Change Score Analysis
David A. Kenny
December 15, 2013
Overview
X as a cause of the change in Y from time 1 to
time 2
Cases
Single Measure vs. Latent Variable
Approaches
Naïve
Repeated Measures ANOVA
Raykov Approach
McArdle’s Latent Change Score Approach
Kenny-Judd Approach
Notation
X as a cause of the change
Single Measure: Y1 and Y2
Multiple Measures
Time 1: Y11, Y12, and Y13
Time 2: Y21, Y22, and Y23
Latent Variables: T1 and T2
Naïve Approach
Compute raw change: Y2 – Y1
Regress change on X
Problems
Loss of information: 2 variables become 1
Specification Error: There might be a different
factor structure at time 2 from that at time 1.
Low reliability of change scores making
measurement of latent change difficult
OK to do, but other options seen as more
acceptable
Repeated Measures ANOVA
X must be categorical (big limitation).
Time as a repeated measure.
The X by Time interaction tests the effect of
X on change in Y.
Model is very similar to the Kenny-Judd
approach discussed later.
Raykov Method
Raykov, T. (1992). Structural models for
studying correlates and predictors of
change. Australian Journal of Psychology,
44, 101-112.
Two Latent Variables
Baseline: Waves one and two measures
load on this latent variable.
Change: Only wave two measures load on
that latent variable.
For a single measure, Y1 and Y2 have no
disturbance.
Similarity to a
Growth Curve Model
Baseline factor is like an Intercept factor.
Change factor is like a slope factor.
Note that error variance in the growth model
is not estimated.
Raykov Method: Measurement
Model
Loadings of the same measure at different
times set equal.
To be safely identified, need at least 3
indicators. If just 2, can set both loadings
to one.
Errors of the same measure correlated over
time.
This same measurement model is estimated
in all cases.
1
c
d
Model Fit
The fit of the models evaluates the fit of the
measurement model.
Same value for the following models.
McArdle’s Latent Change Score
Approach: Measured Variables
The actual LCS model is a more restricted
version of the model that follows and
requires three of more waves.
Y1 causes Y2, and constrain that causal
effect to be 1.
The disturbance in Y2 represents change.
Correlate the disturbance of change with Y1.
Correlate X with Y1 and have it cause
change.
More
Estimate of b the same as for Raykov.
Perhaps a bit more intuitive and definitely
better known than Raykov.
Similar to growth curve model.
Y1 is like an intercept factor
Change factor is like a slope factor.
Latent Difference Score Approach:
Latent Variables
Have latent Y1 cause latent Y2, and
constrain that causal effect to be 1.
The disturbance in latent Y2 represents
change.
Need to correlate the disturbance in change
with latent Y1.
Correlate X with latent Y1 and have it cause
change.
Standard measurement model.
The Actual LCS Model
Intercept
1
0
0
0,
1
1
E1
L1
T1
a
1
0
0,
0
1
1
T2
E2
0
1
D2-1
L2
a
1
0,
E3
T3
1
1
1
1
0
0
0
L3
D3-2
1
Slope
17
Kenny-Judd (1981) Approach
Very similar to repeated measures
analysis of variance.
Have X cause Y1 and Y2.
Correlate the disturbances of Y1 and Y2.
To test the null hypothesis of no effect
of X on Y, test the equality of the
effects of X on Y1 and Y2.
To estimate the effect of X use a
phantom variable.
Summary
Very different methods all give the same
estimate of the effect of X, path b.
No inherent reason to prefer one method
over the other. Chose the method you
and your reviewers feel comfortable with.
In non-randomized situations, strong
assumptions are being made. May want to
consider Standardized Change Score
Analysis.