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：碰撞偏倚发生于对两个变量$X$和$Y$的共同效应（碰撞变量）进行条件化时。本文量化了因对暴露变量与结局变量的二分碰撞变量$S$的某一取值进行选择，所导致的暴露变量$X$与结局变量$Y$之间估计关联中的碰撞偏倚。在逻辑回归情形下，已知暴露-结局回归系数中碰撞偏倚的大小，与碰撞变量在对数可加模型中$X$和$Y$的交互作用强度$\delta_3$成正比：$\mathbb{P} (S = 1 | X, Y) = \exp \left\{ \delta_0 + \delta_1 X + \delta_2 Y + \delta_3 X Y \right\}$。我们证明该结果在暴露-结局关联采用线性或泊松回归模型时同样成立。随后通过模拟说明：即使含交互作用的对数可加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偏倚的大小。最后，我们讨论这些发现对试图校正碰撞偏倚的方法（如通常未在权重模型中纳入变量间交互作用的逆概率加权法）的启示。