Mixture analysis (see "Research" on this site) was used to analyze the tuberculin skin test survey from Tumkur (see previous slide).
The blue **histogram** is the **observed distribution**. It is composed of at least three distributions:
- a distribution among
**persons without** any **mycobacterial infection** (dashed line), usually not exceeding 1 or 2 mm;
- a distribution among
**persons with tuberculous infection** (full line), with an expected close to normal distribution peaking at 16 to 17 mm;
- a distribution among
**persons with infection due to environmental mycobacteria** (dashed-dotted line), probably peaking at 4 to 8 mm.
The table shows the four possibilities that arise when a test (here the tuberculin skin test) with a simple categorical definition into "positive" and "negative" (defined by a cut-off point) is used to identify the presence or absence of a condition (here tuberculous infection).
The number with a **false negative** result is denoted as "**c**", the number with a **false positive** result as "**b**".
The **sensitivity** of the test is the proportion of persons **with tuberculous infection** correctly identified with a positive test: "**a/(a+c)**".
The **specificity** of the test is the proportion of persons **without tuberculous infection** correctly identified with a negative test: "**d/(b+d)**".
The **predictive value of a positive test** is the proportion among all with a positive result who actually have tuberculous infection: "**a/(a+b)**".
The **predictive value of a negative test** is the proportion among all with a negative result who do not have tuberculous infection: "**d/(c+d)**".
At the cut-off defined to denote a positive or a negative test in this example, a certain number of persons (**c**) has a negative test, but has actually tuberculous infection. That is the limitation in the **sensitivity** of the test.
At the cut-off defined to denote a positive or a negative test in this example, a certain number of persons (**b**) has a positive test, but has actually no tuberculous infection. That is the limitation in the **specificity** of the test.
As one moves the cut-off point **to the left**, **sensitivity increases**, and **specificity decreases**.
As one moves the cut-off point **to the right**, **sensitivity decreases**, and **specificity increases**.
The above are the issues one encounters with cut-off points to denote presence or absence of infection. In this example using mixture analysis, the probablity of infection with *M tuberculosis* could be determined with proper credibility intervals using a Bayesian approach to analysis (see "Research" on this site). |