Two-wave Two-variable Models David A. Kenny

The Basic Design 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.

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)

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.

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.

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 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.

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 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.