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: $$y_{it} = N(\mu, \sigma_y) \\ \mu = \mu_0 + \mu_i + \beta_1Time_{it} + \beta_2(Treat_i \times Time_t)$$ $$\mu_i \sim N(0,\sigma)$$


  • $y_{it}$: 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 $\beta_1$).


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.


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.