Relationship between Collider Bias and Interactions on the Log-Additive Scale

TL;DR

This paper investigates how collider bias in statistical models relates to interactions on the log-additive scale, showing that the interaction term indicates the bias magnitude across different regression models and discussing implications for bias adjustment methods.

Contribution

It extends the understanding of collider bias by demonstrating its proportionality to interaction terms in various regression models, including linear and Poisson, beyond logistic regression.

Findings

Collider bias magnitude is proportional to interaction strength in log-additive models.

Interaction terms remain informative about bias even if the model is misspecified.

Implications for bias correction methods like inverse probability weighting are discussed.

Abstract

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.