Standardized Change Score Analysis

David A. Kenny

December 15, 2013

TermsX as a cause of the changeSingle Measure: Y1 and Y2

Multiple MeasuresTime 1: Y11, Y12, and Y13

Time 2: Y21, Y22, and Y23

Latent Variables: T1 and T2

Residualized variance: The variance of a measure after the effects of some variable is removed. In this case, the variance of Y1 and Y2 after the effect of X are removed.

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Model for Change Score Analysis

• Z is the assignment variable that explains the time 1 gap.

• Think of Z as a dichotomy, with levels 1 and 2.• We measure the mean of Y1 and Y2 for

different values of Z.• We compute the gap at each time (controlling

for X).• That gap is the same.

Maybe the gap widens?Do not we sometimes find that “the rich

get richer and the poor get poorer”?There may be diverging slopes and gap

might be getting wider.Why not equate variances of Y1 and Y2

by standardizing?Makes more sense to equate the

residual variances, i.e., the variances of Y1 and Y2 after controlling for Z.

Units of measurement the same?Does it make sense to assume that the

units of Y are the same at each time?In some cases the measures of Y are

different at different times.Again we might want to consider a Z

score transformation to make units comparable.

Does Variance Change over Time?Univariate Test

Correlate (Y2 – Y1) with (Y2 + Y1) controlling for X to see if variance changes.

Latent variable TestRun Kenny-Judd analysis and see if the residual variance in the Time 1 and Time 2 variances are equal.

If variances the same, then use CSA not SCSA.

Standardized Change Scores: How?Create a latent variables for each time, Z1 and Z2,

“standardized” versions of Y1 and Y2.

Have X cause Z1 and Z2, using a phantom variable method to estimate the effect X on standardized change.

Constrain the disturbance variances of Z1 and Z2 to be equal to 1 and correlate.

Have Z1 cause Y1 and Z2 cause Y2 with a free paths and no disturbance.

The paths will be the residual sd’s of Y1 and Y2.

Standardized Change with Latent Variables

• Create a two latent variables for each time, LZ and LY.• Have X cause the LZ’s at each time.• Constrain the residual variances of the LZ’s to be equal

to 1.• Have each LZ cause its LY freeing the path with no

disturbance variance. The path equals the latent sd.• Have each LY cause the indicators in a temporally

invariant way.• Correlate the errors of same indicators over time.

Different Measures of Change• You cannot really compare the standardized

change effect to the other analyses because the units of the outcome, Time 2, are different.– Raw Change: Y2 – Y1

– Standardized Change: Y2/s2.X - Y1/s1.X

– Standardized Change in T2 units: Y2 - (s2.X/s1.X)Y1

Note that the standardized change score is multiplied by s2.X.

Standardized Change Scores in the Units of the Time 2 Latent Variable

• Goal: To rescale the SCSA solution so the outcome is Y2, that variable standardized.

• How– Drop Z2 and Replace Z1 by W1.– Set the variance of the two disturbances equal (v).– Path from W1 to the Y1 when estimated is equal to

the ratio of the Y1 latent residual sd to the Y2 residual sd.

Standardized Change Scores in the Units of the Time 2 Latent Variable

• Goal: To rescale the previous solution so the outcome is raw Latent Time 2.

• How– Drop Z2 and Replace Z1 by W1.– Set the variance of the two disturbances equal (v).– Path from W1 to the Time 1 latent variable is the

ratio of the Time 1 latent residual sd to the Time 2 latent residual sd.