Collider bias occurs when conditioning on a common effect (collider) of two variables $X, Y$. In this manuscript, we quantify the collider bias in the estimated association between exposure $X$ and outcome $Y$ induced by selecting on one value of a binary collider $S$ of the exposure and the outcome. In the case of logistic regression, it is known that the magnitude of the collider bias in the exposure-outcome regression coefficient is proportional to the strength of interaction $\delta_3$ between $X$ and $Y$ in a log-additive model for the collider: $\mathbb{P} (S = 1 | X, Y) = \exp \left\{ \delta_0 + \delta_1 X + \delta_2 Y + \delta_3 X Y \right\}$. We show that this result also holds under a linear or Poisson regression model for the exposure-outcome association. We then illustrate by simulation that even if a log-additive model with interactions is not the true model for the collider, the interaction term in such a model is still informative about the magnitude of collider bias. Finally, we discuss the implications of these findings for methods that attempt to adjust for collider bias, such as inverse probability weighting which is often implemented without including interactions between variables in the weighting model.

翻译：碰撞选择偏倚发生于对两个变量 $X, Y$ 的共同效应（碰撞变量）进行条件化时。本文量化了因选择暴露变量 $X$ 与结局变量 $Y$ 的二分碰撞变量 $S$ 的某个值而导致的暴露-结局关联估计中的碰撞选择偏倚。在逻辑回归中，已知暴露-结局回归系数的碰撞选择偏倚大小与碰撞变量对数加性模型中 $X$ 和 $Y$ 的交互作用强度 $\delta_3$ 成正比：$\mathbb{P} (S = 1 | X, Y) = \exp \left\{ \delta_0 + \delta_1 X + \delta_2 Y + \delta_3 X Y \right\}$。我们证明该结论在线性或泊松回归模型下同样成立。随后通过模拟说明，即使含交互项的对数加性模型并非碰撞变量的真实模型，该模型中的交互项仍能有效反映碰撞选择偏倚的强度。最后，我们讨论了这些发现对尝试校正碰撞选择偏倚方法的启示——例如常忽略变量间交互项的逆概率加权法。