1
AUTOREGRESSIVE & DIFFERENCE
SCORE MODELS of CHANGE
HDFS 597K
February 02, 2009
HDFS 597K
Applied Longitudinal Data Analysis
Overview
1. Twooccasion longitudinal data
2. TypeA AutoRegression Model
3. TypeD difference score models
4. Some additions
2OCCASION LONGITUDINAL DATA
TwoOccasion Data are Valuable
• Twooccasions are the first instance of longitudinal
data
– All longitudinal data collections begin with two occasions,
then they get more … and more … and .
..
Some basic questions of representing change over
• Some basic questions of representing change over
time (
see Nesselroade, 1991
)
–
Change in interindividual differences
–
Interindividual differences in change
•
Different problems
seem to suggest
different
models
and methods of analysis
Example of TwoOccasion Data
• Data from the RIGHT (Research Investigating Growth
and Health Trajectories) Track Research Project
– Focus on the development and developmental trajectories
of early disruptive behavior
• Participants:
– N=431 children, Measured at age 2, 4, 5 & 7 years
• Measures:
– Attention Deficit Hyperactivity Total Scores
• age 4 & age 7 assessments
5
The CORR Procedure
2
Variables:
adhd_to4 adhd_to7
Simple Statistics
Variable
N
Mean
Std Dev
Sum
Minimum
Maximum
adhd_to4
376
13.78
9.29
5182
0
51.00
adhd_to7
328
12.11
10.13
3973
0
54.00
Summary Statistics from RIGHT Track
Pearson Correlation Coefficients
Prob > r under H0: Rho=0
Number of Observations
adhd_to4
adhd_to7
adhd_to4
1.00000
0.60359
<.0001
376
314
adhd_to7
0.60359
1.00000
<.0001
314
328
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There are two approaches to the prediction of
change in such analyses, and in this book (sic)
we use both.
The first approach involves computing a change
score by subtracting the “before” measure from
the “after” measure, and then using the change
score as the “dependent variable.”
Quotes from Bachman et al (2002, p.31)
The second approach uses the “after” measure as
the “dependent variable” and include the “before”
measure as one of the “predictors” (i.e., as a
covariate).
In either case one could say that the earlier score is
being “controlled,” but the means of controlling
differ and the results of the analysis also can
differ  sometimes in important ways.
Bivariate Scatterplot
X=Age 4 vs. Y=Age 7
8
Plotting Individual “Trajectories” (n=50)
9
AUTOREGRESSIVE MODELS
MODELS FOR REPEATED MEASURES
O
A linear model is expressed for i=1 to
N
as
Y[2]
i
=
β
0
+
1
Y[1]
i
+
e
i
• where
0
is the
intercept
term  the predicted score of
Y[2] when Y[1]=0
• where
1
is the
coefficient
term  the change in the
predicted score of Y[2] for a one unit change in Y[1]
Most Common Linear Regression Models
• where
e
is the
residual
score  an unobserved and
random score which is uncorrelated with Y[1] but forms
part of the variance of Y[2]
O
The ratio of the variance of
e
to
Y
[
2
] (
σ
e
2
/
y
2
=
1
R
2
) can be a useful index of forecast
(in)efficiency
Complete CrossProducts based Auto
Regression Model for Repeated Measures
Y[1]
Y[2]
e
e
2
=1
1
2
1
12
1
μ
1
=1
0
Note: The inclusion of constant 1
allows the intercept
0
to be part of
the regression equation for Y[2].
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 Spring '11
 P
 Linear Regression, Regression Analysis, Variance, difference score, Latent Difference Models

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