Standardized Change
Score Analysis
David A. Kenny
December 15, 2013
Terms
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
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.
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.
3
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 Test
Run 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.