# Bayesian calculation with compound independent factors

I have simple model for some medical test characteristics calculation.

• $T$=test
• $D$=Disease
• $D+$= Disease present
• $T+$= Test positive (yes\no)

$P(D+)$ is very small (0.000001).
The problem is I don't know what the probability of Test being positive in the general population. This is a very new test and we don't know the exact profile of this very expensive test yet.

Test result can be affected by many factors. Given this problem definition, how can I incorporate these many factors that may change the test result (cause a false positive or a false positive) into my general basic Bayesian model below?

My model is $P(T+|D+) = P(D+|T+)P(D+) / P(D+)$

Thank you

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First off, don't you want to compute the probability of the disease being present given a positive test result: P(D+|T+)? Either way, Bayes theorem states: P(A | B) = P(B | A) P(A) / P(B) (not what you have listed). Second, you haven't specified a problem. Could you clarify what exactly are you trying to do? –  Nick Oct 19 '11 at 19:39
thanks for the correction. No I don't want to calculate the probability of the disease being present given a positive test result but rather the probability of obtaining a positive test result given the prevalence of disease. My disease prevalence is very low but the test is very specific. –  biomed Oct 20 '11 at 4:09
@Nick has already pointed out that you have not stated Bayes theorem correctly in your model. More pointedly, since $P(D+)$ occurs in both the numerator and denominator of the expression on the right side of your displayed equation, your model simplifies to $$P(T+\mid D+) = P(D+\mid T+)$$ which is surely not what you mean! So what is your model? –  Dilip Sarwate Oct 20 '11 at 20:14