Evidence Against Greenwood et al's Claims of Randomness

I discussed “Patient–physician gender concordance and increased mortality among female heart attack patients” by Brad N. Greenwood, Seth Carnahan, and Laura Huang (henceforth GCH) the other day. Now that their Supplementary Information pdf is available, it is easy to see that their claims about “quasirandom” assignment are nonsense.

To review, GCH claim that female physicians are better at treating female heart-attack victims than male physicians are. Their evidence is that female patients have higher survival rates when treated by female physicians. This would be a plausible argument if patients were assigned randomly to physicians. GCH claim:

In addition, when patients visit the emergency room (ER), they have little agency over their choice of attending physician, allowing for a quasirandom assignment of physician and patient.


Moreover, patients admitted through the ED often arrive via ambulance, and assignments of emergency admittances to attending physicians is usually done in a quasi-random format, where attending physicians who are least busy are assigned newly arriving patients. Thus, it may be possible treat the assignment of a patient to a physician as occurring quasi-randomly.

A plausible argument! But, instead of looking at simple ways of testing this assumption, GCH go down an obscure rabitt-hole. Let’s show them how it should be done.

Consider this summary data from their Supplementary Information (page 10):


Ignore the mean/sd columns. Notice that we have 519,495 total patients and that the male-to-female ratio is 1.43. And that seems reasonable. They have a huge dataset and it seems plausible that there are more male than female heart attack victims during this period.

Before I show you the data for female physicians, predict what it should look like if patient assignment is truly “quasirandom” (or “quasi-random”), if “attending physicians who are least busy are assigned newly arriving patients.” The ratio of male-to-female patients better be very close to 1.43 if the assignment mechanism is close to random. There are 62,350 total treatments of heart-attacks victims by female physicians in the data. If the male-to-female ratio for female physicians were like that for male physicians — as we would expect with quasirandom assignment — we would expect about 25,658 female-physician/female-patient observations. In reality:


First, note how sloppy this is. The first column should be labeled “Female Physician – Male Patient.”

More importantly, there are way too many female patients. The male-to-female patient ratio is 1.14, much lower than the 1.43 value for male physicians. If we treat the total number of patients treated by female physicians as fixed, then they treated about 13% too many women (25,658 expected versus 29,148 actual).

Given that female physicians treated around 3,500 “too many” women to be consistent with random assignment, can we come up with reasons why these 3,500 women might be different than the other women in the sample? Easily! What if women prefer female physicians but only women who are conscious when they come into the emergency room can express that preference? In that case, female physicians might get assigned thousands of more healthy women, women much more likely to survive their heart attack. Unconscious women are both more likely to die and more likely to be assigned randomly.

Summary: GHC fail to notice, much less discuss, the simplest summary data which demonstrates that, whatever else may be said about the assignment of patients to physicians, it is very, very far from random. No meaningful conclusions can be drawn — at least using their procedures — about the causal impact of physician gender on patient outcomes.

David Kane
Preceptor in Statistical Methods and Mathematics
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