  In this chapter, we review the class of causal effects based on incremental
propensity scores interventions proposed by Kennedy [2019]. The aim of
incremental propensity score interventions is to estimate the effect of
increasing or decreasing subjects' odds of receiving treatment; this differs
from the average treatment effect, where the aim is to estimate the effect of
everyone deterministically receiving versus not receiving treatment. We first
present incremental causal effects for the case when there is a single binary
treatment, such that it can be compared to average treatment effects and thus
shed light on key concepts. In particular, a benefit of incremental effects is
that positivity - a common assumption in causal inference - is not needed to
identify causal effects. Then we discuss the more general case where treatment
is measured at multiple time points, where positivity is more likely to be
violated and thus incremental effects can be especially useful. Throughout, we
motivate incremental effects with real-world applications, present
nonparametric estimators for these effects, and discuss their efficiency
properties, while also briefly reviewing the role of influence functions in
functional estimation. Finally, we show how to interpret and analyze results
using these estimators in practice, and discuss extensions and future
directions.
