus) and fit it to observed data, we will get estimates of parameters in the model – in this case, the loadings of the observed variables on the latent factors.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41
Cholesky decomposition
If we specify a model like this (e.g., in Mplus) and fit it to observed data, we will get estimates of parameters in the model – in this case, the loadings of the observed variables on the latent factors.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42
Cholesky decomposition
If we specify a model like this (e.g., in Mplus) and fit it to observed data, we will get estimates of parameters in the model – in this case, the loadings of the observed variables on the latent factors.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43
Cholesky decomposition
If we specify a model like this (e.g., in Mplus) and fit it to observed data, we will get estimates of parameters in the model – in this case, the loadings of the observed variables on the latent factors.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44
Cholesky decomposition
These loadings (i.e., path coefficients in a path diagram) represent the strength of the relationship between the observed variables and the latent factors.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44
Cholesky decomposition
Knowing that the variability that is common to processing speed at ages 10, 12, 14, and 16 is represented by the first latent factor, and the paths between that factor and each of the observed variables...
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41
Cholesky decomposition
... and knowing the estimates of the path loading parameters (λs)...
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41
Cholesky decomposition
... we can quantify the proportion of variance in processing speed at a given time point that is due to factors that cause temporal stability across all four ages.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41
Cholesky decomposition
E.g., at age 12 this proportion is: λ21*var(PS10)*λ21 = λ21*1*λ21 = λ21
2.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41
Cholesky decomposition
E.g., at age 12 this proportion is: λ21*var(PS10)*λ21 = λ21*1*λ21 = λ21
2.
We assume here that the variance of the observedvariable is 1. Otherwise, toobtain the proportion, we have to divide the λ21
2 by the
variance of the observedvariable.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41
Cholesky decomposition
At age 14 it is: λ31*var(PS10)*λ31 = λ31*1*λ31
= λ312.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41
Cholesky decomposition
E.g., if these are the factor loading estimates (below), then the factors that cause stability of processing speed (PS) across all ages explain .52=.25 of the variance in PS at age 12, .32=.09 of the variance in PS at age 14, and .22=.04 of the variance in PS at age 16.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
1 .5 .3 .2
Cholesky decomposition
If there are additional sources of stability arising after the initial age of measurement (age 10), those are quantified by the other path coefficients in the model.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
Cholesky decomposition
For instance, if a factor emerges at age 12, which causes additional temporal stability in processing speed across the ages 12-16, over and above the stability caused by factors present at age 10, the strength of influence of that factor is quantified by the coefficients λ22, λ32, and λ42.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ22 λ32 λ42
Cholesky decomposition
Question: what is the variance in processing speed at age 14 which can be explained by factors that cause temporal stability of processing speed across ages 12-16, but are not yet present at age 10?
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ22 λ32 λ42
Cholesky decomposition
Answer: λ322.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ22 λ32 λ42
Cholesky decomposition
What is the variance in processing speed at age 14 explained by factors causing temporal stability of processing speed across ages 14-16, that are not present at ages 10 and 12?
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44
Cholesky decomposition
Answer: λ332.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44
Cholesky decomposition
What is the variance in processing speed unique to age 16, i.e., not explained by the factors present at previous time points?
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44
Cholesky decomposition
Answer: λ442.
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44
Cholesky decomposition
What happens if we expand the model to the multivariate case?
PSI10 PSI12 PSI14 PSI16
Age 10 Age 12 Age 14 Age 16
PS10
1PS12
1PS14
1PS16
1
λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44
Cholesky decomposition
VCI10
PRI10
WMI10
VCI12
PRI12
WMI12
VCI14
PRI14
WMI14
VCI16
PRI16
WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Cholesky decomposition
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Cholesky decomposition
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Cholesky decomposition
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1
Cholesky decomposition
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM1
0
1
Cholesky decomposition
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM1
0
1
PS12
1WM1
2
1
Cholesky decomposition
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM1
0
1
PS12
1WM1
2
1
PS14
1WM1
4
1
Cholesky decomposition
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM1
0
1
PS12
1WM1
2
1
PS14
1WM1
4
1
PS16
1WM1
6
1
Cholesky decomposition
Question: how much of the variance in processing speed at age 12 is explained by factors that cause the observed stability of both types of cognitive abilities (PS and WM) across all ages?
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
Answer (I will simply highlight the path instead of writing out the coefficient):
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
Question: how much of the variance in processing speed at age 12 is explained by factors that cause temporal stability of processing speed and working memory across ages 12-16, and of working memory, but not of processing speed, at age 10?
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
Answer (I will simply highlight the path instead of writing out the coefficient):
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
Question: how much of the variance in working memory at age 16 is explained by factors that are common to working memory and processing speed at age 16, but do not explain any of the temporal stability at the previous ages?
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
Answer (I will simply highlight the path instead of writing out the coefficient):
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
Note that different measures (e.g., PS and WM) covary not only within a single time point: they also can (and quite often do) covary across different time points (cross-covariance).
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
We can expand the model to include all 4 observed variables (subscales) across all of the 4 time points, in an equivalent manner as we just did for the two variables.
WMI10 WMI12 WMI14 WMI16
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
PS10
1WM10
1PS12
1WM12
1PS14
1WM14
1PS16
1WM16
1
Cholesky decomposition
Cholesky decomposition is not a model; it is simply a variance decomposition, i.e., a different way to express the variance.
As such, we cannot test its fit (the fit will always be perfect).
However, it does allow us to make conclusions of the kind demonstrated above.
Next, we present an actual model.
Overview
Cholesky decomposition Simplex model Latent growth curve model
Simplex model
VCI10
PRI10
WMI10
VCI12
PRI12
WMI12
VCI14
PRI14
WMI14
VCI16
PRI16
WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3
Simplex model
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
One may specify this model and fit it to observed data (e.g., in MPlus, Mx, OpenMx..). This will produce: 1) fit statistics (because simplex is a model), 2) estimates of the parameters below (the βs and the ζs).
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
Interpretation?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
The variance of the variable observed at the first time point (WMI10) is not extensively modeled.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
The variances of the variables at the following time points, however, are decomposed into:
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
The variances of the variables at the following time points, however, are decomposed into:a) a part explained by the variable at the preceding time
point→ stability, expressed by the autoregressive coefficient β
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
The variances of the variables at the following time points, however, are decomposed into:a) a part explained by the variable at the preceding time
point→ stability, expressed by the autoregressive coefficient β
b) a part unique to the time point in question→ innovation, expressed by ζ
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
One may now ask, e.g., how much of the variance in working memory at age 12 is explained by working memory at the preceding age (age 10). The answer is β21*varWMI10*β21 = β21
2*varWMI10.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
Therefore, the variance in working memory at age 12 which is due to temporal stability (transmission from the previous age), can be quantified as β21
2*varWMI10.
Age 10
Age 12
Age 14
Age 16
WMI10 WMI12 WMI14 WMI16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
The covariance between working memory at age 10 (WMI10) and working memory at age 12 (WMI12) can also readily be expressed in terms of model parameters, as β21.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
The part of the variance in working memory at age 12 that is explained by factors unique to age 12 is ζ2.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
Therefore, the entire variance of WMI12 can be expressed asvar(WMI12) = β21
2*varWMI10 + ζ2.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
Therefore, the entire variance of WMI12 can be expressed asvar(WMI12) = β21
2*varWMI10 + ζ2.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
stability
change
Simplex model
Therefore, the entire variance of WMI12 can be expressed asvar(WMI12) = β21
2*varWMI10 + ζ2. Depending on how well the model fits the observed data, this will be an accurate representation of the variance (or not). Unlike Cholesky, this is a model and may be tested!
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
Therefore, a simplex model may be used to estimate:1) variance due to temporal stability (or transmission)2) variance due to change (or innovation)
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
E.g., IQ typically becomes increasingly stabile across childhood and stays very stabile throughout adulthood. In early childhood, therefore, the role of factors that disrupt stability in IQ (ζ) is relatively large, and gradually declines throughout childhood. By ~age 18, it is typically very small, while the role of transmission (as quantified by βs) is very large.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 β2
1
β3
2
β4
3
ζ2 ζ3 ζ4
PSI10 PSI12 PSI14 PSI16
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 βW21
ζ2 ζ3 ζ4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
varWM10 βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
What happens in the multivariate case?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
We can estimate the:1) Covariance between the variables at the first time point2) Covariance between the residuals (innovation variances)
of the variables at all other time points3) Cross-covariances (covariances between one variable at a
given time point and another variable at the adjacent time point)
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
We can, for instance, ask to what extent processing speed at age 10 predicts working memory at age 12.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Simplex model
We can, for instance, ask to what extent processing speed at age 10 predicts working memory at age 12. Answer: the variance in WM at age 12 predicted by PS at age 10 is βW2P1
2*varPS10.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4βW2P1
varWM10
varPS10
Simplex model
Question: what is the variance in processing speed at age 16 explained by working memory at age 14?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4βW2P1βP2W1
βW3P2βP3W2
βW4P3βP4W3
varWM10
varPS10
Simplex model
Answer: βP4W32*varWM14.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4βW2P1βP2W1
βW3P2βP3W2
βW4P3βP4W3
varWM10
varPS10
Simplex model
Question: How much of the variance in working memory at age 14 is due to factors that are specific to that particular age?
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4βW2P1βP2W1
βW3P2βP3W2
βW4P3βP4W3
varWM10
varPS10
Simplex model
Answer: ζW3.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4βW2P1βP2W1
βW3P2βP3W2
βW4P3βP4W3
varWM10
varPS10
Simplex model
Question: What is the covariance between the factors that influence working memory and emerge only at age 12, with factors that affect processing speed, and also only emerge at age 12? (Just point out the arrow.)
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4βW2P1βP2W1
βW3P2βP3W2
βW4P3βP4W3
varWM10
varPS10
Simplex model
Answer:
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4βW2P1βP2W1
βW3P2βP3W2
βW4P3βP4W3
varWM10
varPS10
Simplex model
One can, of course, extend the model to include more observed variables and more time points.
WMI10 WMI12 WMI14 WMI16
Age 10
Age 12
Age 14
Age 16
βW21
ζW2 ζW3 ζW4
PSI10 PSI12 PSI14 PSI16βP21
βW32
βP32
βW43
βP43
ζW2 ζW3 ζW4
varWM10
varPS10
Progress
Up to this point, we have answered 4 out of the 7 example research questions from the beginning of the lecture.
The remaining 3 may be answered using growth curve modeling.
Overview
Cholesky decomposition Simplex model Latent growth curve model
Growth curve model
VCI10
PRI10
WMI10
VCI12
PRI12
WMI12
VCI14
PRI14
WMI14
VCI16
PRI16
WMI16
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Growth curve model
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
Growth curve model
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
I
Growth curve model
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
I
1 1 1 1
Growth curve model
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
I
1 1 1 1
S
Growth curve model
Age 10
Age 12
Age 14
Age 16
PSI10 PSI12 PSI14 PSI16
I
1 1 1 1
S
0 1 2 3
Growth curve model
Score of person i at the various time points can be expressed as:yi1 = 1*Ii + 0*Si + εi1 yi2 = 1*Ii + 1*Si + εi2 yi3 = 1*Ii + 2*Si + εi3 yi4 = 1*Ii + 3*Si + εi4
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
I
1 1 1 1
S
0 1 2 3
ε1 ε2 ε3 ε4
Growth curve model
Score of person i at the various time points can be expressed as:yi1 = 1*Ii + 0*Si + εi1 = Ii + εi1yi2 = 1*Ii + 1*Si + εi2 = Ii + Si + εi2yi3 = 1*Ii + 2*Si + εi3 = Ii + 2Si + εi3yi4 = 1*Ii + 3*Si + εi4 = Ii + 3Si + εi4
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
I
1 1 1 1
S
0 1 2 3
ε1 ε2 ε3 ε4
Growth curve model
Score of person i at the various time points can be expressed as:yi1 = 1*Ii + 0*Si + εi1 = Ii + εi1yi2 = 1*Ii + 1*Si + εi2 = Ii + Si + εi2yi3 = 1*Ii + 2*Si + εi3 = Ii + 2Si + εi3yi4 = 1*Ii + 3*Si + εi4 = Ii + 3Si + εi4
Suppose person i’s score on the I factor is 90, and on the S factor 5:Ii=90, Si=5. Suppose error scores are negligible (for sake of example).
Growth curve model
Score of person i at the various time points can be expressed as:yi1 = 1*Ii + 0*Si + εi1 = Ii + εi1yi2 = 1*Ii + 1*Si + εi2 = Ii + Si + εi2yi3 = 1*Ii + 2*Si + εi3 = Ii + 2Si + εi3yi4 = 1*Ii + 3*Si + εi4 = Ii + 3Si + εi4
Suppose person i’s score on the I factor is 90, and on the S factor 5:Ii=90, Si=5. Suppose error scores are negligible (for sake of example).
yi1 = Ii + εi1 = 90yi2 = Ii + Si + εi2 = 90 + 5 = 95yi3 = Ii + 2Si + εi3 = 90 + 2*5 = 100yi4 = Ii + 3Si + εi4 = 90 + 3*5 = 105
Growth curve model
yi1 = Ii + εi1 = 90yi2 = Ii + Si + εi2 = 90 + 5 = 95yi3 = Ii + 2Si + εi3 = 90 + 2*5 = 100yi4 = Ii + 3Si + εi4 = 90 + 3*5 = 105
Growth curve model
yi1 = Ii + εi1 = 90yi2 = Ii + Si + εi2 = 90 + 5 = 95yi3 = Ii + 2Si + εi3 = 90 + 2*5 = 100yi4 = Ii + 3Si + εi4 = 90 + 3*5 = 105
10 12 14 16 Age
Score
90
95
100
105
Growth curve model
yi1 = Ii + εi1 = 90yi2 = Ii + Si + εi2 = 90 + 5 = 95yi3 = Ii + 2Si + εi3 = 90 + 2*5 = 100yi4 = Ii + 3Si + εi4 = 90 + 3*5 = 105
10 12 14 16 Age
Score
90
95
100
105
Growth curve model
For person j, Ij=100, Sj=2. Error again negligible.
yj1 = Ij + εj1 = 100yj2 = Ij + Sj + εj2 = 100 + 2 = 102yj3 = Ij + 2Sj + εj3 = 100 + 2*2 = 104yj4 = Ij + 3Sj + εj4 = 100 + 3*2 = 106
Growth curve model
yj1 = Ij + εj1 = 100yj2 = Ij + Sj + εj2 = 100 + 2 = 102yj3 = Ij + 2Sj + εj3 = 100 + 2*2 = 104yj4 = Ij + 3Sj + εj4 = 100 + 3*2 = 106
10 12 14 16 Age
Score
90
95
100
105
Person i
Growth curve model
yj1 = Ij + εj1 = 100yj2 = Ij + Sj + εj2 = 100 + 2 = 102yj3 = Ij + 2Sj + εj3 = 100 + 2*2 = 104yj4 = Ij + 3Sj + εj4 = 100 + 3*2 = 106
10 12 14 16 Age
Score
90
95
100
105
Person iPerson j
Growth curve model
Here, we explicitly model individual differences in growth trajectories (specifically, in intercepts and slopes of individual growth curves).
10 12 14 16 Age
Score
90
95
100
105
Person iPerson j
Growth curve model
The intercepts and slopes are said to be random over subjects (i.e., they can vary over subjects; unlike in standard regression).
10 12 14 16 Age
Score
90
95
100
105
Person iPerson j
Growth curve model
One can ask how much a) the intercepts vary over the subjects,b) the slopes vary over the subjects.
10 12 14 16 Age
Score
90
95
100
105
Person iPerson j
Growth curve model
By specifying this model and fitting it to the data, we can estimate the variances of the I and the S factors, which quantify, respectively:a) the variation in intercepts over the subjects,b) the variation in slopes over the subjects.
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
I
1 1 1 1
S
0 1 2 3
ε1 ε2 ε3 ε4
VarI VarS
Growth curve model
One can add additional factors into the model, e.g., a ‘quadratic’ factor (Q, see below). This will make the curve quadratic, i.e., it allows for modeling nonlinearity in growth trajectories.
Age 10 Age 12 Age 14 Age 16
PSI10 PSI12 PSI14 PSI16
I
1 1 1 1
S
0 1 2 3
ε1 ε2 ε3 ε4
Q
0 1 4 9
Growth curve model
E.g., for Ii=90, Si=3, Qi=1, εi=0:yi1 = Ii + εi1 = 90yi2 = Ii + Si + Qi + εi2 = 90 + 3 + 1 = 94 yi3 = Ii + 2Si + 4Qi + εi3 = 90 + 2*3 + 4*1 = 100yi4 = Ii + 3Si + 9Qi + εi4 = 90 + 3*3 + 9*1 = 108
10 12 14 16 Age
Score
90
95
100
105
Overview
Cholesky decomposition Simplex model Latent growth curve model
Final notes
Additional reading:
E.g., chapters on longitudinal models in Hoyle, R.H. (2012). Handbook of Structural Equation Modeling. New York: Guilford Press.
Implementation:
MPlus, Mx, OpenMx, other packages in R...
My contact: