Change Score Analysis

David A. Kenny

December 15, 2013

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

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

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

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.

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.

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.

cd

1

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.

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

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.

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.