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)))