  A fundamental challenge in observational causal inference is that assumptions
about unconfoundedness are not testable from data. Assessing sensitivity to
such assumptions is therefore important in practice. Unfortunately, some
existing sensitivity analysis approaches inadvertently impose restrictions that
are at odds with modern causal inference methods, which emphasize flexible
models for observed data. To address this issue, we propose a framework that
allows (1) flexible models for the observed data and (2) clean separation of
the identified and unidentified parts of the sensitivity model. Our framework
extends an approach from the missing data literature, known as Tukey's
factorization, to the causal inference setting. Under this factorization, we
can represent the distributions of unobserved potential outcomes in terms of
unidentified selection functions that posit an unidentified relationship
between the treatment assignment indicator and the observed potential outcomes.
The sensitivity parameters in this framework are easily interpreted, and we
provide heuristics for calibrating these parameters against observable
quantities. We demonstrate the flexibility of this approach in two examples,
where we estimate both average treatment effects and quantile treatment effects
using Bayesian nonparametric models for the observed data.
