# An Application of Probability to Epidemiology

Bayes rule is used in epidemiology to calculate the probability that an individual has a disease, given that he tests positive on a screening test.

Example (1994): About 1 in 270 fetuses carried by 35 year old women are afflicted with Down syndrome. In a study completed in 1994, a screening test that could be used during pregnancy returned a positive test result with probability about .89 if the fetus was afflicted, and returned a negative test result with probability about .75 if the fetus was not afflicted. Knowing these numbers, we can use Bayes rule to calculate the approximate probability that a fetus is afflicted given he tests positive.

Define two events:

A = has Down syndrome,
B = tests positive.

To calculate the probability of A given B, we use facts about the incidence of Down syndrome in the population and a test used to screen for the disease:

P(A) = 1/270, since 1 in 270 fetuses in 35 year old women have Down syndrome,
P(B|A) ≅ .89, since about 89% of fetusus that have Down syndrome will test positive for the disease,
P(not B|not A) = .75, since about 75% of fetuses that do not have Down syndrome will test negative for the disease.

From these numbers, calculate the following:

P(B) = P(B and A) + P(B and not A) = P(B|A)P(A) + P(B|not A)P(not A) ≅ .89(1/270) + .25(269/270) =.252

Hence, by Bayes rule:

P(A|B) = P(B|A)P(A)/P(B) ≅ .89(1/270)/.252 = .013

The probability that a fetus carried by a 35 year old woman is afflicted is 1/270 = .0037, but for fetuses that test positive the probability is approximately .013. Instead of a chance of less than 4 in 1000, the chance is larger than 1 in 100.

Reference (1994): James E. Haddow, Glenn E. Palomaki, George J. Knight, George C. Cunningham, Linda S. Lustig, and Patricia A. Boyd, Reducing the Need for Amniocentesis in Women 35 Years of Age or Older with Serum Markers for Screening, New England Journal of Medicine, Volume 330:1114-1118, April 21, 1994, Number 16.

Example (2005): In a similar study completed in 2005, a combined screening test returned a positive test result with probability about .95 if the fetus was afflicted, and returned a negative test result with probability about .78 if the fetus was not afflicted. We can use the same methodology as in the first example to calculate the approximate probability that a fetus is afflicted given he tests positive.

P(A) = 1/270, since 1 in 270 fetuses in 35 year old women have Down syndrome,
P(B|A) ≅ .95, since about 95% of fetusus that have Down syndrome will test positive for the disease,
P(not B|not A) = .78, since about 78% of fetuses that do not have Down syndrome will test negative for the disease.

From these numbers, calculate the following:

P(B) = P(B and A) + P(B and not A) = P(B|A)P(A) + P(B|not A)P(not A) ≅ .95(1/270) + .22(269/270) =.2227

P(A|B) = P(B|A)P(A)/P(B) ≅ .95(1/270)/.2227 = .016

The probability that a fetus carried by a 35 year old woman is afflicted given a positive combined screening test result is approximately .016.

Reference (2005): Fergal D. Malone, M.D., Jacob A. Canick, Ph.D., Robert H. Ball, M.D., David A. Nyberg, M.D., Christine H. Comstcok, M.D., Radek Bukowski, M.D., Richard L. Berkowitz, M.D., Susan J. Gross, M.D., Lorraine Dugoff, M.D., Sabrina D. Craigo, M.D., Ilan E. Timor-Tritsch, M.D., Alicja R. Rudnicka, Ph. D., Allan K. Hackshaw, M.Sc., Geralyn Lambert-Messerlian, Ph.D., Nicholas J. Wald, F.R.C.P., and Mary E. D’Alton, M.D., for the First- and Second-Trimester Evaluation of Risk (FASTER) Research Consortium, First-Trimester or Second-Trimester Screening, or Both, for Down’s Syndrome, New England Journal of Medicine, Volume 353:2001-2011, November 10, 2005, Number 19.

Comparison of the 1994 and 2005 tests: Although the 2005 value of .016 is slightly higher than the 1994 value of .013, it is not necessarily true that the 2005 combined screening test is a better diagnostic tool than the 1994 screening test. See the calculation of confidence intervals about the estimates .89 and .95 in this glossary. 