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Say I have a test with the following characteristics:

$P(B|A)$ = positive test in disease population = 0.8

$P(A)$ = incidence of disease = $\frac{1}{5,000}$

$P(B)$ = positive test in general population = 0.3

Thus I end up with the following probability of disease given a positive test: $P (A|B)=\frac{P(B|A) P(A)}{P(B)} =\frac{0.8*\frac{1}{5,000}}{0.3}$ = 0.0005

Now, this only reflects the probability of disease given a positive test, neglecting any other clinical signs.

How can I update this probability given clinical signs as well? In other words, is it possible to extend Bayes into $$P (A|(B,x1,...x_n)$$ somehow, and how can I provide a measure of the added information gained by the test (expensive)?

Say for instance that x1 is ubiquitous for the diagnosis, but that x2 occurs in 75% of patients with the disease. On the other hand, x1 and x2 have incidences of 0.1 and 0.01 in the general population.

Misha
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    Check http://stats.stackexchange.com/questions/185817/interpretation-of-bayes-theorem-applied-to-positive-mammography-results – Tim Nov 23 '16 at 21:36
  • Are you interested in $P(A|x_1,\ldots,x_n)$ or $P(A|B,x_1,\ldots,x_n)$? – Thoth Nov 24 '16 at 02:02
  • See edit. I'm interested in both, but my main interest is as per the edit. – Misha Nov 24 '16 at 11:20

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You started off on the wrong foot, making things much more complicated. Consider a prospective cohort study where there was no over-sampling of diseased cases. Don't even calculate sensitivity and specificity. These are backwards time-order probabilities that just get in the way of understanding. Directly compute probability of disease given patient characteristics. When there is only one patient characteristic (say male vs female) you compute e.g. the proportion of males with disease and the proportion of females with disease. When there are multiple binary characteristics you can simultaneously stratify on them all to get the right probability. But you can handle this and much more, including the use of continuous predictors, by just using a binary logistic model. I go into great detail about all this in Chapter 19 of Biostatistics for Biomedical Research available from http://biostat.mc.vanderbilt.edu/ClinStat

Use Bayes' theorem to turn around backwards probabilities when the setting made it extremely difficult to get the forwards probability at the outset, e.g., when doing a retrospective case-control study that oversampled diseased individuals.

Frank Harrell
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  • I understand. However, this is not part of a study. This is just because I am wondering about the usefulness of the test and all I have is just bits and pieces of the probabilities. The 0.05% probability of disease in the setting above is in the general population with a positive test. But what I am interested in is the probabilty of disease given a positive test in a person with for instance 2 out of 4 symptoms. – Misha Nov 24 '16 at 16:03
  • Did you read the chapter I described? – Frank Harrell Nov 24 '16 at 16:36
  • Yes - and I found your approach to assess the diagnostic yield of a test very helpful. However, not having individual level factual data, only assumptions and clinical intuition that I can quantify as a rough probability, I do not understand how any of the methods provided in your chapter can be employed. The equations in 19.4, would be what I´m looking for, but how can I get the beta-values when I only have rough idea of the probability - can I just substitute the logit of the probability with the beta? – Misha Nov 24 '16 at 18:53
  • I don't know how to get valid estimates without data except in a special case where all the patient characteristics are uncorrelated with each other. – Frank Harrell Nov 25 '16 at 13:14