False Positives and False Negatives

Tests can identify who has the novel Coronavirus (COVID-19). The Department of Health and Social Care updates the number of UK confirmed cases.

Confirmed cases must have a positive test result. Testing is imperfect. This article illustrates how false test outcomes affect our interpretations.

Sensitivity

Imagine one in five people have a virus. We sample, and get a perfect slice of that population. Our sample of 100 people contains 20 patients with the virus. Scientists then conduct tests of our sample:

  • False negative: for 1 in 10 people who have the virus, the test gives a wrong ‘negative’ result. For these people, they have the virus, but the test does not detect it.
  • False positive: for 1 in 10 people who do not have the virus, the test gives an incorrect ‘positive’. For these people, they are not infected, but the test detects the virus.

In each example, I use average false rates. This simplicity is for illustration. In real-world batches, actual numbers of false results will vary.

There are 8 false positive results, and 18 true positive results. (Image: ggplot2/waffle)

Among an average 9 in 10 infected people, the test gives a true ‘positive’ result. This is because the false negative rate is 10%. Another name for the true positive rate is the test’s sensitivity.

Specificity

Different tests have different chances of false results.

In this example, 1 in 80 uninfected people get a false ‘positive’ result. Yet, 3 in 10 infected people receive an incorrect ‘negative’ result.

The asterisk icons show people who have the virus. (Image: ggplot2/waffle)

Whilst 20 people have the virus, there are only 15 positive results.

As the false positive rate is 1.25% (1 in 80), the true negative rate is 98.75%. This true negative rate is also called the test’s specificity.

This test has a high specificity, but low sensitivity. It is like fishing: there may be fish in the river, but you will not catch them every time.

The false positive paradox

Imagine another virus, which 10% of our sample has. Suppose 20% of uninfected people get a false positive result. Among infected people, 1 in 10 receive a false negative result.

The false positives outnumber the false negatives. (Image: ggplot2/waffle)

In this example, there are 18 false positive results. Only nine people have true positive tests.

Given you have a positive result, the probability of having the virus is one in three. A positive test result means it is less than likely that person has the virus. This is sometimes called the false positive paradox.

When identifying if a patient has a rare disease, several tests are often necessary.

The R code for producing the graphs is available on GitHub. There is also a R Pubs document to view.

This blog looks at the use of statistics in Britain and beyond. It is written by RSS Statistical Ambassador and Chartered Statistician @anthonybmasters.

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