Nonetheless, a scale I think should be more often considered because it may sometimes actually have causal meaning, but one not mentioned by VanderWeele and Robins, is the log-complement scale. This is the scale that attempts to capture probabilistic independence among causal processes. Toxicologists developed the concept of “simple independent action” for toxic effects of 2 different chemicals administered simultaneously.6 The idea, which has also been discussed in the context of epidemiology,7 is simple. Suppose that exposure A can cause D and exposure B can cause D by a truly independent pathway, while D can also occur in the absence of either A and B through an independent background cause. The paradigm would be 2 hunters who are independently aiming at the same duck. Of course, the duck could also drop dead from some unrelated and independent background cause, eg, lightning. If the 3 pathways are probabilistically independent, we have:
Hence we have that:
Under probabilistically independent effects then, it follows that there is additivity on the log complement scale:
Thus there is additivity of the effect of A and the effect of B, on the log-complement scale. For a rare outcome, the analogous additive model will approximately hold on the absolute risk scale. However, when the outcome is not rare, additivity of risks is not equivalent to statistical independence (there being a non-negligible chance that both hunters will hit the duck), so is not equivalent to additivity on the log complement scale.
When we study more than one individual, variation across individuals in susceptibility (certain ducks may be easier for the independently shooting hunters to see) can cause violations of probabilistic independence, even if stochastic independence of causes holds for each individual at risk, so even this (to me) appealing formulation does not necessarily permit a biologically meaningful inference.7 Moreover, one must also think differently and in a more complicated way about protective factors, such as vaccines. Nonetheless, although the field may have grown weary of this debate, issues of scale will continue to matter when we try to draw useful inference from models for joint effects.
DAGs have met with a mixed reaction in epidemiology, with some of my colleagues recognizing their importance for analysis of etiologic factors, and others preferring to think about confounding in more classic terms. I have long been one of the boosters.
One practical challenge with DAGs in my experience, however, is the contentiousness of choosing a DAG. These choices can be especially problematic in reproductive epidemiology, where time-related factors become important. For example: Is long interpregnancy interval a cause of change of partner, or is change of partner a cause of long interpregnancy interval? But the hard thinking and very careful consideration of the etiologic context that are needed to decide which DAG is epidemiologically most plausible can be extremely useful, as we try to break through our academically enforced reluctance to think directly about causes.
A different, longstanding problem with DAGs has been that when synergistic effects may be involved in the etiology of the disease, DAGs are annoyingly noncommittal. I'm afraid the paper by VanderWeele and Robins,3 although providing useful categorizations for effect modifiers, has not rescued DAGs from this limitation. There may be more future in rethinking the basics of how we draw the DAGs.
I thank Richard MacLehose and Anne Marie Jukic for their critical reading of the manuscript.
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