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Imagine an essentially random, very large population of people (say 50 million individuals).

A given subset of them have a disease 'D'.

By inspecting this subset, we observe that 40% of the people in it are female (F), 90% have elevated blood pressure (B), and 70% have a history of neck trauma (N).

In the overall population, the prevalence of the above characteristics is: 50% F, 30% B, 15% N.

If $P(D)$ is the probability of having disease 'D' in the overall population, $P(X)$ is the probability of having a characteristic 'X' in the overall population, $P(D|X)$ is the probability of having disease 'D' within the subset of people that have characteristic 'X', and $P(X|D)$ is the probability of having characteristic 'X' within the subset of people that have disease 'D', I think Bayes' theorem can be rearranged to tell us that:

$\frac {P(D|X)} {P(D)} = \frac {P(X|D)} {P(X)}$

If so, in the above example we might conclude that:

$\frac {P(D|F)} {P(D)} = \frac {P(F|D)} {P(F)} = \frac {40%} {50%} = 0.8$

$\frac {P(D|B)} {P(D)} = \frac {P(B|D)} {P(B)} = \frac {90%} {30%} = 3$

$\frac {P(D|N)} {P(D)} = \frac {P(N|D)} {P(N)} = \frac {70%} {15%} = 4.67$

In other words, if you're a female you're slightly less likely than the general population to have the disease; if you have elevated blood pressure you're 3-fold as likely; if you have a history of neck trauma you are almost 5-fold as likely.

First question: is the above reasoning correct?

Second question: if so, is this just an observation applying to the specific population under study, telling us nothing about the probability of future diagnoses of 'D'? Or can we use this knowledge to make predictions, e.g. can we reasonably predict that a male with elevated blood pressure and history of neck trauma is more likely to be diagnosed with 'D' than a female with normal blood pressure and no history of neck trauma?

kjetil b halvorsen
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user6376297
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  • Check https://stats.stackexchange.com/questions/185817/interpretation-of-bayes-theorem-applied-to-positive-mammography-results – Tim Aug 18 '17 at 19:57
  • Thank you @Tim , I had already seen that post and I am familiar with the concept of PPV vs TPR. I don't really see how it answers my questions, though. In the post, the prior probability of the disease is always known, the probability of a positive test is calculated based on it, and further probabilities are calculated (e.g. the PPV). In my case, the question is whether we can make predictions about *odds* (i.e. ratios of probabilities). Are people with elevated blood pressure 3-fold as likely to have disease 'D', compared to the general population, even when they haven't yet been diagnosed? – user6376297 Aug 19 '17 at 09:48

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