I'm doing linear mixed model with an interaction between the time and my exposure. The fixed effects look like this :
$$Y \sim T + T\cdot \text{GRP} + X$$
with $T$ the time, $\rm GRP $ the exposure and $ X $ the adjustments factors.
I am wondering how to see the average effect of my exposure (categorical variable) on my outcome at all times. How can I say that there is an effect at any time (or at some time) of my expositio on the outcome?
By removing the interaction, what can I say?
Interactions between categorical variables and continuous variables behave a little differently from other types of interactions (say an interaction of a continuous variable with a continuous variable). In a linear model like the one you shared, categorical variables are divided into indicator variables where the number of indicator variables is equal to the number of categories minus one (we subtract one category to keep the model identifiable). In your case, the actual model would look something like this: $$ Y = T + T\cdot\text{GRP}_1 + T \cdot\text{GRP}_2 + \cdots + T\cdot\text{GRP}_{p-1} + X$$ where $p$ is number of categories in $\rm GRP$. $\text{GRP}_x$ equals 1 whenever an observation belongs to category $x$, and 0 otherwise. The coefficient for $T$ that is not included in the interaction is, in effect, the estimate of $T$ for the category that was left out. The coefficients for each of the interaction terms would be an estimate of the difference in the effect of $T$ for that category versus the left-out category. This might answer your second question, albeit indirectly, since you would really be testing whether there are differences between your exposures. But maybe that is the question of interest?
If you want to know the average effect of each exposure on your response, I think a model like this would do the trick: $$ Y = T + \text{GRP} + X $$ Without the interaction, you are simply testing the impact of exposure, regardless of the value of time, while controlling for time. I think this is equivalent to the average effect of your exposures (first question), and possibly speaks to your third question as well, though I'm not entirely sure.
Of course, if exposure is simply a categorical version of time, then everything I say can probably be disregarded. I'm assuming that exposure has to do with intensity and not with duration. Just thought I would double-check on that.
