  Among recent developments in definitions and analysis of selection bias is
the potential outcomes approach of Kenah (Epidemiology, 2023), which allows
non-parametric analysis using single-world intervention graphs, linking
selection of study participants to identification of causal effects. Mohan &
Pearl (JASA, 2021) provide a framework for missing data via directed acyclic
graphs augmented with nodes indicating missingness for each sometimes-missing
variable, which allows for analysis of more general missing data problems but
cannot easily encode scenarios in which different groups of variables are
observed in specific subsamples. We give an alternative formulation of the
potential outcomes framework based on conditional separable effects and
indicators for selection into subsamples. This is practical for problems
between the single-sample scenarios considered by Kenah and the variable-wise
missingness considered by Mohan & Pearl. This simplifies identification
conditions and admits generalizations to scenarios with multiple, potentially
nested or overlapping study samples, as well as multiple or time-dependent
exposures. We give examples of identifiability arguments for case-cohort
studies, multiple or time-dependent exposures, and direct effects of selection.
