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I was wondering how the interpretation of a time-varying covariate in a linear mixed effects model differs from from that of a time-independent covariate?

For example, how does the interpretation of $\textrm{Age}_{jt}$ ($j=\textrm{subject}, t=\textrm{time point}$) as a time-varying covariate with an estimated fixed effect coefficient of .64 with respect to a response variable $y$ differs if the same estimate (.64) was from $\textrm{Age}_j$ as a time-independent covariate?

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A time-varying covariate can be thought of as the average association between the time-varying variable and the outcome, averaged across all time points and individuals. It is often a good idea to person mean center time-varying variables by subtracting each person's mean from their time-specific values for the variable. Doing so ensures that the association you are examining is a pure within-person association (i.e., the mean outcome difference as an individual gains 1 unit on the predictor relative to their average value of the predictor). You can recover the pure between-person effect of the predictor by adding the person mean variable to your model. This coefficient tells you, on average and adjusting for other variables in your model, the difference in the outcome mean between persons who differ on the predictor by 1 unit.

That said, a variable such as age is kind of unique in longitudinal models. It is often used to quantify how much the outcome changes for each 1-unit increase in age. You don't typically person-mean center age, but you might center it around some meaningful value. For example, you could subtract from each person's time-varying value of age their age when they were first measured. This ensures that your intercept provides a valid 0-value for age and can be helpful for the random part of the model as well.

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