The magnitude and direction of collider bias for binary variables

Abstract: Suppose we are interested in the effect of variable $X$ on variable $Y$. If $X$ and $Y$ both influence, or are associated with variables that influence, a common outcome, called a collider, then conditioning on the collider (or on a variable influenced by the collider -- its "child") induces a spurious association between $X$ and $Y$, which is known as collider bias. Characterizing the magnitude and direction of collider bias is crucial for understanding the implications of selection bias and for adjudicating decisions about whether to control for variables that are known to be associated with both exposure and outcome but could be either confounders or colliders. Considering a class of situations where all variables are binary, and where $X$ and $Y$ either are, or are respectively influenced by, two marginally independent causes of a collider, we derive collider bias that results from (i) conditioning on specific levels of, or (ii) linear regression adjustment for, the collider (or its child). We also derive simple conditions that determine the sign of such bias.