  It has recently become popular to define treatment effects for subsets of the
target population characterized by variables not observable at the time a
treatment decision is made. Characterizing and estimating such treatment
effects is tricky; the most popular but naive approach inappropriately adjusts
for variables affected by treatment and so is biased. We consider several
appropriate ways to formalize the effects: principal stratification,
stratification on a single potential auxiliary variable, stratification on an
observed auxiliary variable and stratification on expected levels of auxiliary
variables. We then outline identifying assumptions for each type of estimand.
We evaluate the utility of these estimands and estimation procedures for
decision making and understanding causal processes, contrasting them with the
concepts of direct and indirect effects. We motivate our development with
examples from nephrology and cancer screening, and use simulated data and real
data on cancer screening to illustrate the estimation methods.
