*Bounty: 100*

*Bounty: 100*

In a longitudinal study, outcomes $Y_{it}$ of units $i$ are repeatedly measuret at time points $t$ with a total of $m$ fixed measurement occasions (fixed = measurements on units are taken at the same time).

Units are randomly assigned either to a treatment, $G=1$, or to a control group, $G=0$. I want to estimate and test the average effect of treatment, i.e. $$ATE=E(Y | G=1) – E(Y | G=0),$$ where expectations are taken across time and individuals. I consider using a fixed-occasion multilevel (mixed-effects) model for this purpose:

$$Y_{it} = alpha + beta G_i + u_{0i} + e_{it}$$

with $alpha$ the intercept, $beta$ the $ATE$, $u$ a random intercept across units, and $e$ the residual.

Now I am considering alternative model

$$Y_{it} = tilde{beta} G_i + sum_{j=1}^m kappa_j d_{ij} + sum_{j=1}^m gamma_j d_{ij} G_i + tilde{u}*{0i} + tilde{e}*{it}$$

which contains the fixed effects $kappa_j$ for each occasion $t$ where dummy $d_t=1$ if $j=t$ and $0$ else. In addition this model contains an interaction between treatment and time with parameters $gamma$. So this model takes into account that the effect of $G$ may differ across time. This is informative in itself, but I believe that it should also increase precision of estimation of the parameters, because the heterogeneity in $Y$ is taken into account.

However, in this model the $tilde{beta}$ coefficient does not seem to equal the $ATE$ anymore. Instead it represents the ATE at the first occasion ($t=1$). So the estimate of $tilde{beta}$ may be more efficient than $beta$ but it does not represent the $ATE$ anymore.

**My questions are**:

- What is the best way to estimate the treatment effect in this longitudinal study design?
- Do I have to use model 1 or is there a way to use (perhaps more efficient) model 2?
- Is there a way to have $tilde{beta}$ have the interpretation of the $ATE$ and $gamma$ the occasion specific deviation (e.g. using effect coding)?