Properly accounting for quadratic time term in repeated measures I am trying to fit a repeated measures model (random intercept) of the form Score = Time + Treatment + Time * Treatment + (1|Subject). The Time * Treatment term is what I am interested in. I want to know if treatment affects how score measurements change over time. However, I believe that the relationship between score and time is not linear. I'd like to include a quadratic time term but am uncertain of how to properly formulate this in my model.
My thoughts are that I could include the quadratic time term as a main effect and leave the interaction as I have defined it, i.e. Score = Time + Time^2 + Treatment +  Time * Treatment + (1|Subject). However, I am wondering if I should instead  include the higher order time term, or both, in the interaction.
 A: A few thoughts:
a. Yes it is acceptable to fit a linear term for time's interaction with treatment with the quadratic adjustment for time as a main effect. The interaction term is nested within the space of main effects and is a proper interaction. The interpretation of that term is a bit puzzling though; the test is powered to detect whether the instantaneous rate of change at time 0 is statistically significantly different for treated vs untreated individuals, irrespective of the remaining course of treatment. 
I supported a paper dealing with a similar issue where we took the approach of: adjust for time using a very general approach (splines), adjust for exposure and its interaction with all main effects used to model time, then perform the significance test for the nested model with no effects of exposure. It was a 6 degree of freedom test, and we explained at great length why the confidence bounds and parameter estimates were not the right summaries to show the effects of exposure, and opted to clearly explain a null hypothesis and report the p-value. An accompanying graph was used to support the results. The research concerned body weight since birth, which has a paradoxical effect being protective at certain ages, and deleterious at other ages.
b. Suppose the effect in time were not linear, but were somewhat close to linear? You can assess this with splines. Suppose it is bathtub shaped, or logistic. What is the cost of modeling the time effect linearly when the line provides a somewhat reasonable approximation? The consensus seems to be: you don't lose much. While you probably lose a bit of power, you avoid the curse of dimensionality, and the inference is easily recognized and interpreted.
