Last year we discussed an important challenge in causal inference: The standard advice (given in many books, including ours) for causal inference is to control for relevant pre-treatment variables as much as possible. But, as Judea Pearl has pointed out, instruments (as in “instrumental variables”) are pre-treatment variables that we would not want to “control for” in a matching or regression sense.
At first, this seems like a minor modification, with the new recommendation being to apply instrumental variables estimation using all pre-treatment instruments, and to control for all other pre-treatment variables. But that can’t really work as general advice. What about weak instruments or covariates that have some instrumental aspects?
I asked Paul Rosenbaum for his thoughts on the matter, and he wrote the following:
In section 18.2 of Design of Observational Studies (DOS), I [Rosenbaum] discuss “seemingly innocuous confounding” defined to be a covariate that predicts a substantial fraction of the variation in treatment assignment but without obvious importance to the outcomes under study.
The word “seemingly” is important: it may not be innocuous, but only seem so. The example is drawn from a study (Silber, et al. 2009, Health Services Research 44: 444-463) of the timing of the discharge of premature babies from neonatal intensive care units (NICUs). Although all babies must reach a certain level of functional maturity before discharge, there is variation in discharge time beyond this, and we were interested in whether extra days in the NICU were of benefit to the babies who received them. (The extra days are very costly.) It is a long story, but one small part of the story concerns two “seemingly innocuous covariates,” namely the day of the week on which a baby achieves functional maturity and the specific hospital in the Kaiser family of hospitals. A baby who achieves maturity on a Thursday goes home on Friday, but a baby who achieves maturity on Saturday goes home on Tuesday, more or less. It would, of course, be ideal if the date of discharge were determined by something totally irrelevant, but is it true that day-of-the-week is something totally irrelevant?
Should you adjust for the day of the week? A neonatologist argued that day of the week is not innocuous: a doc will keep a baby over the weekend if the doc is worried about the baby, but will discharge promptly if not worried, and the doc has information not in the medical record. Should you adjust for the day of the week? Much of the variation in discharge time varied between hospitals in the same chain of hospitals, although the patient populations were similar. Perhaps each hospital’s NICU has its own culture. Should you adjust for the hospital?
The answer I suggest in section 18.2 of Design of Observational Studies is literally yes-and-no. We did analyses both ways, showing that the substantive conclusions were similar, so whether or not you think day-of-the-week and hospital are innocuous, you still conclude that extra days in the NICU are without benefit (see also Rosenbaum and Silber 2009, JASA, 104:501-511). Section 18.2 of DOS discusses two techniques, (i) an analytical adjustment for matched pairs that did not match for an observed covariate and (ii) tapered matching which does and does not match for the covariate. Detailed references and discussion are in DOS.