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: Y 1 and Y 2 Multiple Measures Time 1: Y 11, Y 12, and Y 13 Time 2: Y 21, Y 22, and Y 23 Latent Variables: T1 and T2

Naïve Approach Compute raw change: Y 2 – Y 1 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, 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, Y 1 and Y 2 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.

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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. Y 1 causes Y 2, and constrain that causal effect to be 1. The disturbance in Y 2 represents change. Correlate the disturbance of change with Y 1. Correlate X with Y 1 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. Y 1 is like an intercept factor Change factor is like a slope factor.

Latent Difference Score Approach: Latent Variables Have latent Y 1 cause latent Y 2, and constrain that causal effect to be 1. The disturbance in latent Y 2 represents change. Need to correlate the disturbance in change with latent Y 1. Correlate X with latent Y 1 and have it cause change. Standard measurement model.

17 The Actual LCS Model

Kenny-Judd (1981) Approach Very similar to repeated measures analysis of variance. Have X cause Y 1 and Y 2. Correlate the disturbances of Y 1 and Y 2. To test the null hypothesis of no effect of X on Y, test the equality of the effects of X on Y 1 and Y 2. 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.