Relationship between collider bias and interactions on the log-additive scale

Collider bias occurs when conditioning on a common effect (collider) of two variables <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi></mml:math>. In this article, we quantify the collider bias in the estimated association between exposure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> and outcome <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> induced by selecting on one value of a binary collider <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>δ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math> between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> in a log-additive model for the collider: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mo>(</mml:mo><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>exp</mml:mi><mml:mspace></mml:mspace><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>X</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>Y</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math>. We show that this result also holds under a linear or Poisson regression model for the exposure-outcome association. We then illustrate numerically 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.