Two-wave Two-variable Models David A. Kenny December 24, 2013 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. 2 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) 3 4 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. 5 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. 6 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 7 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. 8 9 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 10 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. 11