How to assess the validity of random effects assumptions in models with interaction and time covariates? Suppose you had 2 waves of longitudinal data. The univariate outcome $y$ was measured at baseline ($Time=0$) for $N$ units, each unit randomly received a treatment ($Treat=1$) or control ($Treat=0$), then y was measured post-treatment ($Time=1$).
We can estimating the effect of the treatment while using baseline information with a hierarchical model with "random intercepts" for each unit (sometimes called "multilevel ANCOVA") as follows:
$$y_{it} = N(\mu, \sigma_y) \\
\mu = \mu_0 + \mu_i + \beta_1Time_{it} + \beta_2(Treat_i \times Time_t)$$
$$\mu_i \sim N(0,\sigma)$$
Where:

*

*$y_{it}$: the (univariate) outcome of unit $i$ at Time $t$

*$\mu_0$: "grand baseline mean" outcome across all observations

*$\mu_i$: "random intercepts" capturing time-invariant unit-specific deviations from the grand mean

*$\beta_1$: mean control group ($Treat=0$) change in $y$ between $t=1$ and $t=2$

*$\beta_2$: Average treatment effect. i.e., the mean treated group ($Treat=1$) deviation from the control group's change (i.e. deviation from $\beta_1$).

Question(s)
My understanding is that one of the assumptions of this model is that the individual-specific errors ($\mu_i$) are not correlated with the other covariates (Time, Treat*Time). However, I'm struggling to understand how to substantively interpret the assumption to assess its validity because one of the covariates is an interaction and "time" is fairly abstract as a covariate.

*

*Is this assumption the same as saying unobserved, time-invariant between-unit differences affecting $y$ cannot vary with (a) time or (b) magnitude of treatment effects? (a) is ensured since the intercepts are time-invariant, but I'm quite uncertain about (b).


*Is it possible for $Corr(Time, \mu_i) \neq 0$? What does it mean if they are correlated - that units with higher/lower intercepts have systematically different changes from $t=0$ to $t=1$?


*Is it possible for $Corr(Time*Treat, \mu_i) \neq 0$? Treat is randomly assigned, so it is uncorrelated with all time-invariant features, but I have no idea how to think about the interaction.

This assumption would be violated when the standard deviation of the random intercepts $\sigma$ depends on covariates. For example, if the variance of $\mu_i$ depends on the treatment,
$$\left \{
\begin{array}{l}
\mu_i \sim N(0, \sigma_0), \quad \mbox{if Treat = 0},\\\\
\mu_i \sim N(0, \sigma_1), \quad \mbox{if Treat = 1}.
\end{array}
\right.$$
Something similar would hold if the variance of $\mu_i$ is associated with time.
