One form of Bayes's theorem is to calculate the post-test odds for a disorder from the pre-test odds and performance characteristics for the test.
odds ratio =
= (probability of event) / (1 - (probability of event))
likelihood ratio =
= (probability of a test result in a person with the disease) / (probability of a test result in a person without the disease)
post-test odds =
= (pre-test odds) * (likelihood ratio)
where:
• The disease prevalence in the population can be used as the pretest odds.
• Likelihood ratios can be expressed in terms of the sensitivity and specificity of the test for the diagnosis.
• The positive likelihood ratio is the likelihood ratio for a positive test result; it is the true-positive rate divided by the false-positive rate, or (sensitivity) / (1 - (specificity))
• The negative likelihood ratio is the likelihood ratio for a negative test result; it is the false negative rate divided by the true negative rate, or (1 - (sensitivity)) / (specificity)
post-test odds that the person has the disease if there is a positive test result =
= (pre-test odds) * (positive likelihood ratio)
post-test odds that the person has the disease if there is a negative test result =
= (pre-test odds) * (negative likelihood ratio)
Calculating Post-Test Odds
Step 1: Calculate the positive and negative likelihood ratios for the test
positive likelihood ratio =
= (sensitivity) / (1 - (specificity))
negative likelihood ratio =
= (1 - (sensitivity)) / (specificity)
Step 2: Convert the prior probability to prior odds:
((prior probability) * 10) : ((1 - (prior probability)) * 10)
Step 3: Multiply the prior odds by the likelihood ratios to obtain the post-test odds
((positive likelihood ratio) * (prior probability) * 10) : ((1 - (prior probability)) * 10)
((negative likelihood ratio) * (prior probability) * 10) : ((1 - (prior probability)) * 10)
Step 4: Convert the post-test odds to post-test probabilities
positive post-test probability =
= ((positive likelihood ratio) * (prior probability) * 10) / (((positive likelihood ratio) * (prior probability) * 10) + ((1 - (prior probability)) * 10)) =
= ((positive likelihood ratio) * (prior probability)) / (((positive likelihood ratio) * (prior probability)) + (1 - (prior probability)))
negative post-test probability =
= ((negative likelihood ratio) * (prior probability) * 10) / (((negative likelihood ratio) * (prior probability) * 10) + ((1 - (prior probability)) * 10)) =
= ((negative likelihood ratio) * (prior probability)) / (((negative likelihood ratio) * (prior probability)) + (1 - (prior probability)))
If you expand the likelihood ratios in terms of sensitivity (S) and specificity (E), you obtain the expression for Baye's theorem given in the previous section.
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)))
