Cross-lagged Panel
Correlation (CLPC)
David A. Kenny
December 25, 2013
Example
• Depression and Marital
Satisfaction measured at two
points in time.
• Four measured variables S1,
S2, D1, and D2.
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Causal Assumptions
• Most analyses of longitudinal variables
explain the correlation between two
variables as being due to the variables
causing each other: S  D and D  S.
• CLPC starts by assuming that the correlation
between variables is not due to the two
variables causing one another.
• Rather it is assumed that some unknown
third variable, e.g., social desirability, brings
out about the relationship.
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Model of Spuriousness
• Assume that a variable Z explains the
correlation between variables at each time.
The variable Z is changing over-time.
• The model is under-identified as a whole, but
the squared correlation between Z1 and Z2 is
identified as rD1S2rD2S1 /(rD1S1rD2S2).
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Ruling out Spuriousness
• The strategy developed by Kenny in the
1970s in a series of paper is to assume
stationarity.
• Requires at least three variables measured
at each time.
• Stationarity
– Define how much variance for a given a given
variable, say D, is available to correlate.
– Define the ratio of variance, time 2 divided by
time 1.
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Stationarity
• Define how much variance for a given a
given variable, say XA, is available to
correlate.
• Define the ratio of variance, time 2 divided
by time 1 for XA, to be denoted as kA2.
• Given stationarity, the covariance between
XA and XB at time 2 equals the time 1
covariance times kAkB.
• Also C(XA1,XB2)kB = C(XA2,XB1)kA where C is
a covariance.
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Basic Strategy
• Test for stationarity of cross-sectional
relationships.
o df = n(n – 3)/2
• If met, test for spuriousness.
o df = n(n – 1)/2
• Mplus syntax can be downloaded at
www.handbookofsem.com/files/ch09/index.html
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Example Data
Dumenci, L., & Windle, M. (1996). Multivariate
Behavioral Research, 31, 313-330.
Depression with four indicators (CESD)
PA: Positive Affect (lack thereof)
DA: Depressive Affect
SO: Somatic Symptoms
IN: Interpersonal Issues
Four times separated by 6 months
Use waves 1 and 2 for the example
433 adolescent females
Age 16.2 at wave 1
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Example
• Test for stationarity of cross-sectional
relationships:
o c2(2) = 5.186, p = .075
• Because stationarity is met, test for
spuriousness:
o c2(6) = 2.534, p = .865
• Evidence consistent with spuriousness.
• Mplus syntax can be downloaded at
• www.handbookofsem.com/files/ch09/index.html
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Why is this strategy not adopted?
• Most researchers are interested in
estimating a causal effect, not in showing
you do not need to estimate any causal
effects.
• Also, CLPC was initially proposed as way of
determining causal effects, not as a way of
testing of spuriousness.
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In principle…
• Researchers should show that spuriousness
can plausibly explain the covariation in their
data.
• CLPC has a use.
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