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). |
