# 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.

1. 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).

2. 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 $$t=1$$?

3. 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.