Question

The sensitivity and specificity of a diagnostic test in health care are defined as: • Sensitivity = probability that the diagnostic test result is positive IF the patient has the disease; • Specificity = probability that the diagnostic test result is negative IF the patient does not have the disease. Suppose that two tests for the disease TB are applied as follows. Test A is applied to the full population, and anyone found positive according to test A is treated. Everyone else is given test B, and those positive according to test B are also treated, whilst those negative on both tests are not treated. The sensitivities and specificities for the two tests when applied in this way are as follows:

	Sensitivity	Specificity
Test A	0.6	0.95
Test B	0.7	0.9

Suppose that the population of a particular region in Tibet is 10,000 and that 30% of people have TB. (i) How many people will be treated for TB after just Test A has been applied, and how many of them will have TB?

(ii) How many more people will be treated for TB after Test B has been applied, and how many of them will have TB?

Answer 1

Population in the given region is 10000 and 30% of them have TB

So , prevalence = 0.3

(i)

Sensitivity of test A is 0.6

Specificity of test A is 0.95

No. of people treated for TB just after test A = 10000*[sensitivity*prevalence+(1-specificity)*(1-prevalence)]

= 10000*[0.6*0.3+0.05*0.7]

= 2150

Number of people among them that actually have disease =

2150*sensitivity*prevalence/[sensitivity*prevalence+(1-specificity)*(1-prevalence)]

=2150*0.6*0.3/0.215

=1800

(ii)

Sensitivity of test B is 0.7

Specificity of test B is 0.9

No. of people treated for TB just after test B = 10000*[sensitivity*prevalence+(1-specificity)*(1-prevalence)]

= 10000*[0.7*0.3+0.1*0.7]

= 2800

Number of people among them that actually have disease =

2800*sensitivity*prevalence/[sensitivity*prevalence+(1-specificity)*(1-prevalence)]

=2800*0.7*0.3/0.28

=2100