# #StackBounty: What is the best way to estimate the average treatment effect in a longitudinal study?

### 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)?

Get this bounty!!!

This site uses Akismet to reduce spam. Learn how your comment data is processed.