Does abduction need an adjustment set? If so, what are the rules?
It is good to review, an adjustment set is a: “set of covariates such that adjustment, stratification, or selection (e.g. by restriction of matching) will minimize bias when estimating the causal effect of the exposure on the outcome (assuming that the causal assumptions encoded in the diagram hold).” (Drawing and Analyzing Causal DAGs with DAGitty, User Manual for Version 2.2, Johannes Textor)
Based on the post on deduction I am sure you can imagine that abduction also needs an adjustment set. After all, it is simply the use of the same type of causal inferences resulting in conditionals (If X, then Y) as in deduction. So at the very least the same adjustment set as induction is necessary for considering the premise in the abduction with the conditional. The most common “form” of abduction is the fallacy “affirming the consequent.”
If X, then Y
This is not an accepted form for deduction, nor is it valid. Y as an effect may be caused by another variable (another cause). So the abduction adjustment set must include all of the variables in the inductive adjustment set for the first premise (If X, then Y) and also any variable that can directly cause Y. Therefore the abduction and deduction adjustment sets are equal. It is important to point out that even with an adjustment set for abduction and with complete adjustment on the variables of that set, abduction still cannot be justified (valid and sound) as deduction can. Meaning, with abduction, we cannot say that if the premises are true the conclusion necessarily follows (is logically entailed) even when the adjustment set is fully adjusted.
Thinking about abduction adjustment sets has been fruitful and I expect more to come. Despite what I say above I do believe there are at least categories, or perhaps classifications, within abductive inference whereby we can be “near justified” if we have knowledge about the adjustment set. The most clear cut case would be when we have conditional premise whereby we know If X then Y and Y if and only if X. With such a conditional situation (Y if and only if X, abbreviated iff) I believe we can say we have justified abduction, we know it is true. Of course it raises the question of what it takes to conclude through an inductive process that Y iff X. Coming up with other classifications of abduction is another matter altogether but could be helpful and could make the process of identifying (logically) the clinical prediction rules for diagnostic or prognostic accuracy most likely to be empirically fruitful (i.e. highest likelihood ratios).