Non-random selection with and without bias due to selecting on an exposure

Non-random selection represents a potential threat to valid estimation of causal effects. However, non-random selection does not always lead to bias. Whether bias occurs depends on the causal structure and the estimand. Although settings when non-random selection is expected not to result in bias have been discussed in the epidemiologic literature, such settings are underexplored due to the emphasis on when bias will occur rather than when it will not. Identifying when selection bias will not occur can deepen understanding of selection bias, including how to prevent it; can promote more rigorous study design and data analysis; or can aid in responding to peer reviews. Thus, we use motivating examples involving selecting on an exposure level, causal diagrams, and simulations to illustrate scenarios where non-random selection is not expected to result in bias. SAS and R code are provided to facilitate reproducibility and more in-depth understanding.