Applied Longitudinal Data Analysis: An Introductory Course

(1)

Applied Longitudinal Data Analysis:

An Introductory Course

Emilio Ferrer

UC Davis

The Risk and Prevention in Education Sciences (RPES)

Curry School of Education - UVA

(2)

Acknowledgments

z

Materials for this workshop are the result of work

interactions with:

Jack McArdle

John Nesselroade

Aki Hamagami

Kevin Grimm

Nilam Ram

Sy Miin Chow

(3)

Course Overview – Day 1

z

Basis of latent growth curve and mixed-effects

models

z

Linear and nonlinear modeling

z

Programming and fitting linear LGC models

z

Programming and fitting nonlinear LGC

(4)

z

Incomplete data, exogenous variables, and

multiple groups

z

Multivariate models

z

Programming and fitting multiple groups

(5)

z

Introduction to dynamic systems and its

application to developmental research

z

Models for the analysis of individual processes

z

Programming and fitting dynamic models 1:

univariate models

z

Programming and fitting dynamic models 2:

(6)

Statistical Methods to Represent

Growth and Change – 1:

Introduction to Growth Curve

Modeling

(7)

Overview

z

Introduction to growth curve modeling

z

Basics of GCM

z

Specification, estimation, and evaluation

z

Examples

(8)

z

1. Identification of

intra-

individual change

z

2. Direct identification of

inter-

individual

differences in intraindividual change

z

3. Analysis of

interrelationships

in change

z

4. Analysis of

causes

(

determinants

) of

intra-individual change

z

5. Analysis of causes (

determinants

) of

inter-individual differences in intra-change

Objectives of Longitudinal Research

(9)

z

Some of the same entities (at least some of

them) are observed at repeated occasions

z

The measurement and scaling of observations

are known

z

The ordering or time underlying the observations

is known

(10)

Growth Curve Models

z

Class of techniques used to study change

z

They allow explicit testing of hypotheses regarding

the structure of longitudinal data

z

Step 1: A model of change is specified

z

Step 2: Expectations about means and covariances

are generated based on the specified model

z

Step 3: Parameters are estimated

z

Step 4: Model fit is evaluated (discrepancy

(11)

Development

z

Origins

- Rao (1958), Tucker (1958, 1966),

Meredith & Tisak (1984)

z

Expansions

- Browne & DuToit, (1991), McArdle (1988),

McArdle & Epstein (1987)

z

Overviews

-

McArdle & Nesselroade (2003), Singer & Willet (2003),

(12)

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(13)

Longitudinal Individual Data

z

WISC-R data from

N

=204 children

z

Repeated measurements at grades 1, 2, 4, and 6

z

WISC total means = 18.8, 26.6, 36.0, and 47.3

z

WISC total SDs = 6.4, 7.3, 7.7 and 10.4

(14)

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(15)

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Describing the Growth: Initial Level

(16)

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Describing the Growth: Slope

µ

₀

+

σ

₀

(17)

e

_y_[3]

e

_y_[1]

e

_y_[2]

e

_y_[4]

e

_y_[5]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

σ

_s2

σ

₀2

1

1 1

σ

_e

σ

_e

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

β

₁

β

₂

β

₃

β

4

β

5

β

₆

Basic Growth Model – Factor Model

(18)

Basic Growth Model – Factor Model

F

₀

F

s

u

1

u

2

u

3

u

4

u

5

u

6

Y

1

β

1

u

1

σ

_u2

0

0 0

Y

2

1

β

2

u

2

0

σ

_u2

0

Λ

= Y

3

1

β

3

Ψ

= u

3

0

σ

_u2

0

0 0

Y

4

1

β

4

u

4

0

σ

_u2

0 0

y

i

=

µ

+

Λ

f

i

+ u

F

2

σ

0s

σ

s

2

Ε

=

ΛΦΛ

' +

Ψ

(19)

1

e

_y_[3]

e

_y_[1]

e

_y_[2]

e

_y_[4]

e

_y_[5]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

Basic Growth Model – With Means

β

₁

β

₂

β

₃

β

4

β

(20)

Basics of Growth Models

z

First level model

Y

[

t

]

n

= y

0n

+ B

[

t

]

y

sn

+ e

[

t

]

n

y

₀

= latent score representing an individual’s initial level

B

[

t

]

= group “basis” parameters represent timing

y

s

= latent slopes for the individual change over time

e

[

t

]

= errors of measurements

z

Second level model

y

0n

=

µ

01

+

e

0n

y

sn

=

µ

s1

+

e

1n

the levels and slope scores have means (

µ

i,j

) and residuals

(

e

1

), and the residuals have variance components (

σ

i

2

)

(21)

z

“Fixed” or “group” terms:

1.

µ

₀

= the mean of the initial level scores

y

₀

2.

µ

s

= the mean of the slope scores

y

s

3.

B

[

t

]

= the basis coefficients of the slope scores

y

s

z

“Random” or “individual” terms:

4.

σ

e

2

= the variance of the residual score

e

[

t

]

5.

σ

₀

2

= the variance of the initial level scores

y

0

6.

σ

s

2

= the variance of the slope scores

y

s

7.

σ

0s

= the covariance of the level and slope scores

(22)

z

These techniques go by a number of different names:

Mixed-effects models (SAS PROC MIXED,

NLMIXED, MIXNOR, MIXREG)

Multi-level models (SPSS HLM, MLn)

Random coefficient models (VARCL)

Hierarchical linear models (SPSS HLM)

Latent growth models (SEM LISREL, Mx,

AMOS, etc.)

z

These models are algebraically identical with varied

statistical computations

(23)

LGC vs. RM (M)ANOVA

z

Group effects vs. individual change or growth

z

MANOVA needs balanced designs

– same number of observation per subject

– same interval across assessments (and across subjects)

z

MANOVA can’t handle missing data

z

Time is treated as a categorical variable

z

Limited handling of covariates

(24)

Growth Hypotheses

z

Level Only Model

Y

[

t

]

n

= y

0n

+ e

[

t

]

n

z

Linear Slope Model

Y

[

t

]

n

= y

0n

+ B

[

t

]

y

sn

+ e

[

t

]

n

with

B

[

t

]

fixed = 0, 1, 2, …t

z

Latent Slope Model

Y

[

t

]

n

= y

0n

+ B

[

t

]

y

sn

+ e

[

t

]

n

with

B

[

t

]

free

z

More complex functional relations

(25)

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(26)

“Level Only” Growth Model

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

(27)

µ

=

1

Y

[1]

µ

₀

Y

[2]

µ

₀

Y

[4]

µ

₀

Y

[6]

µ

₀

(28)

Σ

=

Y

[1]

Y

[2]

Y

[4]

Y

[6]

Y

[1]

σ

e

2

Y

[2]

0

σ

e

2

Y

[4]

0

σ

e

2

Y

[6]

0

σ

e

2

(29)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

No-Growth Model (with

σ

₀

2

)

y

₀*

(30)

µ

=

1

Y

[1]

µ

₀

Y

[2]

µ

₀

Y

[4]

µ

₀

Y

[6]

µ

₀

(31)

Σ

=

Y

[1]

Y

[2]

Y

[4]

Y

[6]

Y

[1]

σ

₀

2

+

σ

e

2

Y

[2]

σ

₀

2

σ

0

2

+

σ

e

2

Y

e

2

(32)

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Linear Slope

(

Y

[

t

]

n

= y

0n

+ B

[

t

]

y

sn

+ e

[

t

]

n

)

(33)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

Linear Growth

β

₁

β

₂

β

₃

β

4

β

5

β

₆

(34)

Linear Growth Model

z

The mean at any time is:

µ

[

t

]

=

µ

₀

+

µ

1

B

[

t

]

µ

₀

= mean of the initial level. It is usually

scale-dependent

µ

1

= mean of the slope. It is the average group

change per unit of the basis

B

[

t

]

B

[

t

]

= basis coefficients of the slope scores. The

value of these coefficients define the shape of the

average growth curve

(35)

µ

=

1

Y

[1]

µ

₀

+

µ

s

·

β

1

Y

[2]

µ

₀

+

µ

s

·

β

2

Y

[4]

µ

₀

+

µ

s

·

β

4

Y

[6]

µ

₀

+

µ

s

·

β

6

(36)

Σ

=

Y

[1]

Y

[2]

Y

[4]

Y

[6]

Y

[1]

σ

₀

2

+

σ

e

2

+

λ

1

2

σ

s

2

+ 2

σ

0s

λ

1

Y

[2]

σ

₀

2

σ

0

2

+

σ

e

2

+

λ

1

σ

s

2

λ

2

+

λ

2

σ

s

2

+2

λ

1

σ

0s

λ

2

+ 2

σ

0s

λ

2

Y

[4]

σ

₀

2

σ

0

2

σ

0

2

+

σ

e

2

+

λ

1

σ

s

2

λ

4

+

λ

2

σ

s

2

λ

4

+

λ

4

2

σ

s

2

+2

λ

1

σ

0s

λ

4

+2

λ

2

σ

0s

λ

4

+ 2

σ

0s

λ

4

Y

[6]

σ

₀

2

σ

0

2

σ

0

2

σ

0

2

+

σ

e

2

+

λ

1

σ

s

2

λ

6

+

λ

2

σ

s

2

λ

6

+

λ

4

σ

s

2

λ

6

+

λ

6

2

σ

s

2

+2

λ

1

σ

0s

λ

6

+2

λ

2

σ

0s

λ

6

+2

λ

4

σ

0s

λ

6

+ 2

σ

0s

λ

6

(37)

Latent Slope

(

Y

[

t

]

n

= y

₀

n

+ B

[

t

]

y

sn

+ e

[

t

]

n

)

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ISC

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(38)

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ISC

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Quadratic Slope

(

Y

[t]n

= y

₀

n

+ B

1[

t

]

y

s1

n

+B

2[

t

]

y

s2

n

+ e

[t]

n

)

(39)

Extension Variables

z

Initial Latent Growth Model

Y

[

t

]

n

= y

0n

+ B

[

t

]

y

sn

+ e

[

t

]

n

z

Prediction of individual level scores

y

₀

n

= G X

n

+ H Z

n

+ e

₀

n

z

Prediction of individual slope scores

y

sn

= J X

n

+ K Z

n

+ e

sn

z

Exactly the same logic as what are now termed

(40)

Latent Growth in Groups

z

Latent growth model with

groups

Y

(1)

[

t

]

n

= L

(1)

n

+ B

(1)

[

t

]

S

(1)

n

+ U

(1)

[

t

]

n

Y

(2)

[

t

]

n

= L

(2)

n

+ B

(2)

[

t

]

S

(2)

n

+ U

(2)

[

t

]

n

Y

(g)

[

t

]

n

= L

(g)

n

+ B

(

g

)

[

t

]

S

(

g

)

n

+ U

(

g

)

[

t

]

n

(41)

Statistical Methods to Represent

Growth and Change – 2:

(42)

Nonlinear Models

z

Most psychological phenomena are nonlinear

in nature

z

Most psychological theories are described with

nonlinear relationships

z

Y

=

f

(

X

), with the function

f

changing at

different levels of

X

z

Some classic examples include learning curves,

developmental stages, or the inverted

∩

function of arousal and performance

z

Some more recent examples include nonlinear

dynamics

(43)

(44)

Empirical Nonlinear Data

0 10 20 30 40 50 60 70 80 90 100 440 460 480 500 520 540 560

Age

W

J F

lui

d A

bi

li

ty

(45)

Fitted Curves of Fluid and Crystallized WJ-R Factors

0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60

General Fluid Ability (Gf) score as a function of Age

Age-at-Testing G e ne ra l F lui d A b ili ty s c o re 0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60

General Crystallized Ability (Gc) score as a function of Age

Age-at-Testing G e n e ra l C ry s ta lliz e d A b ili ty s c o re

(46)

Nonlinear Models – Exponential functions

Cross-sectional data Age (yr) 6 8 10 12 Visual M at ch in g Scores 0 10 20 30 40 50 60 70 Cross-sectional data Age (yr) 6 8 10 12 Cros s Out Sc ores 0 10 20 30 40

Y = a – be

–c*age

(47)

Nonlinear Models

z

There are some theoretical nonlinear curves

such as Verhulst’s logistic, Gompertz, von

Bertalanffy (competition)

z

Rao (1958) and Tucker (1966) principal

components of repeated measures

z

There are also mathematical (nonlinear)

functions that can be fitted to the data – with

no theoretical basis

z

An alternative approach is to estimate a set of

latent coefficients based on the data

(48)

Nonlinear Models – Fixed Coefficients

z

One option is to use the basis coefficients to specify a

particular function

Λ

[t] = [1, 1, 2, 2, 3, 3] for “steps”

Λ

[t] = [1, 2, 3, 3, 2, 1] for “up and down”

Λ

[t] = [1, -1, 1, -1, 1, -1] for “cycles”

z

Another possibility is to specify the basis coefficients as

unknown but functionally related constants

Λ

[t] =

q

[t]

z

In all these cases, the parameter estimates may be altered

but other features remain the same: the value of the

model expectations, the goodness-of-fit, and the change

in goodness-of-fit due to a latent slope

(49)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

Nonlinear Growth – Fixed Coefficients

1

2

(50)

Nonlinear Models – Polynomials

z

Quadratic model

Y

[

t

]

n

= y

0n

+ B

1[

t

]

y

s

1

n

+ B

2[

t

]

y

s

2

n

+ e

[

t

]

n

y

0n

= latent score representing an individual’s initial level

B

1[

t

]

= fixed linear weights with slopes

y

s

1

n

B

2[

t

]

= fixed quadratic weights with slopes

y

s

2

n

e

[

t

]

= errors of measurements

z

Second level model

y

0n

=

µ

01

+

e

0n

y

s

1

n

=

µ

s1

+

e

1n

y

s

2

n

=

µ

s2

+

e

2n

z

the levels and slope scores have means (

µ

ij

) and residuals

(51)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s1

y

₀*

y

s1*

ρ

_0s1

σ

_s1

σ

₀

µ

_s1

µ

₀

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

Y

_[3]

Y

_[5]

Quadratic Growth

y

_s2

y

_s2*

σ

_s2

ρ

_s1,s2

ρ

_0s2

µ

_s2

B

_[_t_] 1/2

B

[t]2

(52)

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Quadratic Slope

(

Y

[

t

]

n

= y

₀

n

+ B

1[

t

]

y

s1

n

+B

2[

t

]

y

s2

n

+ e

[t]

n

)

(53)

Nonlinear Models – Splines

z

By defining a “knot” point

k

, time can be divided in

segments and a nonlinear curve expressed as

Y

[

t

]

n

= y

0n

+ B

1[

t

]

y

s

1

n

+ B

2[

t

]

y

s

2

n

+ e

[

t

]

n

where B

1[

t

]

= T – k

, iff

t

<

k, and

B

2[

t

]

= T – k

, iff

t

>

k

y

0n

= intercept – the predicted score of

Y

[0]

at

k

(

B

1[

t

]

= B

2[

t

]

=

0)

y

s

1

= slope term before

k

– change in the predicted score of

Y

[

t

]

for one unit change in

B

1[

t

]

before

k

y

s2

= slope term after

k

– change in the predicted score of

Y

[

t

]

for

one unit change in

B

2[

t

]

after

k

e

[

t

]

= errors of measurements – the part of

Y

[

t

]

that unpredicted

and independent of the specification

B

[

t

]

(54)

z

Linear spline model (piecewise model)

Y

tn

=

y

₀

n

+

B

1

(

t<k) y

s1n

+

B

2

(

t >k) y

s

2

n

+

e

t

n

z

For example, given

T

= 6 and

k

= 4

B

1[

t

]

= [-3, -2, -1, 0, 0, 0], and

B

2[

t

]

= [ 0, 0, 0, 0, 1, 2], and

z

Y

₀

n

= intercept at

k

= 4

(

B

1[

t

]

= B

2[

t

]

=

0)

z

This model can be compared with a single-slope

model via

χ

2

and df

z

It is possible to find

k

from the data, with

individual differences (Cudeck & Klebe, 2002)

(55)

0 10 20 30 40 50 60 70 80 90 100 440 460 480 500 520 540 560

Age

W

J F

lui

d A

bi

li

ty

WJ Fluid Ability as a Function of Age

(56)

2 4 6 8 10 12 14 16 18 20 22 24 40 50 60 70 80 90 100 110 120 130 140 Time (seconds) H ea rt R ate

Heart Rate During Gazing Task -- Non-Attached

2 4 6 8 10 12 14 16 18 20 22 24 40 50 60 70 80 90 100 110 120 130 140 Time (seconds) H ea rt R ate

Heart Rate During Gazing Task -- Attached

Nonlinear Models – Splines

(57)

2 4 6 8 10 12 14 16 18 20 22 24 80 90 100 110 Time (s) HR -- - Non-A tt ac hed

Heart Rate During Gazing Task

2 4 6 8 10 12 14 16 18 20 22 24 80 90 100 110 Time (s) H R -- A tt ac he d r_t_1,_t₁= .22 rt1,t1= .17ns r_t_1,_t₁= .43 r_t_1,_t₁= .58

(58)

Nonlinear Models – Residuals

z

It is possible to model the structure of the

residuals

z

This is often used to account for changes in the

individual differences (covariances) that are

not reflected in the group trends (means) over

time

z

This approach uses time-series concepts about

changes over time and can easily improve the

fit

z

It is easy to apply with current programs but it

is important to evaluate its use

(59)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

β

₁

β

₂

β

₃

β

4

β

5

β

₆

Nonlinear Models – Residuals AR(1)

e

_y_[2]

σ

_e

σ

_e

e

_y_[4]

(60)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

β

₁

β

₂

β

₃

β

4

β

5

β

₆

Nonlinear Models – Residuals AR(2)

e

_y_[2]

σ

_e

σ

_e

e

_y_[4]

β

₁

β

₁

β

₁

β

₁

β

₁

(61)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

e

_y_[6]

Y

_[6]

1

Y

_[3]

Y

_[5]

β

₁

β

₂

β

₃

β

4

β

5

β

₆

Nonlinear Models – Residuals (other)

e

_y_[2]

e

y[4]

β

₁

β

₁

β

₁

β

₁

β

₁

β

₂

β

₂

β

₂

β

₂

(62)

Nonlinear Models – Latent Coefficients

z

It is also possible to estimate the basis coefficients

as latent values (based on the data) as in a

common factor model

(see Rao, 1958, Tucker, 1966,

Meredith & Tisak, 1990, McArdle, 1986)

z

This requires identification constraints, e.g.,

Λ

[

t

] = [0

=

,

β

2

,

β

3

,

β

4

,

β

5

, 1

=

]

z

The fixed values are used for centering (

β

1

=0) and

scaling (

β

1

=1), and the other coefficients are

estimated from the data to define the best

generalized curve

z

This model is exploratory but comparable with

other alternatives via goodness-of-fit

(63)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

Nonlinear Models – Latent Basis

e

_y_[2]

σ

_e

σ

_e

e

_y_[4]

β

0

β

₂

.4

1

.8

β

₄

(64)

Latent Slope

(

Y

[

t

]

n

= y

0n

+ B

[

t

]

y

sn

+ e

[

t

]

n

)

1 2 3 4 5 6 0 10 20 30 40 50 60 70 80

Grade at Testing

W

ISC

Sc

o

re

(65)

µ

=

1

Y

[1]

µ

₀

+

µ

s

·

β

1

Y

[2]

µ

₀

+

µ

s

·

β

2

Y

[4]

µ

₀

+

µ

s

·

β

4

Y

[6]

µ

₀

+

µ

s

·

β

6

(66)

Σ

=

Y

[1]

Y

[2]

Y

[4]

Y

[6]

Y

[1]

σ

₀

2

+

σ

e

2

+

λ

1

2

σ

s

2

+ 2

σ

0s

λ

1

Y

[2]

σ

₀

2

σ

0

2

+

σ

e

2

+

λ

1

σ

s

2

λ

2

+

λ

2

σ

s

2

+2

λ

1

σ

0s

λ

2

+ 2

σ

0s

λ

2

Y

[4]

σ

₀

2

σ

0

2

σ

0

2

+

σ

e

2

+

λ

1

σ

s

2

λ

4

+

λ

2

σ

s

2

λ

4

+

λ

4

2

σ

s

2

+2

λ

1

σ

0s

λ

4

+2

λ

2

σ

0s

λ

4

+ 2

σ

0s

λ

4

Y

[6]

σ

₀

2

σ

0

2

σ

0

2

σ

0

2

+

σ

e

2

+

λ

1

σ

s

2

λ

6

+

λ

2

σ

s

2

λ

6

+

λ

4

σ

s

2

λ

6

+

λ

6

2

σ

s

2

+2

λ

1

σ

0s

λ

6

+2

λ

2

σ

0s

λ

6

+2

λ

4

σ

0s

λ

6

+ 2

σ

0s

λ

6

(67)

8 6 17 .10 (.06) 1.00 1.00 1677/5 (1.28) 61/2 (.381) 6 8 78 .20 (.00) .98 .98 1616/3 (1.63) 3 11 1694 .70 (.00) .45 .45 ----Goodness-of-Fit Parameters Degrees of freedom Likelihood Ratio L2

RMSEA e_a(p-close fit) CFI TLI Fit Changes ∆χ2_/_∆_{df (RMSEA}_∆₎ .55 (6) .65 (6) 0 (=) Correlation ρ_0s 5.61 (17) 5.27 (12) ----2.95 (29) 5.63 (17) 4.85 (11) ----3.19 (29) 3.68 (5) 0 (=) ----12.9 (35) Deviations/Variances Level σ₀ Slope σ_s Mother’s ED σ_s2 Unique Deviation σ_e ----Regressions from X Level γ_0x Slope γ_sx 18.8 (42) 28.6 (61) ----19.7 (46) 27.7 (62) ----32.2 (62) 0 (=) ----Means/Intercepts Level µ₀/ γ₀₁ Slope µ_s/ γ_s1 Mother’s ED µ_x 0 (=) .27 (30) .4 (=) .60 (68) .8 (=) 1.0 (=) 0 (=) .2 (=) .4 (=) .6 (=) .8 (=) 1.0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) Slope Loadings β_[0] β_[1] β_[2] β_[3] β_[4] β_[5] Latent Linear Level Parameters & Fit Indices

(68)

Nonlinear Models – Exponential Models

z

Another possibility is to specify the basis coefficients as unknown

but functionally related constants

Λ

[t] =

q

[t]

z

Setting

β

[

t

]=exp[(-t-1)

π

] gives a nonlinear exponential shape

with rate of change

π

to be estimated

(McArdle & Hamagami,

1996)

z

Double-exponential model

(McArdle et al., 2002)

Y

tn

=

y

0n

+

β

(Age

t

)

⋅

y

s

(

τ

1

,

τ

2

)

n

+

e

t

n

with

β

[

t

] = exp(-

π

b

⋅

Age

t

) - exp(-

π

a

⋅

Age

t

)

β

[

t

] = the accumulation of a latent age basis,

π

b

= latent rate “before” the age peak,

π

a

= latent rate “after” the age peak, and

y

s

(

τ

1

,

τ

2

)

n

= the combined latent slope for person

n

Dual nonlinear exponential shape with two rates of change (

π

a

,

π

a

)

representing competing forces

(69)

1

e

_y_[1]

e

_y_[2]

e

_y_[4]

Y

_[1]

Y

_[2]

Y

_[4]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

e

_y_[6]

Y

_[6]

σ

_e

1

Y

_[3]

Y

_[5]

Nonlinear Models – Exponential Models

e

_y_[2]

σ

_e

σ

_e

e

_y_[4]

β

e

(-0π)

e

(-1π)

e

(-2π)

e

(-3π)

e

(-4π)

e

(-5π)

(70)

Nonlinear Growth (SAS-NLMIXED)

TITLE

: ‘Double Exponential Model’;

PROC NLMIXED

;

PARMS m_level=-80 m_slope=120 m_rate_a=.100 m_rate_b =.001

v_error=20 v_level=80 v_slope=10

c_levslo=-.01 ;

level = m_level + d_level ;

slope = m_slope + d_slope ;

rate_a = m_rate_a ;

rate_b = m_rate_b;

traject = level+slope*(EXP(-rate_b*age)-EXP(-rate_a*age));

MODEL

y01 ~ NORMAL(traject, v_error);

RANDOM d_level d_slope ~ NORMAL([0,0],

[v_level, c_levslo, v_slope]) SUBJECT=id;

(71)

Raw Data – Longitudinal

0 10 20 30 40 50 60 70 80 90 100 440 460 480 500 520 540 560

Age

W

J F

lui

d A

bi

li

ty

(72)

0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60 Age at Testing F lui d R eas oni ng (G f) s co re 0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60 Age at Testing C ry st alliz ed K no w le dg e ( G c) s co re 0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60 Age at Testing P roc es si ng S peed ( G s) s co re 0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60 Age at Testing S ho rt -T erm M em ory (G sm ) s co re

50

Du

al

-E

xp

Age-at-Testing

(a)

(b)

(c)

(d)

(74)

Double-Exponential Model

0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60

Age-at-Testing G e ne ra l F lui d A b ili ty s c o re 0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60

General Crystallized Ability (Gc) score as a function of Age

Age-at-Testing G e n e ra l C ry s ta lliz e d A b ili ty s c o re

(75)

Growth Curve of Fluid Reasoning “Gf”

0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60

Age-at-Testing G e n e ra l F lu id A b ilit y s c o re

(76)

0 10 20 30 40 50 60 70 80 90 100 -60 -40 -20 0 20 40 60

Predicted change in Broad Cognitive Ability (BCA) score as a function of first Age of testing

Age-at-Testing (first age is real data, second age is predicted scores)

P redi ct ed B roa d C ogni tiv e A bi lit y s core

Individual Modeling

(77)

time in days

18 151 121 91 61 31 1

p

o

si

tive

a

ff

e

ct

5 4 3 2 1 0

(78)

(79)

(80)

12 8 22 .09 (.05) 1.00 1.00 ----8 6 17 .10 (.06) 1.00 1.00 1677/5 (1.28) 61/2 (.381) 6 8 78 .20 (.00) .98 .98 1616/3 (1.63) 3 11 1694 .70 (.00) .45 .45 ----Goodness-of-Fit Parameters Degrees of freedom Likelihood Ratio L2

RMSEA e_a(p-close fit) CFI TLI Fit Changes ∆χ2_/_∆_{df (RMSEA}∆₎ .52 (5) .55 (6) .65 (6) 0 (=) Correlation ρ_0s 4.57 (16) 5.12 (12) 7.28 (10) 2.95 (29) 5.61 (17) 5.27 (12) ----2.95 (29) 5.63 (17) 4.85 (11) ----3.19 (29) 3.68 (5) 0 (=) ----12.9 (35) Deviations/Variances Level σ₀ Slope σ_s Mother’s ED σ_s2 Unique Deviation σ_e 1.21 (9) 0.47 (3) ----Regressions from X Level γ_0x Slope γ_sx 5.78 (4) 23.5 (13) 10.8 (57) 18.8 (42) 28.6 (61) ----19.7 (46) 27.7 (62) ----32.2 (62) 0 (=) ----Means/Intercepts Level µ₀/ γ₀₁ Slope µ_s/ γ_s1 Mother’s ED µ_x 0 (=) .27 (30) .4 (=) .60 (68) .8 (=) 1.0 (=) 0 (=) .27 (30) .4 (=) .60 (68) .8 (=) 1.0 (=) 0 (=) .2 (=) .4 (=) .6 (=) .8 (=) 1.0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) Slope Loadings β_[0] β_[1] β_[2] β_[3] β_[4] β_[5] Latent with Exogenous Latent Linear Level Parameters & Fit Indices

(81)

Fitting Latent

Growth Models

(82)

Fitting Latent Growth

Models 1:

(83)

Different Input and Output

z

Slightly different data inputs required for different

computer programs

z

Assuming

N

individuals on

T

repeated occasions

z

Most programs’ input is based on

fla

t (

N

x

T

) raw

data matrix or

T

means and (

T

x

T

) covariances

z

Many mixed models (e.g., SAS PROC MIXED)

use

relational

input of

T

vectors (rows) per person

(

T

x

N

) with same ID code

z

Outputs also differ, but basic model parameters

(84)

Example (McArdle & Epstein, 1987)

z

Data from longitudinal study of WISC-R by on

N

=204 children (Osborne & Suddick, 1972)

z

Repeated measurements at grades 1, 2, 4, and 6

z

WISC total means = 18.8, 26.6, 36.0, and 47.3

z

WISC total SDs = 6.4, 7.3, 7.7 and 10.4

z

WISC total correlations = .765 - .867

(85)

WISC Data (Individual Scores)

1 2 3 4 5 6 0 10 20 30 40 50 60 70 80

Grade at Testing

W

ISC

Sc

o

re

(86)

“Level Only” Growth Model

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

₀*

σ

₀

µ

₀

1

1 1

σ

_e

σ

_e

σ

_e

σ

_e

σ

_e

e

_y_[5]

Y

_[5]

σ

_e

1

Y

_[2]

Y

_[4]

(87)

Level Only Model (AMOS input)

Sem.BeginGroup "wisc.sav"

Sem.Structure "total1 = (0) + (1) LEVEL + (1) E1 "

Sem.Structure "total5 = (0) + (1) LEVEL + (1) E5 "

Sem.Mean "LEVEL", "mn_level"

Sem.Structure "E1 (v_uniq) "

Sem.Structure "E6 (v_uniq) “

(88)

Level Only Model (AMOS output)

Regression Weights: Estimate S.E. C.R. Label --- --- --- --- ---total1 <--- LEVEL 1.000 total2 <--- LEVEL 1.000 total3 <--- LEVEL 1.000 total4 <--- LEVEL 1.000 total5 <--- LEVEL 1.000 total6 <--- LEVEL 1.000

Means: Estimate S.E. C.R. Label --- --- --- --- ---LEVEL 32.164 0.519 61.953 mn_leve

Variances: Estimate S.E. C.R. Label --- --- --- --- ---LEVEL 13.305 5.927 2.245 E1 165.646 9.493 17.450 v_uniq E2 165.646 9.493 17.450 v_uniq E3 165.646 9.493 17.450 v_uniq E4 165.646 9.493 17.450 v_uniq E5 165.646 9.493 17.450 v_uniq E6 165.646 9.493 17.450 v_uniq Chi-square = 1697.040 Degrees of freedom = 11 Probability level = 0.000

(89)

Estimates from a“Level Only” Model

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

₀*

1

1 1

12.9

e

_y_[5]

Y

_[5]

1

Y

_[2]

Y

_[4]

12.9

32.2

3.68

χ

2

(11) = 1697

(90)

1 2 3 4 5 6 0 10 20 30 40 50 60 70 80

Grade at Testing

W

ISC

Sc

o

re

(91)

Linear Growth Model

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

0

.2

.4

.6

1

σ

_e

σ

_e

σ

_e

σ

_e

σ

_e

e

_y_[5]

Y

_[5]

σ

_e

1

.8

Y

_[2]

Y

_[4]

(92)

Linear Growth Model (AMOS input)

Sem.Structure "total1 = (0) + (1) LEVEL + (

0

)

SLOPE

+ (1) E1 "

.2

)

SLOPE

+ (1) E2 "

.4

)

SLOPE

+ (1) E3 "

.6

)

SLOPE

+ (1) E4 "

.8

)

SLOPE

+ (1) E5 "

1

)

SLOPE

+ (1) E6 "

Sem.Mean "LEVEL", "mn_level"

Sem.Mean "

SLOPE

", "mn_slope"

Sem.Structure "LEVEL<>SLOPE (c_ls) "

Sem.Structure "E6 (v_uniq) "

(93)

Linear Growth Model (AMOS output)

Regression Weights: Estimate S.E. C.R. Label --- --- --- --- ---total1 <--- LEVEL 1.000 total2 <--- LEVEL 1.000 total3 <--- LEVEL 1.000 total4 <--- LEVEL 1.000 total5 <--- LEVEL 1.000 total6 <--- LEVEL 1.000 total1 <--- SLOPE 0.000 total2 <--- SLOPE 0.200 total3 <--- SLOPE 0.400 total4 <--- SLOPE 0.600 total5 <--- SLOPE 0.800 total6 <--- SLOPE 1.000

Means: Estimate S.E. C.R. Label --- --- --- --- ---LEVEL 19.698 0.430 45.784 mn_leve SLOPE 27.704 0.447 62.008 mn_slop Covariances: Estimate S.E. C.R. Label --- --- --- ---

---LEVEL <---> SLOPE 17.637 2.877 6.130 c_ls Variances: Estimate S.E. C.R. Label --- --- --- --- ---LEVEL 31.580 3.753 8.414 SLOPE 23.395 4.198 5.573

(94)

Estimates from a Linear Slope Model

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

_s

y

₀*

y

s*

0

.2

.4

.6

1

e

_y_[5]

Y

_[5]

.8

Y

_[2]

Y

_[4]

3.19

19.7

5.63

χ

2

(8) = 79

∆χ

2

(3) = 1616

27.7

4.85

.65

(95)

1 2 3 4 5 6 0 10 20 30 40 50 60 70 80

Grade at Testing

W

ISC

Sc

o

re

Linear Slope

(

Y

[

t

]

n

= y

0n

+ B

[

t

]

y

sn

+ e

[

t

]

n

)

(96)

Latent Growth Model

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

_s

y

₀*

y

s*

ρ

_0s

σ

_s

σ

₀

µ

_s

µ

₀

1

1 1

0

β

₂

.4

1

σ

_e

σ

_e

σ

_e

σ

_e

σ

_e

e

_y_[5]

Y

_[5]

σ

_e

1

.8

Y

_[2]

Y

_[4]

β

₄

(97)

Latent Growth Model (AMOS input)

Sem.Structure "total1 = (0) + (1) LEVEL + (0) SLOPE + (1) E1 "

b_1

) SLOPE + (1) E2 "

Sem.Structure "total3 = (0) + (1) LEVEL + (.4) SLOPE + (1) E3 "

b_2

) SLOPE + (1) E4 "

Sem.Mean "LEVEL", "mn_level"

Sem.Mean "SLOPE", "mn_slope"

Sem.Structure "LEVEL<>SLOPE (c_ls) "

End Sub

(98)

Latent Growth Model (AMOS output)

Regression Weights: Estimate S.E. C.R. Label --- --- --- --- ---total1 <--- LEVEL 1.000 total2 <--- LEVEL 1.000 total3 <--- LEVEL 1.000 total4 <--- LEVEL 1.000 total5 <--- LEVEL 1.000 total6 <--- LEVEL 1.000 total1 <--- SLOPE 0.000 total2 <--- SLOPE 0.271 0.009 30.002 b_1 total3 <--- SLOPE 0.400 total4 <--- SLOPE 0.597 0.009 67.809 b_2 total5 <--- SLOPE 0.800 total6 <--- SLOPE 1.000 Means: Estimate S.E. C.R. Label --- --- --- --- ---LEVEL 18.813 0.443 42.441 mn_leve SLOPE 28.590 0.471 60.734 mn_slop Covariances: Estimate S.E. C.R. Label --- --- --- ---

---LEVEL <---> SLOPE 16.298 2.919 5.583 c_ls

Variances: Estimate S.E. C.R. Label --- --- --- --- ---LEVEL 31.289 3.681 8.499 SLOPE 27.676 4.433 6.243 E1 8.675 0.609 14.248 v uniq

(99)

Estimates from a Latent Slope Model

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

_s

y

₀*

y

s*

0

.27

.4

.60

1

e

_y_[5]

Y

_[5]

.8

Y

_[2]

Y

_[4]

2.95

18.8

5.61

χ

2

(6) = 17

∆χ

2

(2) = 61

28.6

5.27

.55

(100)

Latent Slope Model with Exogenous

Variable (Mother’s Education)

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

_s

z

_y0*

z

ys*

ω

_0s

ω

_s

ω

₀

γ

_s1

γ

₀₁

1

1 1

β

₀

β

₁

β

₂

β

3

β

₅

σ

_e

σ

_e

σ

_e

σ

_e

σ

_e

e

_y_[5]

Y

_[5]

σ

_e

1

β

₄

X

µ

_x

γ

_0x

γ

_sx

σ

_x2

Y

_[2]

Y

_[4]

(101)

Sem.Structure "total2 = (0) + (1) LEVEL + (b_1) SLOPE + (1) E2 "

Sem.Structure "total4 = (0) + (1) LEVEL + (b_2) SLOPE + (1) E4 "

Sem.Structure "LEVEL = (int_level) +

(mom_l) momed

+ (1) var_level "

Sem.Structure "SLOPE = (int_slope) +

(mom_s) momed

+ (1) var_slope "

Sem.Structure "momed = (int_momed) +

(1) var_momed

Sem.Structure "var_level<>var_slope (c_ls) "

End Sub

(102)

Regression Weights: Estimate S.E. C.R. Label --- --- --- ---

---LEVEL <--- momed 1.206 0.134 9.012 mom_l SLOPE <--- momed 0.475 0.168 2.821 mom_s total1 <--- LEVEL 1.000 total2 <--- LEVEL 1.000 total1 <--- SLOPE 0.000 total2 <--- SLOPE 0.271 0.009 29.995 b_1 total3 <--- SLOPE 0.400 total4 <--- SLOPE 0.597 0.009 67.843 b_2 total5 <--- SLOPE 0.800 total6 <--- SLOPE 1.000

Intercepts: Estimate S.E. C.R. Label --- --- --- --- ---momed 10.811 0.189 57.228 int_mom LEVEL 5.776 1.494 3.867 int_lev SLOPE 23.457 1.875 12.508 int_slo Covariances: Estimate S.E. C.R. Label --- --- --- ---

---var_level <---> var_slope 12.151 2.404 5.054 c_ls Variances: Estimate S.E. C.R. Label --- --- --- --- ---var_momed 7.245 0.719 10.075 var_level 20.756 2.644 7.851 var_slope 26.034 4.274 6.090 E1 8.676 0.609 14.248 v_uniq E2 8.676 0.609 14.248 v_uniq

(103)

Estimates from a Latent Slope Model

with Mother’s Education

1

e

_y_[2]

e

_y_[0]

e

_y_[1]

e

_y_[3]

e

_y_[4]

Y

_[0]

Y

_[1]

Y

_[3]

y

₀

y

_s

z

_y0*

z

ys*

e

_y_[5]

Y

_[5]

X

1.21

0.47

7.25

Y

_[2]

Y

_[4]

0

.27

.4

.60

1

.8

2.95

5.78

4.57

χ

2

(8) = 22

23.5

5.12

.52

10.8

(104)

LDS Fit Statistics

Goodness

Model 1 Model 2 Model 3 Model 4

of Fit

Level

Linear Latent Mothed

LRT (

χ

2

)

1694

78

17

22

df

11

8

6

8

RMSEA

.70

.20

.10

.09

p

-close fit

.00

.06

.05

(105)

Conclusions

z

A latent model with unequal growth over time

seems more reasonable for these data than models

with flat or linear trajectories

z

Mother’s education have a positive influence on

both the level and slope

z

Other modeling alternatives are possible (e.g., age

(106)

Other Programs:

Mplus

(107)

Linear Growth Model (Mplus input)

TITLE

: Linear Growth Models --WISC Data

DATA

: FILE IS wiscraw.dat;

VARIABLE

:

NAMES

ARE id wisc1 wisc2 wisc4 wisc6;

USEVAR

= wisc1 wisc2 wisc4 wisc6;

ANALYSIS

: TYPE = MEANSTRUCTURE;

MODEL

: !creating latent variables to deal with incomplete data

lwisc1 by wisc1@1;

lwisc2 by wisc2@1;

lwisc3 by wisc1@0;

lwisc4 by wisc4@1;

lwisc5 by wisc2@0;

lwisc6 by wisc6@1;

(108)

Linear Growth Model (Mplus input cont.)

!level loadings fixed at 1

level BY lwisc1-lwisc6@1 ;

!slope loadings fixed at linear estimates (0-1); relax this for a latent model (*)

slope BY lwisc1@0 [email protected] [email protected] [email protected] [email protected] lwisc6@1;

!level and slope means with starting values; other means set to 0

[level*19 slope*27 wisc1-wisc6@0 lwisc1-lwisc6@0];

!level and slope variances and covariance (r= cov/sd*sd)

level*25 slope*25 ; level with slope*17 ;

!equal unique variances

wisc1-wisc6*10 (1);

!latent variances to 0

lwisc1-lwisc6@0 ;

(109)

Linear Growth Model (Mplus output)

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value 79.101 Degrees of Freedom 8 P-Value 0.0000 RMSEA (Root Mean Square Error Of Approximation)

Estimate 0.209

90 Percent C.I. 0.168 0.252 Probability RMSEA <= .05 0.000

MODEL RESULTS

Estimates S.E. Est./S.E. Means LEVEL 19.698 0.429 45.895 SLOPE 27.704 0.446 62.164 Variances LEVEL 31.584 3.744 8.435 SLOPE 23.391 4.187 5.587 SLOPE BY LWISC1 0.000 0.000 0.000 LWISC2 0.200 0.000 0.000 LWISC3 0.400 0.000 0.000 LWISC4 0.600 0.000 0.000 LWISC5 0.800 0.000 0.000 LWISC6 1.000 0.000 0.000 LEVEL WITH SLOPE 17.629 2.870 6.143 Residual Variances

WISC1 10.104 0.707 14.283 WISC2 10.104 0.707 14.283

(110)

Latent Growth Model (Mplus output)

Chi-Square Test of Model Fit

Value 17.485 Degrees of Freedom 6 P-Value 0.0076 RMSEA (Root Mean Square Error Of Approximation)

Estimate 0.097 90 Percent C.I. 0.046 0.151 Probability RMSEA <= .05 0.063 Means LEVEL 18.813 0.442 42.544 SLOPE 28.590 0.470 60.884 Variances LEVEL 31.292 3.673 8.520 SLOPE 27.677 4.422 6.259 SLOPE BY LWISC1 0.000 0.000 0.000 LWISC2 0.271 0.009 30.079 LWISC3 0.400 0.000 0.000 LWISC4 0.597 0.009 67.978 LWISC5 0.800 0.000 0.000 LWISC6 1.000 0.000 0.000 SLOPE WITH LEVEL 16.296 2.912 5.596 Residual Variances WISC1 8.675 0.607 14.283 WISC2 8.675 0.607 14.283 WISC4 8.675 0.607 14.283 WISC6 8.675 0.607 14.283

(111)

Other Programs:

SAS

(112)

TITLE

'Making a Multiple-Record Data File'

;

DATA

temp1;

SET

wiscraw;

age1=

6

; age2=

6.95

; age4=

8.8

; age6=

10.8

;

grade1=

0

; grade2=

1

; grade4=

3

; grade6=

5

;

FILE

outfile

LRECL

=

200

LINESIZE

=

200

;

PUT

#

1

id age1 grade1 verbal1 nv1 wisc1 mothed

#

2

id age2 grade2 verbal2 nv2 wisc2 mothed

#

3

id age4 grade4 verbal4 nv4 wisc4 mothed

#

4

id age6 grade6 verbal6 nv6 wisc6 mothed ;

RUN

;

DATA

temp2;

INFILE

outfile

LRECL

=

200

LINESIZE

=

200

;

INPUT

id age grade verbal nv wisc mothed ;

agec=age-

6

; agec2=agec*agec; age2=age*age;

RUN

;

(113)

Age-Based Linear Models (SAS Mixed)

TITLE

: ‘Initial Baseline Variance';

PROC MIXED

NOCLPRINT COVTEST; CLASS id;

MODEL

wisc = / SOLUTION;

RUN

;

TITLE

: ‘No Growth';

PROC MIXED

MODEL

wisc = / SOLUTION;

RANDOM

INTERCEPT / SUBJECT=id TYPE=UN GCORR;

RUN

;

TITLE

: 'Linear Age';

PROC MIXED

MODEL

wisc = age / SOLUTION;

(114)