In medical literature, the "anomaly that wouldn't go away" refers to a finding published in 1978 by a group of Welsh doctors (Cochrane, St Leger, and Moore). They had set out to examine the relationship between health services and mortality in the major developed countries, but in doing so they came across a correlation that surprised them — the more doctors there were per capita, the higher was the rate of infant mortality.
The correlation wasn't a weak one. In fact, for infant mortality it was the strongest correlation in their study. The number of doctors per capita seemed to have a stronger negative impact on infant mortality than did the level of cigarette or alcohol consumption in the population.
Obviously the researchers found the correlation unsettling since, ideally, more doctors should result in fewer, not more, infants dying.
So why would more doctors correlate with higher infant mortality? The three doctors did their best to figure this out:
As the above passage indicates, they didn't think it was plausible that doctors themselves were somehow responsible for the elevated infant mortality, but nor could they come up with a satisfactory explanation for the correlation. So they called it "the anomaly that wouldn't go away."
I'm not sure if the correlation still holds true. I believe it still did about twenty years ago. Unfortunately much of the relevant literature is locked behind paywalls.
Over the years there have been quite a few attempts to explain the anomaly. I've listed two below. Again, I'm not sure if one has been accepted as THE explanation. So the anomaly may still persist.
C Buck & V Bacsi, "The doctor anomaly," Journal of Epidemiology and Community Health, 1979, 33:307.
It occurred to us that some of the countries richly endowed with physicians may obtain their large supplies by having bigger medical schools, larger classes, and thus less individual instruction of the medical student. The consequence could be a poorer standard of medical practice, the influence of which would be evident in the mortality of the younger age groups where the outcome of disease is most susceptible to the physician's skill.
F.W. Young, "An explanation of the persistent doctor-mortality association," Journal of Epidemiology and Community Health, 2001, 55:80-84.
The explanation proposed here is that, as compared with other regions, the expectation of opportunities in the growing industrial cities initially attracts an over supply of doctors. Once in practice, doctors in new regions enjoy fewer economies of scale, which means that they are more numerous as compared with the mature regions. These same industrialising cities attract rural immigrants whose health habits and supports break down in the context of city life. Thus, the places with the most doctors also have the highest death rates, but the two variables are associated only by common location.
More info (pdf): Cochrane, Leger, & Moore, "Health service 'input' and mortality 'output' in developed countries."