Bayes's Theorem

post-test probability disease present given a positive test result =

= ((pretest probability that disease present) * (probability test positive if disease present)) / (((pretest probability that disease present) * (probability test positive if disease present)) + ((pretest probability that disease absent) * (probability test positive if disease absent)))

post-test probability disease present given a negative test result =

= ((pretest probability that disease present) * (probability test negative if disease present)) / (((pretest probability that disease absent) * (probability disease absent when test negative)) + ((pretest probability that disease present) * (probability test negative if disease present)))

Restated in terms of sensitivity (S) and specificity (E):

post-test probability disease present given a positive test result =

= ((pretest) * S) / (((pretest) * S) + ((1 – E) * (1 – (pretest))))

post-test probability disease present given a negative test result =

= ((pretest) * (1 – S)) / ((E * (1 – (pretest))) + ((pretest) * ( 1 – S)))

Limitations of Bayes's theorem:

• Bayes's theorem assumes test independence, which may not occur if multiple tests are used for diagnosis.

Relation to positive and negative predictive values (PPV and NPV):

• If the prevalence of disease in the study used to derive S and E is the same as in the study population [prevalence in the population = (a + c) / (a + b + c + d)], then the PPV and NPV can be used to express Baye's probabilities.

• The PPV describes the post-test probability given a positive result.

• (1 - NPV) describes the post-test probability given a negative result; NPV describes the probability that a negative result indicates a true negative result.