Selection Bias with Outcome-dependent Sampling

In a seminal paper, Hernan et al. 2004 provided a systematic classification of selection biases, for scenarios where the selection is a collider between the exposure and the outcome. Hernan 2017 discussed another scenario, where the selection is statistically independent of the exposure, but associated with the outcome through common causes. In this note, we extend the discussion to scenarios where the selection is directly influenced by the outcome, but not by the exposure. We discuss whether these types of outcome-dependent selections preserve the sharp causal null hypothesis, and whether or not they allow for estimation of causal effects in the selected sample and/or in the source population.