Specificity and Sensitivity

Definition

In medical diagnostic tests, the accuracy of said diagnostic tests is often assessed with two conditional probabilities:

1. Sensitivity: Probability the test is positive, given the subject actually has the disease
2. Specificity: Probability the test is negative, given the subject actually does not have the disease

Let

• $X$ : Actual state of the subject. $X=1$ is diseased, $X=2$ is healthy
• $Y$ : Outcome of test. $X=1$ is positive, $X=2$ is negative

Then

$$\begin{array}{rl}\text{sensitivity}& =P(Y=1|X=1)={\pi }_{1|1}=\frac{{\pi }_{11}}{{\pi }_{1+}}\\ \text{specificity}& =P(Y=2|X=2)={\pi }_{2|2}=\frac{{\pi }_{22}}{{\pi }_{2+}}\end{array}$$

The higher the sensitivity AND specificity, the better the diagnostic test.

Example

Consider the following diagram