A diagnotic implication of Bayes Rule

A doctor diagnoses illness based on symptoms. Bayes rule is the statistical basis of that kind of diagnosis. It states that the probability of a disease given a symptom is equal to the a priori probability of the disease in the population times a ratio. The ratio is the probability of the symptom given the disease versus the general probability of the symptom in the population.

I remember an older physician in Cali, Colombia who remembered that early in his career he would diagnose malaria simply on the basis of a patient presenting with fever. Prior to effective anti-malarial campaigns malaria was so common in the city that even a symptom common to many diseases sufficed to allow a diagnosis of malaria that would very probably be right. (Of course, he may also have had so much experience with the disease that he unconsciously recognized other symptoms of the condition, and lacking any of those might have gone more into detail in his diagnostic procedure.)

In a contrasting example, there was a case described in a recent newspaper article of a physician who suffered from scabies (a skin problem caused by a microscopic mite) who suffered for a year and was misdiagnosed not only by himself but also be a couple of dermatologists. While scabies was once common, the incidence is now very low in the United States. The symptoms would have had to be very strongly indicative of scabies to justify that diagnosis, and many skin problems like itching and rashes are often not specific to a single disease. Once a dermatologist thought to check for mites with a microscope the diagnosis was made quickly and accurately, but that test was not made quickly because it was so unlikely to prove informative for the average presenting dermatological patient.

Of course, it is also the case that physicians, even dermatologists, don't see many cases of scabies and thus may not be that familiar with their symptoms. In the past, when scabies was common, the case would almost certainly have been quickly diagnosed and successfully treated. Indeed, the treatment was quickly efficacious once the correct diagnosis was made. Still, the underlying epidemiological situation made the mis-diagnosis not unreasonable.

I remember being surprised when I first learned how small a portion of patients presenting with acute problems for primary care actually received a specific diagnosis of their disease. In retrospect that situation is understandable. First, as a physician friend used to say often, the right prescription for simple health problems is often "the tender elixir of time". Bodies often heal themselves, and doing nothing does not bring side effects. Moreover, placebos work, in that many patients receiving treatments that should not be efficacious seem to cure themselves more rapidly simple due to their belief in the cure. In addition, many efficacious treatments are broad spectrum; many bacterial infections may be cured by the same antibiotic.

The physician is faced with a huge number of possible presenting diseases, each with a variety of possible presenting symptoms. Bayes theorem provides a basis for intuitive understanding as to why they occasionally make the mistake of missing a rare disease, even one with quite specific symptoms. It also helps to intuit why they are so often effective in treating the common conditions that come before them. While the physician does not actually calculate the probabilities using Bayes theorem, even now that he is using a computer with each patient, it is likely that his intuitive grasp of the likelihood of diseases implements Bayes approach.

Can we guard against the kind of diagnostic failure that made the physicians life a misery for a year? Perhaps it is useful to stop and consider whether we are jumping to conclusions to fast, and where the consequences justify that approach add additional diagnostic steps to check for unlikely causes.