Ding and Miratrix [1] recently concluded that adjustment on a pre-treatment covariate is almost always preferable to reduce bias. I extend the examined parameter space of the models considered by Ding and Miratrix, and consider slight extensions of their models as well. Similar to the conclusion by Pearl [7], I identify constellations in which bias due to adjustment, or failing to adjust is symmetrical, but also confirm some findings of Ding and Miratrix.Keywords: M-bias, causal inference, butterfly bias Ding and Miratrix [1], henceforth DM, recently examined bias in graphical causal models. Their main interest was to explore whether it is beneficial to adjust on a variable that may have both confounder properties (being a common cause of two variables), or collider properties (being a common effect of two variables). This is an important question, as e.g., evidenced by the debate of Rubin [2], and Pearl [3,4], and I applaud the authors to tackle it, especially using the methodology expressed in Pearl [5], and Chen and Pearl [6].The main conclusion of DM was that typically adjustment is preferable, even if a variable may have bias-inducing properties. The authors justified this conclusion based on the fact that in a majority of their conditions for which they derived asymptotic bias, adjustment yielded smaller biases. DM however did note that some of their analyses were incomplete, because certain path coefficients were always fixed to be of equal magnitude, and importantly of equal sign. They explicitly left exploration of a broader parameter space (including negative correlations) to future studies.In a comment, Pearl [7] rebutted the claim of the authors, and argued that in the case of the M-bias structure with correlated causes, bias due to adjustment should be comparable in size to bias due to failing to adjust for a confounder. In the case of so-called butterfly bias, in which both confounding and M-bias are present, Pearl [7] also conjectured that bias due to adjustment and lack of adjustment would be largely symmetrical, an argument based on the equal prevalence of positive and negative correlations of unobserved causes in the M-bias structure.This comment extends the work of DM, by examining the same models, or slight extensions thereof, but also varying involved path coefficients over a much larger, and arguably more complete, parameter space.
Asymptotic biasesI used the same methodology as DM to derive the asymptotic bias in the model depicted in Figure 1.1 Note that all models of DM are included in this model as special cases. If path coefficients e and f are set to 0,