# Testing simultaneous and lagged effects in longitudinal mixed models with time-varying covariates

I was recently told that it was not possible to incorporate time-varying covariates in longitudinal mixed models without introducing a time lag for these covariates. Can you confirm / deny this? Do you have any references on this situation ?

I propose a simple situation to clarify. Suppose that I have repeated measures (say over 30 occasions) of quantitative variables (y, x1, x2, x3) in 40 subjects. Each variable is measured 30 times in each subject by a questionnaire. Here the final data would be 4 800 observations (4 variables X 30 occasions X 40 subjects) nested in 40 subjects.

I would like to test separately (not for model comparison) for :

• simultaneous (synchronous) effects : the influence of x1, x2, and x3 at time t on y at time t.
• lagged effects : the influence of x1, x2, and x3 at time t-1 on y at time t.

I hope everything is clear (I'm not a native English speaker !).

For instance, in R lmer{lme4}, the formula with lagged-effects is :

lmer(y ~ lag1.x1 + lag1.x2 + lag1.x3 + (1|subject))


where y is the dependent variable at time t, lag1.x1 is the lagged independent variable x1 at the individual level, etc.

For simultaneous effects, the formula is :

lmer(y ~ x1 + x2 + x3 + (1|subject))


Everything is running well and it gives me interesting results. But is it correct to specify a lmer model with synchronous time-varying covariates or have I missed something ?

Edit: Moreover, is it possible to test both simultaneous and lagged effects at the same time ? , For instance :

lmer(y ~ x1 + x2 + x3 + lag1.x1 + lag1.x2 + lag1.x3 + (1|subject))


Theoretically, it makes sense to test competition between concurrent vs. lagged effects. But is it possible with lmer{lme4} in R, for example ?

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