  Most causal inference methods consider counterfactual variables under
interventions that set the treatment deterministically. With continuous or
multi-valued treatments or exposures, such counterfactuals may be of little
practical interest because no feasible intervention can be implemented that
would bring them about. Furthermore, violations to the positivity assumption,
necessary for identification, are exacerbated with continuous and multi-valued
treatments and deterministic interventions. In this paper we propose
longitudinal modified treatment policies (LMTPs) as a non-parametric
alternative. LMTPs can be designed to guarantee positivity, and yield effects
of immediate practical relevance with an interpretation that is familiar to
regular users of linear regression adjustment. We study the identification of
the LMTP parameter, study properties of the statistical estimand such as the
efficient influence function, and propose four different estimators. Two of our
estimators are efficient, and one is sequentially doubly robust in the sense
that it is consistent if, for each time point, either an outcome regression or
a treatment mechanism is consistently estimated. We perform a simulation study
to illustrate the properties of the estimators, and present the results of our
motivating study on hypoxemia and mortality in Intensive Care Unit (ICU)
patients. Software implementing our methods is provided in the form of the open
source \texttt{R} package \texttt{lmtp} freely available on GitHub
(\url{https://github.com/nt-williams/lmtp}).
