  There are limited options to estimate the treatment effects of variables
which are continuous and measured at multiple time points, particularly if the
true dose-response curve should be estimated as closely as possible. However,
these situations may be of relevance: in pharmacology, one may be interested in
how outcomes of people living with -- and treated for -- HIV, such as viral
failure, would vary for time-varying interventions such as different drug
concentration trajectories. A challenge for doing causal inference with
continuous interventions is that the positivity assumption is typically
violated. To address positivity violations, we develop projection functions,
which reweigh and redefine the estimand of interest based on functions of the
conditional support for the respective interventions. With these functions, we
obtain the desired dose-response curve in areas of enough support, and
otherwise a meaningful estimand that does not require the positivity
assumption. We develop $g$-computation type plug-in estimators for this case.
Those are contrasted with g-computation estimators which are applied to
continuous interventions without specifically addressing positivity violations,
which we propose to be presented with diagnostics. The ideas are illustrated
with longitudinal data from HIV positive children treated with an
efavirenz-based regimen as part of the CHAPAS-3 trial, which enrolled children
$<13$ years in Zambia/Uganda. Simulations show in which situations a standard
g-computation approach is appropriate, and in which it leads to bias and how
the proposed weighted estimation approach then recovers the alternative
estimand of interest.
