Monday, 10 August 2009

The Answer: Bayesian probability

This post answers the question posed by the previous post, namely:

A patient, known to have condition "A", presents with a new complaint. The complaint isn't typically associated with condition "A". Is the patient more likely to have:
  • a rare manifestation of condition "A", or
  • a common manifestation of a second condition ("B")?

To make this example more concerete, let's take a patient who is known to have Cushing's Syndrome. She now complains of polyuria. This is a known but rather rare complication of Cushing's (only occuring in about 20% of cases). More to the point, however, only a tiny minority of polyuria cases are caused by Cushing's in the general population - most organic ones are due to diabetes mellitus. Without consulting a textbook or journal, do you think this patient's symptom is more likely to be due to her Cushing's, or due to diabetes?

Proving the answer requires us to use a form of probability calculation known as Bayesian probability (after Thomas Bayes, the British mathematician who founded the field). I'm no statistician (and I'd bet you aren't either!), so let's keep it simple. The fundamental path we will be following is this:
  1. Choose the end scenarios that we'll eventually be able to pick between. In our case, these are polyuria in this particular patient caused by either Cushing's syndrome or diabetes.
  2. Identify the starting probabilities of the two sets of conditions. In our case, we need to know the probability that the patient has Cushing's vs the probability that patient has diabetes.
  3. Identify the probabilty that each condition will lead the the outcome (i.e. the stroke). In other words, what is the probability that (a) a patient who has Cushing's will develop a polyuria vs (b) a patient who has diabetes will develop a polyuria).
  4. Thus we will have two numbers for each scenario (a probability of having a condition and a probability for that condition causing a stroke). Multiplying the two numbers together will give the probability that the particular scenario did actually occur. We can compare the figures for the two scenarios and see which is the more likely.

Does that make sense? Let's collect the required figures.
  • The probability that the average member of the population has Cushing's syndrome (we'll exclude those cases arising from prescribed steroid medication) is very small - about 13 per million. However, we know that this patient has the disease. Therefore the probability that this patient has Cushing's syndrome is 100% (i.e. 1.0).
  • We've already seen that the probability of a patient with Cushing's getting polyuria is approximately 20%, or 0.20.
  • The probability that the average member of the population, over 20 years of age, has diabetes is about 8.5% (i.e. 0.085).
  • The probability that a diabetic will have polyuria is roughly 90% (0.90).

OK, now we're ready to plug the numbers into the equation. In each case, we're multiplying the probability that the patient has the relevant underlying condition by the probability that that underlying condition is causing polyuria. We'll round off to two significant figures.

The probability that the patient's polyuria is caused by her Cushing's syndrome is 1 × 0.2 = 0.2
The probability that the patient's polyuria is caused by diabetes is 0.085 × 0.9 = 0.077

Therefore the patient's polyuria is roughly 2.5 times more likely to be due her Cushing's than due to diabetes.

Now at this point you may be asking why we bothered to plug in all these numbers, since we almost never have them at hand when we need them. The point, however, was to prove a general principle, namely that a symptom is almost always more likely to be due to a manifestation of a disease that the patient has, rather than a second disease that the patient might have. This holds true even if the symptom would be a very rare complication of the known disease, but a common manifestation of another disease, as in the above case.

Another even more convincing example can be taken from the Oxford Handbook of Clinical Medicine (6th edn., Longmore, Wilkinson, Rajagopalan):

A 50-yr-old man with known carcinoma of the lung has some transient neurological symptoms and a normal [brain] CT scan. Are these symptoms due to secondaries in the brain or to transient ischaemic attacks (TIAs)?
  • The chance of secondaries in the brain which cause transient neurological symptoms is 0.045 given carcinoma of the lung.
  • The chance of such secondaries not showing up on CT scan is 0.1. ∴ the chance of this cluster of symptoms is 0.0045 (ie 0.045 × 0.1).
  • The chance of a normal CT + transient CNS symptoms given a TIA is 0.9.
  • The chance of a 50-yr-od man developing a TIA is 0.0001. ∴ the odds ratio is 0.0045/(0.9 × 0.0001). This equals 50.
That is, the odds ratio is ~50 to 1 in favour of secondaries in the brain.

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