change score analysis david a. kenny december 15, 2013
TRANSCRIPT
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Change Score Analysis
David A. Kenny
December 15, 2013
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OverviewX 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
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NotationX 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
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Naïve ApproachCompute raw change: Y2 – Y1
Regress change on X
ProblemsLoss 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
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Repeated Measures ANOVAX 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.
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Raykov MethodRaykov, T. (1992). Structural models for studying
correlates and predictors of change. Australian Journal of Psychology, 44, 101-112.
Two Latent VariablesBaseline: 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.
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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.
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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.
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cd
1
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Model Fit
The fit of the models evaluates the fit of the measurement model.
Same value for the following models.
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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.
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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.
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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.
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17
The Actual LCS Model
Intercept
Slope
0
T1
0
T2
0
T3
0,
E2
0,
E31
1
0,
E11
0
L1
0
L2
0
L3
1
1
1
1
1
0
D2-1
0
D3-2
1
1
a
a
1
1
1
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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.
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SummaryVery 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.