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As a statistician, I was given a task to fit a mixed effects model where the right-hand side independent variables include a time variable, a time-varying variable, and the interaction of the time and the time-varying variable. I do not quite understand the model specification and am not sure how to interpret the results. The mixed effects models I have come across usually only use baseline covariates (i.e., covariates at time = 0) if time is included in the model, and I know how to interpret the interaction of time and the baseline covariate.

Can anybody help me understand the model specification with the interaction of time and a time-varying variable?

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  • $\begingroup$ A time-varying variable ($X_t$) has (or may have) different values at different points in time, but it's effect may simply be a constant ($\beta_{X}$). A time varying effect ($\beta_{X,t}$) has (or may have) different estimated values at different points in time, whether or not its variable changes values over time or is constant over time. Both the variable and the effect can thus be modeled as time-varying. $\endgroup$
    – Alexis
    Commented Mar 28, 2022 at 23:52
  • $\begingroup$ Thank you Alexis! In my mixed effects model, the independent variables are t (the variable indicating time length, e.g., number of days), X (the time-varying variable), and t * X (the interaction of t with X). I guess my question is how to interpret the β for t * X, not the β for X. $\endgroup$
    – Lily
    Commented Mar 29, 2022 at 0:14
  • $\begingroup$ $\beta_{tX}$ is how much $\beta_{X}$ changes by at time $t$ (assuming the structure of your model for time). $\endgroup$
    – Alexis
    Commented Mar 29, 2022 at 21:00
  • $\begingroup$ Thank you Alexis. So if the treatment dose is time varying, then the coefficient on the interaction of treatment dose and time would tell us how the rates of change in outcome differ between the treatment groups? $\endgroup$
    – Lily
    Commented Apr 5, 2022 at 17:07

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This is similar to having time-varying covariates in proportional-hazards survival modeling. The inherent assumption is that the association of the independent variable with outcome at a given time only depends on the current value of that variable at that time. The past history of that variable doesn't matter.

The interaction with time just allows the association of that independent variable (whatever its current value) with outcome to change over time since the start of the study. That isn't fundamentally different from any interaction term in a regression, as @Alexis notes in comments.

Whether that type of model makes sense in terms of the subject matter is another question. You might want to discuss that with those gave you this task.

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  • $\begingroup$ Thank you EdM. If we have time-varying treatment dose, then the interaction of treatment dose and time would tell us for each treatment dose, how the association of the independent variable with outcome changes over time, correct? $\endgroup$
    – Lily
    Commented Apr 5, 2022 at 17:14
  • $\begingroup$ @Lily that's correct, based on the assumption that the outcome depends at any time only on the current value of dose. Whether that's a reasonable assumption depends on your and your colleagues' understanding of the subject matter. $\endgroup$
    – EdM
    Commented Apr 5, 2022 at 17:51

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