Two-wave Two-variable
Models
David A. Kenny
December 24, 2013
The Basic Design
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Two variables
Measured at two times
Gives rise to 4 variables
Say Depression and Marital Satisfaction are
measured for wives with a separation of one
year.
• We have D1, D2, S1, and S2.
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Standard Cross-lagged
Regression Model
• Time 1 variables cause Time 2 variables.
– Two stabilities
• S1  S2
• D1  D2
– Two cross-lagged effects
• S1  D2
• D1  S2
• Time 2 disturbances correlated.
• Inadvisable to have paths between Time 2
variables (S2  D2 or D2  S2)
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Causal Preponderance
• Is S a stronger cause of D than is D of
S?
• No easy way because the units of
measurement of S and D are likely very
different.
• Can standardize all the variables, but
as will be seen this is more difficult
when S and D are latent.
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Assumptions
• No measurement error in S1 and D1. (Ironically,
OK if there is measurement error in S2 and D2.)
• Nothing that causes both the time 1 variable and
the time 2 variables. Such a variable is sometimes
called a confounder. So if there is a gender (and
gender is not controlled) difference at time 1, once
we control for S1 and D1, there are no remaining
gender differences at time 2 in S or D.
• The lagged effect of variables is exactly the length
of measurement.
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What to Do about the
Assumptions?
• Measurement error in S1 and D1:
– Latent variable analysis (discussed in a
latter slide).
• Confounders
– Measure them.
– Sensitivity analysis: See how the results
change assuming confounders.
• Wrong lag
– Multi-wave study can be used to establish
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the optimal lag.
Latent Variables
• Can have as few as two indicators per latent
variable.
• Correlate errors of the same indicator measured at
different times.
• Test to see if loadings do not change over time.
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Causal Preponderance
• Note that even if the Time 1 latent variables
are standardized, the Time 2 ones are not.
– One can standardize disturbances (U and
V in the figure), but cannot standardize
latent endogenous variables (S2 and D2).
• One can through a series on non-linear
constraints standardized latent endogenous
variables, it is very complicated.
• However, the SEM program laavan does
have an option to standardize all latent
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variables (std.lv=TRUE).
Depression and Marital
Satisfaction Example
• Gustavson, K. B. et al. (2012). Reciprocal
longitudinal associations between
depressive symptoms and romantic partners'
synchronized view of relationship quality.
Journal of Social and Personal Relationships
29, 776- 794.
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