Modeling longitudinal data Sanja Franić Vrije Universiteit Amsterdam

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

VCI10

PRI10

WMI10

VCI12

PRI12

WMI12

VCI14

PRI14

WMI14

VCI16

PRI16

WMI16

Age 10

Age 12

Age 14

Age 16

PSI10 PSI12 PSI14 PSI16


Age 10

Age 12

Age 14

Age 16



Age 10

Age 12

Age 14

Age 16

PS10

1



Age 10

Age 12

Age 14

Age 16

PS10

1PS12

1




Age 10

Age 12

Age 14

Age 16

PS10

1PS12

1

PS14

1



Age 10

Age 12

Age 14

Age 16

PS10

1PS12

1

PS14

1

PS16

1


- 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


Age 10 Age 12 Age 14 Age 16

PS10

1PS12

1PS14

1PS16

1


- this factor (PS10) represents what is common to all four observed variables



PS10

1PS12

1PS14

1PS16

1


→ factor that causes stability of the observed measure (perceptual speed) across all four ages



PS10

1PS12

1PS14

1PS16

1


- this factor (PS12) represents what is common only to the last three observed variables



PS10

1PS12

1PS14

1PS16

1


→ factor that causes stability of the observed measure (over and above the stability caused by the first factor, PS10) across the last three ages



PS10

1PS12

1PS14

1PS16

1


- the third factor (PS14) represents what is common only to the last two observed variables



PS10

1PS12

1PS14

1PS16

1


- the last factor (PS16) represents the variation that is unique to the last variable (PSI16)



PS10

1PS12

1PS14

1PS16

1


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.



PS10

1PS12

1PS14

1PS16

1





PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41





PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42





PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43





PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44


These loadings (i.e., path coefficients in a path diagram) represent the strength of the relationship between the observed variables and the latent factors.



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44


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...



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41


... and knowing the estimates of the path loading parameters (λs)...



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41


... 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.



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41


E.g., at age 12 this proportion is: λ21*var(PS10)*λ21 = λ21*1*λ21 = λ21

2.



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41


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.



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41


At age 14 it is: λ31*var(PS10)*λ31 = λ31*1*λ31

= λ312.



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41


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.



PS10

1PS12

1PS14

1PS16

1

1 .5 .3 .2


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.



PS10

1PS12

1PS14

1PS16

1


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.



PS10

1PS12

1PS14

1PS16

1

λ22 λ32 λ42


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?



PS10

1PS12

1PS14

1PS16

1

λ22 λ32 λ42


Answer: λ322.



PS10

1PS12

1PS14

1PS16

1

λ22 λ32 λ42


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?



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44


Answer: λ332.



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44


What is the variance in processing speed unique to age 16, i.e., not explained by the factors present at previous time points?



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44


Answer: λ442.



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44


What happens if we expand the model to the multivariate case?



PS10

1PS12

1PS14

1PS16

1

λ11 λ21 λ31 λ41λ22 λ32 λ42λ33 λ43 λ44


VCI10

PRI10

WMI10

VCI12

PRI12

WMI12

VCI14

PRI14

WMI14

VCI16

PRI16

WMI16

Age 10

Age 12

Age 14

Age 16



WMI10 WMI12 WMI14 WMI16

Age 10

Age 12

Age 14

Age 16




Age 10

Age 12

Age 14

Age 16


PS10

1



Age 10

Age 12

Age 14

Age 16


PS10

1WM1

0

1



Age 10

Age 12

Age 14

Age 16


PS10

1WM1

0

1

PS12

1WM1

2

1



Age 10

Age 12

Age 14

Age 16


PS10

1WM1

0

1

PS12

1WM1

2

1

PS14

1WM1

4

1



Age 10

Age 12

Age 14

Age 16


PS10

1WM1

0

1

PS12

1WM1

2

1

PS14

1WM1

4

1

PS16

1WM1

6

1


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?




PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1


Answer (I will simply highlight the path instead of writing out the coefficient):




PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1


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?




PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1






PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1


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?




PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1






PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1


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).




PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1


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.




PS10

1WM10

1PS12

1WM12

1PS14

1WM14

1PS16

1WM16

1


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


Simplex model

VCI10

PRI10

WMI10

VCI12

PRI12

WMI12

VCI14

PRI14

WMI14

VCI16

PRI16

WMI16

Age 10

Age 12

Age 14

Age 16


Simplex model


Age 10

Age 12

Age 14

Age 16

Simplex model


Age 10

Age 12

Age 14

Age 16

varWM10

Simplex model


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

Simplex model


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

Simplex model


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

Simplex model


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

ζ2

Simplex model


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

ζ2 ζ3

Simplex model


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).


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

ζ2 ζ3 ζ4

Simplex model

Interpretation?


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.


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:


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


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


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.


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


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.


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.


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.


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

ζ2 ζ3 ζ4

Simplex model


2*varWMI10 + ζ2.


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

ζ2 ζ3 ζ4

stability

change

Simplex model


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!


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)


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.


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?


Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

ζ2 ζ3 ζ4

Simplex model



Age 10

Age 12

Age 14

Age 16

varWM10 β2

1

β3

2

β4

3

ζ2 ζ3 ζ4


Simplex model



Age 10

Age 12

Age 14

Age 16

varWM10 βW21

ζ2 ζ3 ζ4

PSI10 PSI12 PSI14 PSI16βP21

βW32

βP32

βW43

βP43

Simplex model



Age 10

Age 12

Age 14

Age 16

varWM10 βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43

ζW2 ζW3 ζW4

Simplex model



Age 10

Age 12

Age 14

Age 16

varWM10 βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43

ζW2 ζW3 ζW4varPS10

Simplex model



Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


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)


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


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.


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


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.


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


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?


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43

ζW2 ζW3 ζW4βW2P1βP2W1

βW3P2βP3W2

βW4P3βP4W3

varWM10

varPS10

Simplex model

Answer: βP4W32*varWM14.


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43


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?


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43


βW3P2βP3W2

βW4P3βP4W3

varWM10

varPS10

Simplex model

Answer: ζW3.


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43


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.)


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43


βW3P2βP3W2

βW4P3βP4W3

varWM10

varPS10

Simplex model

Answer:


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


βW32

βP32

βW43

βP43


βW3P2βP3W2

βW4P3βP4W3

varWM10

varPS10

Simplex model

One can, of course, extend the model to include more observed variables and more time points.


Age 10

Age 12

Age 14

Age 16

βW21

ζW2 ζW3 ζW4


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


Growth curve model

VCI10

PRI10

WMI10

VCI12

PRI12

WMI12

VCI14

PRI14

WMI14

VCI16

PRI16

WMI16

Age 10

Age 12

Age 14

Age 16


Growth curve model

Age 10

Age 12

Age 14

Age 16


Growth curve model

Age 10

Age 12

Age 14

Age 16


I

Growth curve model

Age 10

Age 12

Age 14

Age 16


I

1 1 1 1

Growth curve model

Age 10

Age 12

Age 14

Age 16


I

1 1 1 1

S

Growth curve model

Age 10

Age 12

Age 14

Age 16


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



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



I

1 1 1 1

S

0 1 2 3

ε1 ε2 ε3 ε4

Growth curve model


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


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


Growth curve model


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


10 12 14 16 Age

Score

90

95

100

105

Person i

Growth curve model


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.



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.



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


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:

[email protected]

Modeling longitudinal data Sanja Franić Vrije Universiteit Amsterdam

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