EX 2.31 Incidence of a rare disease

• Only 1 in 1000 adults is afflicted with a rare disease for which a diagnostic test has been developed.

• The test is such that when an individual actually has the disease,

  • a positive result will occur $99\%$ of the time,
  • whereas an individual without the disease will show a positive test result only $2\%$ of the time
    • the sensitivity of this test is $99\%$
    • and the specificity is $98\%$ ;
    • in contrast, the Sept. 22, 2012 issue of The Lancet reports that the first at-home HIV test has a sensitivity of only $92\%$ and a specificity of $99.98\%$

• If a randomly selected individual is tested and the result is positive,

  • what is the probability that the individual has the disease?

• To use Bayes’ theorem, let

  • $A_1=$ individual has the disease,
  • $A_2=$ individual does not have the disease,
  • $B=$ positive test result.

• Then

  • $P\left(A_1\right)=.001$ ,
  • $P\left(A_2\right)=.999$,
  • $P\left(B\mid A_1\right)=.99$ ,
  • $P\left(B\mid A_2\right)=.02$ .

• The tree diagram for this problem is in Figure 2.12.

Figure 2.12 Tree diagram for the rare-disease problem

• Next to each branch corresponding to a positive test result,
  • the multiplication rule yields the recorded probabilities.
• Therefore, $P\left(B\right)=.00099+.01998=.02097$ ,
  • from which we have

$$P\left(A_1\mid B\right)=\frac{P\left(A_1\cap B\right)}{P\left(B\right)}=\frac{.00099}{.02097}=.047$$

• This result seems counterintuitive;
  • the diagnostic test appears so accurate that we expect someone with a positive test result to be highly likely to have the disease,
  • whereas the computed conditional probability is only .047 .
• However, the rarity of the disease implies that most positive test results arise from errors rather than from diseased individuals.
• The probability of having the disease has increased by a multiplicative factor of 47 (from prior .001 to posterior .047);
  • but to get a further increase in the posterior probability, a diagnostic test with much smaller error rates is needed.