Bayes Theorem

Bayes Theorem is defined as "a theorem in probability theory named for Thomas Bayes (1702-1761). In epidemiology, it is used to obtain the probability of disease in a group of people with some characteristic on the basis of the overall rate of that disease and of the likelihoods of that characteristic in healthy and diseased individuals. The most familiar application is in clinical decision analysis where it is used for estimating the probability of a particular diagnosis given the appearance of some symptoms or test result".

Calculations

Two-by-two table for a diagnostic test

		Disease
		Present	Absent
Test result	Positive	Cell A	Cell B	Total with a positive test
	Negative	Cell C	Cell D	Total with a negative test
		Total with disease	Total without disease

Sensitivity and specificity

The sensitivity and specificity of diagnostic tests are defined as "measures for assessing the results of diagnostic and screening tests. Sensitivity represents the proportion of truly diseased persons in a screened population who are identified as being diseased by the test. It is a measure of the probability of correctly diagnosing a condition. Specificity is the proportion of truly nondiseased persons who are so identified by the screening test. It is a measure of the probability of correctly identifying a nondiseased person. (From Last, Dictionary of Epidemiology, 2d ed)."

${\mbox{Sensitivity of a test}}=\left({\frac {\mbox{Total with a positive test}}{{\mbox{Total }}without{\mbox{ disease}}}}\right)=\left({\frac {\mbox{Cell A}}{{\mbox{Cell A}}+{\mbox{Cell C}}}}\right)$

${\mbox{Specificity of a test}}=\left({\frac {\mbox{Total with a negative test}}{{\mbox{Total }}without{\mbox{ disease}}}}\right)=\left({\frac {\mbox{Cell D}}{{\mbox{Cell B}}+{\mbox{Cell D}}}}\right)$

Predictive value of tests

The predictive values of diagnostic tests are defined as "in screening and diagnostic tests, the probability that a person with a positive test is a true positive (i.e., has the disease), is referred to as the predictive value of a positive test; whereas, the predictive value of a negative test is the probability that the person with a negative test does not have the disease. Predictive value is related to the sensitivity and specificity of the test."

${\mbox{Positive predictive value}}=\left({\frac {{\mbox{Total }}with{\mbox{ disease and a positive test}}}{\mbox{Total with a positive test}}}\right)=\left({\frac {\mbox{Cell A}}{{\mbox{Cell A}}+{\mbox{Cell B}}}}\right)$

${\mbox{Negative predictive value}}=\left({\frac {{\mbox{Total }}without{\mbox{ disease and a negative test}}}{\mbox{Total with a negative test}}}\right)=\left({\frac {\mbox{Cell D}}{{\mbox{Cell C}}+{\mbox{Cell D}}}}\right)$